REGULAR SOLUTIONS
1
REGULAR SOLUTIONS FOR WAVE EQUATIONS WITH
SUPER-CRITICAL SOURCES AND
EXPONENTIAL-TO-LOGARITHMIC DAMPING
Lorena Bociu
Department of Mathematics, NC State University
Raleigh, NC 27695, USA
Petronela Radu
Department of Mathematics, University of Nebraska-Lincoln
Lincoln, NE 68588, USA
Daniel Toundykov
Department of Mathematics, University of Nebraska-Lincoln
Lincoln, NE 68588, USA
(Communicated by the associate editor name)
Abstract. We study regular solutions to wave equations with super-critical
source terms, e.g., of exponent p > 5 in 3D. Such high-order sources have been
a major challenge in the investigation of finite-energy (H 1 × L2 ) solutions to
wave PDEs for many years. The well-posedness question has been answered
in part, but even the local existence, for instance, in 3 dimensions requires the
relation p ≤ 6m/(m + 1) between the exponents p of the source and m of the
viscous damping.
We prove that smoother initial data (H 2 × H 1 ) yields regular solutions
that do not necessitate a correlation of the source and the damping. Local
existence of such solutions is shown for any source exponent p ≥ 1 and any
monotone damping including: exponential, logarithmic, or none at all in dimensions 3 and 4 (and with some restrictions on p in dimensions n ≥ 5). This
result extends the known theory which in the context of supercritical sources
predominantly focuses on damping of polynomial growth and guarantees local
smooth solutions without correlating the damping and the source only if p < 5
n+2
if n = 3, 4).
in 3D (or p < n−2
Furthermore, if we assert the classical condition that the damping grows at
least as fast the source, then these regular solutions are global.
1. Introduction.
2000 Mathematics Subject Classification. Primary: 35L05; Secondary: 35L20.
Key words and phrases. wave equation, regular solutions, critical exponent, super-critical, nonlinear damping.
The second author is supported by NSF grant DMS-0908435.
The third author is supported by NSF grant DMS-1211232.
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LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV
1.1. Motivation. The primary goal is to expand results on the long-standing wellposedness problem for a damped wave equation perturbed by a super-critical source
term, e.g., of order strictly above 5 in 3 dimensions. To date this question has generated a vast library of research work that is reviewed in more detail below. However,
the well-posedness for solutions of finite energy in the presence of high-exponent
sources was established relatively recently in the series of papers [4, 7, 6, 8, 23, 26].
For decades the wave equation with nonlinear damping (of exponent “m”) and
source (of exponent “p”), the first one extending the lifespan of solutions and the
other potentially shortening it, has been a benchmark prototype for many 2ndorder hyperbolic problems. It has been long known that the relation p ≤ m is
necessary for global existence of finite-energy solutions [28, 29]. But already for
source exponents above the critical Sobolev embedding level (p > 3 in 3 dimensions),
even local existence of finite energy solutions has only been verified [26, 23, 4] under
the following condition:
6m
.
p≤
m+1
It is not presently known if this assumption can be removed for weak solutions.
However, we prove that the condition is not needed to obtain local existence of
solutions with higher regularity.
1.2. Synopsis of new results. In this work we demonstrate local existence of
regular, in appropriate sense, solutions for any source exponent p and any damping
in dimensions 3 and 4, thus extending the existing research library which provides
analogous results for smooth solutions only if p < 5.
Moreover, the new theorem also holds for damping of non-polynomial growth e.g.
logarithmic or exponential, or none at all for that matter if we are concerned with
local existence only. Such general feedbacks naturally arise in sub-gradient interpretation of the damping, e.g. see [2, Section 4.3], but this framework is not directly
applicable to supercritical sources since they don’t correspond to even locally Lipschitz operators on the state space (an extended version of this method appeared in
[4], but still for polynomial-like damping only). To our knowledge this is the first
paper to deal with non-polynomial damping in context of supercritical sources.
And if the damping dominates the source, e.g. grows faster than an m-degree
polynomial for some m ≥ p, then these regular solutions are global.
1.3. Model. Let Ω ⊂ Rn be an open bounded domain of class C 2 with boundary
Γ. The following wave equation with interior interaction of a nonlinear source and
monotone viscous damping is the object of our discussion:


in Ω × [0, ∞)
utt + g(ut ) = ∆u + f (u)
(1)
u=0
in Γ × [0, ∞)


u(0) = u0 and ut (0) = u1
The feedback map s 7→ g(s) is monotone increasing and represents the effect of
viscous dissipation (energy loss due to friction). For global well-posedness we will
eventually require a quantifiable lower bound on the growth of g e.g.: c|s|m ≤ |g(s)|.
The map s 7→ f (s) models nonlinear amplitude-modulated forces that could
either have a dissipative effect (e.g. nonlinear Hooke’s law) or a destabilizing one.
The latter is the most interesting scenario, namely when the term f (u) is “energybuilding” and thus counteracts the effect of the damping g(ut ). The dynamics of
REGULAR SOLUTIONS
3
the model is fundamentally affected by the behavior of the nonlinear term f (u);
henceforth, its upper bound will be of polynomial order p, via the estimate
|f (s)| ≤ C|s|p .
1.4. Significance of the source exponent p. The well-posedness analysis of (1)
may become extremely challenging due to the presence of f which in our scenario is
neither monotone, nor dissipative, nor locally Lipschitz on the natural energy space,
which for this equation means (u, ut ) ∈ H 1 (Ω) × L2 (Ω). In general, such a source
can be classified into one of three categories exemplified here for a 3-dimensional
scenario:
Sub-critical & critical: This range corresponds to p ≤ 3, with equality being
the critical case. The criticality comes from the (3D) Sobolev embeddings
∗
H 1 (Ω) → L2 =6 (Ω). When f is sub-critical, the map z → f (z) may be
regarded (under some differentiability conditions on f ) as a locally Lipschitz
compact operator H 1 (Ω) → L2 (Ω). Consequently, if we recast (1) as a first
order evolution problem y 0 = Ay on the space H 1 (Ω) × L2 (Ω), then the
nonlinear operator induced by f would be locally Lipschitz on this space.
At the critical level p = 3, the locally Lipschitz property persists, but the
compactness is lost.
Super-critical: In the range 3 < p ≤ 5, the Nemytski operator associated to f
is no longer locally Lipschitz on the finite energy
space, however the potential
R
ˆ
energy associated to the source, given by Ω f (u) dΩ, with fˆ denoting an
anti-derivative of f , is still well-defined for solutions (u, ut ) ∈ H 1 (Ω) × L2 (Ω).
Upper bound 5 also marks the threshold up to which one may apply the
potential well theory (e.g. see [22, 9]) to this problem in 3 space dimensions.
Super-super-critical: For 5 < p < 6 the source is no longer tractable in the
framework of the potential well theory; however, the nonlinear function f (u)
is still in L1 (Ω) for finite energy solutions.
In the presence of sources that are super-critical and above, the role of feedback
g(ut ), modeling frictional dissipation, is two-fold: it is intended not only to stabilize
(as in the classical control theory of dissipative systems), but also to ascertain
existence, uniqueness, and the life-span of solutions by preventing finite-time blowup. The idea of using damping to positively influence the well-posedness of solutions
was introduced over 20 years ago in the context of control theory for dealing with
boundary or point controls, which by themselves without any damping do not lead
to well-posed dynamics [18]. General “energy-building” sources, even if locally
Lipschitz (and conducive to local existence) may cause blow-up of solutions in finite
time unless counteracted by sufficiently strong nonlinear damping [4, 5, 13, 22, 30].
1.5. Previous results on well-posedness. Full analysis of Hadamard wellposedness for weak (finite energy H 1 (Ω)×L2 (Ω)) solutions for (1) (with u0 ∈ H 1 (Ω), and
u1 ∈ L2 (Ω)) was provided recently in [4, 6, 7]. In fact, the authors consider a more
general case of super-critical sources that are present simultaneously in the interior and on the boundary of the domain. It was shown that solutions exist locally,
are unique, and depend continuously on initial data in the finite-energy topology.
These papers introduced new techniques that rely on monotonicity methods combined with suitable truncations-approximations of nonlinear terms, rather than on
compactness arguments which had limited the results in the previous attempts [30].
This strategy made it possible to extend the range of Sobolev exponents for which
4
LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV
the analysis is applicable (5 ≤ p < 6 for the exponent of the interior source and
3 ≤ k < 4 for the boundary in 3 dimensions), and extend previously available results
on this problem for the case of interior super-critical sources and boundary sources
[13, 12, 3, 26, 23, 17, 30, 10]. Moreover, the results in [4, 5] provide sharp range
of parameters for the damping and sources that characterize the exact borderline
between well-posedness and blow-up for finite energy solutions, thereby completing
the picture for Hadamard well-posedness of weak solutions. This theory has been
complemented by [9] which demonstrates (i) global existence of potential well solutions without exploiting interior and boundary overdamping (i.e. without requiring
m ≥ p and q ≥ k); (ii) blow-up in finite time for potential well solutions for initial
data of non-negative energy that complements the blow up result in [5] for initial
data of negative energy. For an extension of the Hadamard well-posedness of weak
solutions to the Cauchy problem on R, see [8].
In this context we will focus on well-posedness of regular solutions:
(u, ut ) ∈ H 2 (Ω) × H 1 (Ω) .
Such a setting was considered before, e.g. [11, Thm. 2.1] (p ≤ 3) and [3, Thm. 1.10]
(with p < 5). More recently, in [24] the author proved local existence of solutions
n+2
, m ≥ 0 in n = 3, 4. The long-standing restriction
for the entire range 1 < p < n−2
that governed all previous existence results for weak solutions
p
2n
n
n+2
p+
<
for
<p<
m
n−2
n−2
n−2
was eliminated via a new methodology that takes into account higher order estimates, by using first- and second-order potential well arguments. The techniques
of [24] also apply to source terms of order p ≥ 5 with defocusing “good” signs that
contribute to a decrease of the energy. Thus, the author proved that the presence
of damping terms is not essential to handling source terms (even if they are energyaccretive) locally in time: the results apply to systems with no damping (g ≡ 0)
as well as just, with linear damping (m = 1). Another feature of this approach is
that it works on unbounded domains well, as it is based on the patching argument
previously developed for wave equations in [23, 24].
1.6. Goals of this paper. Our focus is on the well-posedness of the system (1)
with smoother initial data u0 ∈ H 2 (Ω) ∩ H01 (Ω), and u1 ∈ H 1 (Ω). The goal
is to show that the correlation between damping and the source can be relaxed,
especially in dimensions above 2. This result was demonstrated in [24] provided
n+2
p < n−2
(n = 3, 4). The argument presented here eliminates this restriction and
accommodates any p ≥ 1, except in high-dimensional cases n ≥ 5 where p has to
be restricted.
A summary of previous work in 3 space dimensions is illustrated in Figure 1(A):
• existence of local weak solutions was proven for the range of exponents to the
left of the dashed curve in [26, 23, 24, 4, 6, 7, 8],
• existence of global weak solutions was proven for the region above the line
m = p, for p < 5 in [26, 4],
• existence of local strong solutions was established for the entire box 1 < p <
5, m ≥ 0 in [25] .
In comparison, the new results of this paper demonstrate:
• existence of local regular solutions for all interior sources with exponent p ≥ 1
without correlation with the damping. For example, if |g(s)| ∼ sm then Figure
REGULAR SOLUTIONS
(A)
5
(B)
Figure 1. Assuming dimension n = 3 and that the damping has
a polynomial like growth |g(s)| ∼ |s|m , these graphs depict various
ranges of the source and damping exponents (p and m) as described
in Section 1.6. The sloped line corresponds to p = m and the
6m
dashed curve in (A) to p = m+1
for p ≥ 3 .
1(B) shows that we obtain existence of local regular solutions for all values
of m > 0. Moreover, the conclusion holds for damping of exponential or
logarithmic growth, or even without any damping at all (g ≡ 0).
• existence of global regular solutions whenever the damping grows at least as
fast as the source, for example, when m ≥ p if g has a lower polynomial bound
of order m.
1.7. Notation.
• The form (·, ·)Ω will indicate the L2 (Ω) inner product with the corresponding
norm denoted k·k. Inner products and norms on other spaces will be indicated
by suggestive subscripts.
• The space Cw ([0, T ], Y ) will denote weakly continuous functions with values
in a Banach space Y , namely such functions v : [0, T ] → Y that the map
t 7→ hv(t), y ∗ i is continuous for every y ∗ ∈ Y ∗ .
• For two functions the relation a(s) . b(s) indicates that a(s) ≤ Cb(s) for a
constant C > 0 independent of s. Whenever invoked it will also be assumed
that C is independent of the solution trajectories in question. If a(s) . b(s)
and b(s) . a(s) we may write a(s) ∼ b(s).
• The calculations will often take place on a space-time cylinder (0, T )×Ω where
T > 0 will be understood from the context. For a shorthand we will denote
QT := (0, T ) × Ω.
2. Preliminaries on Orlicz spaces. To concisely formulate the analysis involving
the monotone feedback map g we, inspired by the approach in [26], will work in
Orlicz spaces which extend reflexive Lebesgue space. The benefit of this framework
6
LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV
is that it does not necessitate to impose a polynomial growth condition on the
function g.
Here we summarize some of the basic facts concerning these spaces. For a more
extensive reference on the background theory see [15] or [1, Ch. 8].
Let g be a continuous strictly monotone increasing function vanishing at 0 and
having lims→∞ g(s) = ∞. Then we can define the following N -functions:
Z t
Z t
g −1 (s)ds for t ≥ 0 .
(2)
g(s)ds and Ψ(t) =
Φ(t) =
0
0
For T > 0 and a function v(x, t) on QT introduce the nonlinear functionals
Z
Z
ρ(u; Φ) :=
Φ(|u|)dQ and ρ(u; Ψ) :=
Ψ(|u|)dQ .
QT
(3)
QT
It can be shown that Φ and Ψ are complementary N -functions, in particular they
are convex conjugates of each other. The functions for which the quantity ρ is finite
form the Orlicz classes K Φ (QT ) and K Ψ (QT ) respectively. The linear hulls (under
pointwise addition and scalar multiplications) of these classes are the Orlicz spaces
LΦ (QT ) and LΦ (QT ), which are Banach spaces of measurable functions under with
the norms
Z
Z
uv .
uv and kukΨ = sup kukΦ = sup ρ(v;Ψ)≤1
QT
ρ(v;Φ)≤1
QT
The Orlicz class K Φ is always contained in the Orlicz space LΦ ([15, (9.12), p. 23])
as readily follows from
kukΦ ≤ ρ(u, Φ) + 1 .
(4)
The closure of bounded functions in LΦ (QT ) yields a subspace E Φ (QT ). In general
we have the strict inclusions
E Φ (QT ) ⊂ K Φ (QT ) ⊂ LΦ (QT ) ,
but the equivalence holds provided the following condition is satisfied:
Definition 1 (∆2 -condition [15, p. 23]). A function F : R+ → R+ is said to
satisfy the ∆2 -condition if there exists C > 0 such that for all large s > 0 we have
F (2s) ≤ CF (s).
Under the ∆2 -condition the Orlicz classes and Orlicz spaces coincide. Moreover,
if both Φ and Ψ have the ∆2 property then the spaces LΦ (QT ), LΨ (QT ) are reflexive
Banach spaces dual to each other. However, if g is exponential or logarithmic, then
respectively either g or g −1 violates the ∆2 -condition.
Henceforth we will not a priori assume the ∆2 condition, neither on g nor on g −1
as neither is necessary for local existence. In the result concerning global existence,
the asserted sufficiently rapid growth of g at infinity will, however, imply that g −1
should satisfy the ∆2 requirement.
The main properties of Orlicz spaces are summarized in the next proposition.
Proposition 1 (On Orlicz spaces). Let Φ and Ψ be complementary N -functions.
Then
(a) A set RU is bounded in LΦ (QT ) if and only if it is bounded in the class K Φ (QT ),
i.e. { QT Φ(|u|) dQ : u ∈ U } is bounded.
REGULAR SOLUTIONS
7
(b) The Orlicz norm on LΦ (QT ) is equivalent to the Luxemburg norm [15, p. 80]
Z
−1
kukΦ ∼ inf k > 0 :
Φ k |u| ≤ 1 .
(5)
QT
(c) The inequality
Z
QT
uv ≤ kukΦ kvkΨ
(6)
holds for any pair of functions u ∈ LΦ (QT ) and v ∈ LΨ (QT ). [15, Thm. 9.3,
p. 74]. From here follows an embedding LΨ (QT ) ⊂ [LΦ (QT )]∗ . But integration
against an LΨ function is not the general form of a bounded linear functional
on LΦ unless both the N -functions satisfy the ∆2 condition. However, it is the
general form of such a functional if restricted to the subspace E Φ (QT ) as the
next property states.
(d) The space E Φ (QT ) is separable and its dual is isomorphic and homeomorphic
to LΨ (QT ) [1, Thm. 8.19, p. 273]. For a functional ` ∈ [E Φ (QT )]∗ the norm
therefore is
k`k[E Φ ]∗ =
sup
|`(u)| .
kukΦ ≤1, u∈E Φ
Ψ
Φ
(e) Orlicz space L (QT ) is E -weakly complete and E Φ -weakly compact. In particular, a sequence {ψn } that is bounded in LΨ (QT ) has a E Φ (QT )-weakly convergent subsequence {ψnk } in the sense that for a unique χ ∈ LΨ (QT )
Z
Z
χv as k → ∞
ψnk v →
QT
QT
for every v ∈ E Φ (QT ) [15, Thm. 14.4, p. 131]. (And analogously LΦ (QT ) is
E Ψ -weakly complete and E Ψ -weakly compact).
(f ) If Ψ and Φ both satisfy the ∆2 condition, then the corresponding Orlicz classes
and Orlicz spaces coincide with E Ψ and E Φ respectively.
Remark 1 (Lebesgue spaces). If g(s) = |s|m−1 s, m > 1, then the Orlicz spaces
m+1
LΨ (QT ) and LΦ (QT ) are topologically isomorphic to the Lebesgue spaces L m (QT )
m
and L (QT ) respectively.
3. Assumptions and definitions. Here we present and explain the assumptions
on the maps f , g and specify a variational interpretation of solutions.
Assumption 1. With reference to system (1) suppose that
(g-1) g is a continuous odd strictly monotone increasing function with g(0) = 0 and
lims→∞ g(s) = ∞.
(g-2) Let the N -functions Φ and Ψ be defined using g as in (2) . Assume that for
any set V of measurable functions on QT the boundedness of scalars
Z
g(v)v : v ∈ V
QT
implies that the family {g(v) : v ∈ V} is bounded in LΨ (QT ) (see Section 3.1
for examples).
(f-1) f ∈ C 1 (R), f (0) = 0 (the latter is inessential and merely for convenience).
(f-2) f satisfies the following growth condition:
|f 0 (s)| . |s|p−1 ,
where p ≥ 1 for dim(Ω) = n ≤ 4, and 1 ≤ p <
n−2
n−4
for dim(Ω) = n ≥ 5.
8
LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV
Remark 2. The condition on p < n−2
n−4 in dimensions n ≥ 5 can, in fact, be relaxed
as it will be demonstrated in Theorem 4.3.
3.1. Explaining the conditions (g-1), (g-2) on damping. The benefit of the
N -function characterization of g(s) is that it naturally accommodates bounds arising from the monotonicity of this map without the need to relate its growth to
Lebesgue spaces (analogous approach, but for Φ, Ψ both satisfying the ∆2 -condition,
was employed in [26, Sec. 3]).
The odd property of g mentioned in (g-1) can be relaxed to |g(s)| ∼ |g(−s)|.
Regarding the property (g-2) we give several examples.
3.1.1. Polynomially bounded dmaping. The assumption (g-2) is satisfied in the canonical case g(s) = sgn(s)|s|m for m > 0. Then
|v|m
Z g(|v|)
m+1 m
m
=
τ m |v|m+1
Ψ(|g(v)|) =
τ 1/m dτ =
m
+
1
m
+
1
0
0
Z |v|
Z |v|
=m
τ m dτ = mΦ(|v|) = m
g(τ )dτ
0
0
≤ mg(|v|)|v| = mg(v)v .
R
Integrating over QT shows that QT g(v)v being bounded also yields an estimate on
g(v) in the class K Ψ (QT ), which in turn gives a bound on the norm in LΨ (QT ).
This scenario illustrates the classical estimate for polynomially growing damping
of degree m:
Z
m+1
g(v)v ≤ C =⇒ v ∈ Lm+1 (QT ) and g(v) ∈ L m (QT ) .
QT
3.1.2. Exponential damping. Let g(s) = sgn(s)(e|s| −1) and g −1 (s) = sgn(s) ln(|s|+
1). Given that
Z
Z
g(v)v =
(|v|e|v| − |v|)
(7)
QT
QT
is bounded we would like to deduce a bound on g(v) in LΨ . Let’s directly estimate
g(v) in the Orlicz class K Ψ :
Z
|g(v)|
Z
g −1 (s)dsdQ =
ρ(g(v), Ψ) =
QT
0
Z
Z
e|v| −1
ln(s + 1)dsdQ
QT
Z
=
0
(|v|e|v| − e|v| + 1)dQ
QT
|s|
|s|
|s|
Because (|s|e − |s|) ∼ (|s|e − e + 1) for |s| 1, then (since QT is a bounded
domain) the boundedness of (7) gives a bound on ρ(g(v), Ψ), which controls via (4)
the norm kg(v)kΨ .
3.1.3. Logarithmic damping. Let g(s) = sgn(s) ln(|s|+1) whence g −1 (s) = sgn(s)(e|s| −
1) for s ≥ 0. Starting with a bound on
Z
Z
g(v)v =
ln(|v| + 1)|v|
(8)
QT
QT
REGULAR SOLUTIONS
9
compute
Z
Z
ρ(g(v), Ψ) =
QT
|g(v)|
g −1 (s)dsdQ =
0
Z
Z
QT
Z
=
ln(|v|+1)
(es − 1)dsdQ
|v| + 1 − ln(|v| + 1) dQ .
0
QT
Since ln(|s| + 1)|s| dominates |s| + 1 − ln(|s| + 1) as |s| → ∞, then from a bound on
(8) we infer a bound on g(v) in the class K Ψ .
3.2. Weak and regular solutions. Consider the Laplace operator on functions
with vanishing traces:
Au = −∆u with D(A) := u ∈ H 2 (Ω) | u = 0 on Γ = H 2 (Ω) ∩ H01 (Ω) . (9)
This operator is positive self-adjoint on L2 (Ω) and maximal accretive L2 (Ω) →
L2 (Ω). So we can define fractional powers of A and identify via the topological
isomorphism D(A1/2 ) ≈ H01 (Ω).
Definition 2 (Weak solution). Weak solutions will be considered when the exponent p of the source satisfies 1 ≤ p < ∞ if n ≤ 2, or 1 ≤ p ≤ 2∗ if n ≥ 3 where
∗
2n
2∗ = n−2
is the critical Sobolev exponent for the embedding H 1 (Ω) → L2 (Ω). By
a weak solution of (1) defined on some interval [0, T ], we mean functions
u ∈ Cw [0, T ]; H01 (Ω) ∩ L2 0, T ; H01 (Ω) , ut ∈ Cw [0, T ]; L2 (Ω) ∩ L2 (QT )
with the following properties:
(i) Let r be sufficiently large to ensure the embedding D(Ar ) ⊂ L∞ (Ω). For any
test-function φ ∈ H 1 (0, T ; D(Ar )) the following identity must hold
T Z
Z
Z
f (u)φ dQ (10)
g(ut )φ dQ = −(ut , φ)Ω +
(−ut φt + ∇u · ∇φ) dQ +
QT
QT
0
QT
(ii) In addition, for ψ0 ∈ H −1 (Ω), ψ1 ∈ L2 (Ω)
lim hu(t) − u0 , ψ0 i = 0
t→0
and
lim (ut (t) − u1 , ψ1 )Ω = 0 .
t→0
For weak solutions we introduce the finite energy functional on H01 (Ω) × L2 (Ω)
E(u, v) := 12 k∇uk2 + 21 kvk2 ,
with
E(t) := E(u(t), ut (t)) .
(11)
Definition 3 (Regular Solution). Regular solutions can be defined for a larger
range of source exponents: 1 ≤ p < ∞ if n ≤ 4, or 1 ≤ p ≤ 2∗∗ if n > 4, where
∗∗
2n
is the critical Sobolev exponent for the embedding H 2 (Ω) → L2 (Ω). A
2∗∗ = n−4
regular solution of (1) is a weak solution in the sense of Definition 2 with the
additional regularity
(u, ut ) ∈ Cw [0, T ]; H 2 (Ω) × H 1 (Ω) ∩ L∞ 0, T ; H 2 (Ω) × H 1 (Ω) .
Remark 3 (Test functions). If g(ut ) ∈ LΨ (QT ), then from the variational identity
(10) it follows that the space of test functions can be closed in the graph topology of
H 1 0, T ; L2 (Ω) ∩ L2 0, T ; H 1 (Ω) ∩ L2 (0, T ; Lq (Ω)) ∩ LΦ (QT )
where q is the conjugate exponent of either 2∗ /p (for weak solutions) or 2∗∗ /p (for
regular solutions). In lower dimensions the membership in L2 (0, T ; Lq (Ω)) is automatic as follows from Sobolev embeddings. The choice of a sufficiently large parameter r in Definition 2 ensures that the original space H 1 (0, T ; D(Ar )) is contained in
10
LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV
L∞ (QT ). Consequently, the closure in the above topologies (LΦ (QT ) in particular)
would ensure that the resulting test elements belong to the subspace E Φ (QT ) whose
dual is isomorphic to LΨ (QT ).
4. Main Results. Our main theorem demonstrates propagation of the initial regularity which allows for local regular solutions. The solutions are also global under
the additional assumption that the damping dominates the source at infinity.
Theorem 4.1 (Local existence of unique regular solutions). Consider equation (1) under the Assumption 1. Suppose u0 ∈ H 2 (Ω) ∩ H01 (Ω), u1 ∈ H01 (Ω).
Then there is TM > 0 such that (1) has a unique regular solution on [0, T ] for any
T < TM . The result holds even without damping (g ≡ 0).
Theorem 4.2 (Global existence of regular solutions). With Assumption 1
suppose further
|s|m . |g(s)| for all |s| ≥ 1
for some m ≥ p. Then the regular solution obtained in Theorem 4.1 is global.
Recall that in dimensions n ≥ 5 the Assumption 1 asks for p <
context of local well-posedness this result can be further extended:
n−2
n−4 .
In the
Theorem 4.3 (Relaxing p < n−2
n−4 in n ≥ 5 for local wellposedness). Consider
equation (1) under Assumption 1, but replace part (f -2) with:
(i)
|f 00 (s)| . |s|p−2 for all s ∈ R.
(ii)
f (0) = 0 (necessary whereas in Assumption 1 it was just for convenience).
Then in dimensions n ≥ 5 for any
2n − 4
n
p<
=1+
n−4
n−4
there exists a sufficiently large m > 0 such that if
|s|m . |g(s)|
for all
|s| ≥ 1 ,
then the local existence result of Theorem 4.1 holds. More precisely it requires:
p(n − 4) + 4
2m(n − 2) − 4
m>
or p <
.
(2n − 4) − p(n − 4)
(n − 4)(m + 1)
If in addition g satisfies
|s − r|m+1 . (g(s) − g(r))(s − r)
(12)
then this solution is unique.
5. Proofs. The above theorems are verified in a series of steps:
I. The Galerkin approximations used to construct local regular solutions are
presented in Section 5.1. The investigation of the limit of these approximations
is given in Sections 5.2 and 5.3 (with 5.3.1 handling the nonlinear damping
term).
II. The uniqueness of regular solutions is addressed in Section 5.5.
III. The alternate version of local existence and uniqueness (Theorem 4.3) is outlined in Section 5.6.
IV. Local regular solutions are extended globally in Section 5.7.
V. Finally, Section 5.8 mentions an alternative approach to local existence of
regular solutions using semigroup theory in the simpler scenario when the
function g is linear.
REGULAR SOLUTIONS
11
5.1. Galerkin approximations and a priori bounds. We consider the standard
Galerkin scheme that constructs solutions of (1) via a limit of finite-dimensional
approximations based on the eigenfunctions of the Laplacian.
The set {ek }k∈N of eigenfunctions of the Dirichlet Laplacian A = −∆ forms
an orthonormal basis forPL2 (Ω), and orthogonal basis for every D(Ar ) ⊂ H 2r (Ω),
n
r > 0. We let un (t) := k=1 un,k (t)ek , where un (t) satisfies the following system
of ordinary differential equations:
(untt , v)Ω + (∇un , ∇v)Ω + (g(unt ), v)Ω = (f (un ), v)Ω
(un (0), v)Ω = (u0 , v)Ω ,
(unt (0), v)Ω = (u1 , v)Ω
(13)
(14)
for all v ∈ span{ek }nk=1 . By our choice of ek it follows that
un (0) → u0
strongly in H01 (Ω)
unt (0) → u1
strongly in L2 (Ω) .
(15)
Moreover, since un (0) and unt (0) are projections of smooth data u0 and u1 , then
the former are bounded in H 2 and H 1 respectively.
Note that (13)–(14) is an initial value problem for a system of n second-order
ordinary differential equations with continuous nonlinearities in the n unknown
functions un,k and their derivatives. From the Cauchy-Peano theorem it follows
that for every n ≥ 1, (13)–(14) has a solution un,k ∈ C 2 [0, Tn ], for some Tn > 0.
Proposition 2 (A priori bounds on approximate solutions). There exists TM > 0
such that for any T < TM the functions {(un, unt )} are defined for t ∈ [0, T ] and
bounded in the space L∞ 0, T ; D(A) × H01 (Ω) . In addition, {unt } are bounded in
LΦ (QT ) and {untt } are bounded in Orlicz-Bochner space LΨ (0, T ; [D(Ar )]∗ ) for r
such that D(Ar ) ⊂ L∞ (Ω).
Proof. Step 1: un and unt . Pick n > 0 and let [0, T̄n ) be the maximal right-interval
of existence for the solution to (13). Let’s temporarily fix some T ∈ (0, T̄n ). Define
the higher-order energy functional of n-th approximation:
E 1,n (t) := 21 kun (t)k2H 2 (Ω) + 21 kunt (t)k2H 1 (Ω) .
Use the test function v = −∆unt in (13) and integrate over t ∈ [0, T ]. If g 0 (unt )
exists and is integrable then
−(g(unt ), ∆unt )Ω = (g 0 (unt ), |∇unt |)Ω .
Hence the multiplier −∆unt ultimately gives the identity
Z T
Z T
E 1,n (T ) +
(g 0 (unt ), |∇unt |2 )Ω = E 1,n (0) +
(−f (un ), ∆unt )Ω ,
0
0
where the second term on the left is non-negative and we get (17) listed below.
If g 0 does not exist a.e. or has no suitable regularity, then we can approximate it
by smooth functions taking advantage of the fact that g corresponds to a maximal
monotone graph in R × R. Define the operator B : D(B) ⊂ L2 (Ω) → L2 (Ω) by
Bu(x) = g(u(x))
2
2
a.e. x ∈ Ω
(16)
with D(B) = {u ∈ L (Ω) | g(u) ∈ L (Ω)}. Since g is continuous monotone increasing with g(0) = 0 then |g ◦ (I + g)−1 (s)| . |s|. Consequently, the operator (I + B)−1
maps L2 (Ω) into D(B), and thus B is a maximal monotone operator (e.g. see [2,
12
LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV
Thm. 1.2 on p. 43, or p. 82]). In addition, the domain of B contains L∞ (Ω). The
Yosida approximations of B are defined by
Bλ = λ−1 I − (I + λB)−1 = B(I + λB)−1 for λ > 0 ,
and (Bλ u)(x) coincides a.e. x ∈ Ω with gλ (u(x)) where gλ are regularizations of g:
gλ (s) = λ−1 s − (I + λg)−1 (s) .
Each gλ is a monotone increasing globally Lipschitz function. For v ∈ D(B) we
have Bλ (v) → B(v) in L2 (Ω) as λ & 0 [2, Prop. 1.3, p. 49]. Consequently, since unt
is smooth (in particular in L∞ (Ω) ⊂ D(B)) then for any fixed n and a.e. t ∈ [0, T ]
(g(unt ), −∆unt )Ω =(B(unt ), −∆unt )Ω = lim (Bλ (unt ), −∆unt )Ω
λ&0
= lim (gλ (unt ), −∆unt )Ω
λ&0
= lim (gλ0 (unt ), |∇unt |2 )Ω ≥ 0 ,
λ&0
which likewise leads to the inequality
E
1,n
(T ) ≤ E
1,n
Z
f (un )∆unt
(0) −
(17)
QT
bypassing the direct integration by parts of the product (g(unt ), −∆unt )Ω .
Next, from (17) we proceed to estimate the product involving the source term.
We demonstrate it in the case of n ≥ 5 (lower dimensions don’t place any restrictions
2n
on p). Let q = n−2
be the critical Sobolev exponent for the embedding H 1 (Ω) →
q
L (Ω). A Hölder estimate with three conjugate exponents 2, q and 2q/(q − 2) gives
Z
n
n
−(f (u ), ∆ut ) =
f 0 (un )∇un · ∇unt . kun kp−1
k∇un kLq k∇unt k . (18)
L2q(p−1)/(q−2)
Ω
By the choice of q we have k∇un kLq (Ω) ≤ kun kH 2 (Ω) . Note also that in dimensions
2q(p−1)
below 5 the L q−2 (Ω) norm can be estimated via the H 2 (Ω) norm. Whereas if
n ≥ 5 this estimate holds provided
2q(p − 1)
2n
≤
,
q−2
n−4
or equivalently if p ≤
n−2
n−4
as a fortiori guaranteed by (f -2). Thus
−(f (un ), ∆un ) . [E 1,n ](p+1)/2
and
E 1,n (T ) ≤ E 1,n (0) + C
Z
T
[E 1,n (t)]
p+1
2
dt.
(19)
0
The resulting optimal bound on E 1,n (t) may blow up in finite time since 21 (p+1) > 1;
however, a comparison theorem (see for instance [19, Thm. 1.4.1, p. 15]) shows
that E 1,n (t) can blow up no faster than the solution to z 0 = cz (p+1)/2 where c
and the initial condition z(0) depend only on C, p and an upper bound on E 1,n (0).
Thus, there exists TM > 0 such that if T < TM then E 1,n (t) must remain a priori
bounded on [0, T ]. In particular, the time-dependent
coefficients of un,k are a priori
Pn
n
n,k
bounded on [0, T ], and therefore u (t, x) = j=k u (t)ek (x) is bounded on spacetime domain QT . Consequently, the maximal right-time of existence T̄n must be
at least TM and the energy E 1,n is uniformly bonded on [0, T ] for any T < TM
independently of n.
REGULAR SOLUTIONS
13
Step 2: unt in Orlicz space. RTo prove that unt is bounded in LΦ (QT ) uniformly
in n it suffices to prove that for QT Φ(|unt |). Recall the quadratic energy (11) and
let E n (t) := E(un (t), unt (t)). Use unt as the test function in (13) to obtain
Z
Z
E n (T ) +
g(unt )unt dQ = E n (0) +
f (un )unt dQ .
(20)
QT
QT
n
ut it
n
to the uniform bounds on on u and
follows that the product
RAccording
n n
g(u
)u
is
bounded
uniformly
in
n.
On
the
other
hand,
since g is odd and
t
t
QT
monotone increasing then
Z |v|
g(τ )dτ ≤ g(|v|)|v| = g(v)v ,
Φ(|v|) =
0
whence
Z
Φ(|unt |) ≤
QT
Z
g(unt )unt ≤ C(u0 , u1 ) ,
QT
from which the desired conclusion follows.
Step 3: bound on untt . We proceed by duality. Let r be large enough to
guarantee the continuous embedding
D(Ar ) ⊂ L∞ (Ω) .
(21)
Consider a test φ function in a D(Ar )-valued Orlicz-Bochner space
φ ∈ X := LΦ (0, T ; D(Ar )),
RT
consisting of Bochner-measurable functions such that 0 Φ(kkf kD(Ar ) )dt < ∞ for
some k > 0. Then it forms a Banach space under the Luxemburg norm
)
(
Z T
kφkX = inf k > 0 :
Φ k −1 kφkD(Ar ) dt ≤ 1 .
0
Due to (21) we also have the inclusion
X ⊂ L1 (0, T ; D(Ar )) ∩ LΦ (QT ) .
(22)
r
Next, since {ek } form an orthogonal basis for D(A ) we can define orthogonal
projections:
Pn : D(Ar ) → span{ek : k = 1, 2, . . . , n} ⊂ D(Ar ) .
In particular, {Pn } are bounded on D(Ar ) uniformly in n and hence bounded uniformly on LΦ (0, T ; D(Ar )). Then from the Galerkin variational identity (13), by
virtue of (22) we have
Z
Z
Z
Z
n n
n
n
utt φ = −
∇u · ∇φ −
g(ut )(Pn φ) +
f (u )(Pn φ)
QT
QT
QT
QT
. kun kL∞ (0,T ;H 1 (Ω)) kφkL1 (0,T ;H 1 (Ω))
+ kg(unt )kLΨ (QT ) kPn φkLΦ (QT )
+ kf (un )kL∞ (0,T ;L2∗∗ /p (Ω)) kPn φkL1 (0,T ;L{2∗∗ /p}0 (Ω))
!
.
n
ku kL∞ (0,T ;H 1 (Ω)) +
kg(unt )kΨ
n
+ kf (u )k
2∗∗
L∞ 0,T ;L p (Ω)
kφkX .
∗∗
The restriction on p in (f -2) a fortiori implies that f (un ) is bounded in L2 /p (Ω)
provided un is bounded in H 2 (Ω). Then the previously verified bounds on un and
unt imply that integration against untt defines a bounded family of functionals in
14
LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV
X ∗ . In particular, these functionals are bounded when restricted to the subspace
E Φ (0, T ; D(Ar )). Since D(Ar )∗ is reflexive, then it has the Radon-Nikodym property (e.g. [20, Cor. 3.6.12, p. 203]), and so [E Φ (0, T ; D(Ar ))]∗ is topologically
isomorphic to LΨ (0, T ; [D(Ar )]∗ ) via [14, Thm. 2] which yields the desired conclusion.
5.2. Limits of the approximations. The a priori bounds shown in the previous
section lead via compactness to the following convergence results:
Proposition 3. Let TM be as in Proposition 2. For any T < TM there exists a
function u ∈ L∞ (0, T ; H 2 (Ω) ∩ H01 (Ω)) with time derivative ut ∈ L∞ (0, T ; H01 (Ω))
such that on a subsequence (re-indexed again by n) the approximate solutions defined
by (13), (14) satisfy:
strongly in L∞ (0, T ; H 2−ε (Ω) ∩ H01 (Ω))
un → u
unt → ut
strongly in Lq (0, T ; H01−ε (Ω))
any
ε>0
and
(23)
1 ≤ q < ∞ . (24)
Proof. From Proposition 2 it follows on a subsequence (reindexed for convenience
again by n we have
un
*
u
unt
*
ut
weakly* in L∞ 0, T ; H 2 (Ω) ∩ H01 (Ω))
weakly* in L∞ 0, T ; H01 (Ω) .
The extension due to Simon [27] of Aubin’s compactness result now readily implies
(23). For the convergence of ut use the fact that untt is bounded in
LΨ (0, T ; [D(Ar )]∗ ) ⊂ L1 (0, T ; [D(Ar )]∗ )
for some r. Then the same compactness theorem implies that unt converges to some
v ∈ Lq (0, T ; H01−ε (Ω)) any finite q ≥ 1. In particular, from the identity
Z t
un (t) = un (0) +
unt (s)ds in L2 (Ω) for a.e. t ∈ [0, T ]
0
it readily follows that u has an absolutely continuous in time t version u ∈ C([0, T ]; L2 (Ω))
with the time derivative a.e. equal to v, which confirms (24).
5.3. Limit n → ∞ in the approximate equation. Let N ∈ N and consider
φ ∈ C 1 [0, T ]; span{ek }N
(25)
k=1
then for all n ≥ N the Galerkin approximation (13) satisfies
t=T Z T
Z T
Z
Z
n
(u , φ)
−
(unt , φt )Ω +
(∇un , ∇φ)Ω +
g(unt )φ =
t=0
0
0
QT
f (un )φ . (26)
QT
From the convergence results in Proposition 3 we have as n → ∞ (on a subsequence)
Z T
Z T
Z T
Z T
(unt , φt )Ω →
(ut , φt )Ω ,
(∇un , ∇φ)Ω →
(∇u, ∇φ)Ω
(27)
0
0
0
0
Because the exponent p in assumption (f -2) is sub-critical (in dimensions ≤ 4
that is automatic) while un bounded in H 2 it follows that (on a subsequence) f (un )
converges weakly∗ to some η in L∞ (0, T ; Lq (Ω)), q > 1. Since QT is a bounded
domain then f (un ) → η also weakly in L1 (QT ). But from the convergence of un we
REGULAR SOLUTIONS
15
have f (un ) → f (u) pointwise a.e. (x, t), and hence in measure. Consequently f (un )
converges strongly in L1 (QT ) and therefore the weak limit η agrees with f (u):
Z
Z
n
f (u )φ →
f (u)φ .
(28)
QT
QT
Moreover, from (15) we get
(unt (0), φ(0))Ω → (ut (0), φ(0))Ω .
(29)
Finally, the uniform bound on untt ∈ LΨ (0, T ; [D(Ar )]∗ ) ≈ [E Φ (0, T ; D(Ar ))]∗
(via [14, Thm. 2] and the reflexive property of [D(Ar )]∗ ) as verified in Proposition 2 permits to work with a subsequence weakly* convergent to some element
ζ ∈ LΨ (0, T ; [D(Ar )]∗ ). All functions in LΨ (0, T ; [D(Ar )]∗ ) are weakly* scalarly
measurable since [D(Ar )]∗ is separable. From here we conclude that for a smooth
space-dependent test function ψ = ψ(x) the duality pairing hψ, ζ(·)i is integrable.
So for t ∈ [0, T ] the identity
Z t
(n)
(n)
(30)
(ut (t), ψ)Ω = (unt (0), ψ)Ω +
hψ, utt (s)ids
0
in the limit n → ∞ implies that the mapping t 7→ (ut (t), ψ)Ω is well-defined and
continuous on [0, T ]. Choosing ψ = φ(T ) (which is smooth since φ comes from (25))
implies
(n)
(ut (T ), φ(T ))Ω → (ut (T ), φ(T ))Ω as n → ∞ .
(31)
5.3.1. Passing to the limit in the damping term. The limits (27)–(31) addressed
almost all of the terms in the approximate variational formulation (26). It remains
to analyze
the products involving the damping g as n → ∞. From (20) we see
R
that QT g(unt )unt is bounded. Then {g(unt )} is bounded in LΨ (QT ) according to
Assumption (g-2). Using E Φ -weak compactness and completeness of LΨ (QT ) conclude that there is a unique function χ ∈ LΨ (QT ) such that (on a subsequence):
Z
Z
n
χφ dQ .
g(ut )φ dQ =
lim
n→∞
QT
QT
for every φ ∈ E (QT ) (a fortiori satisfied if φ ∈ L∞ (QT )) . Moreover, because
unt converges to ut pointwise a.e.-(t, x) in the bounded domain QT , then g(unt )
converges in measure to g(ut ). Hence [15, Thm. 14.6, p. 132] the E Φ -weak limit χ
must coincide with g(ut ) as desired.
Φ
5.4. Obtaining local regular solutions. Let TM be the maximal time of existence provided by Proposition 3. As it was verified above, the passage to the limit
n → ∞ (on a subsequence) in Galerkin approximation gives for any T < TM
t=T Z T
Z T
Z
Z
(u, φ)
−
(ut , φt )Ω +
(∇u, ∇φ)Ω +
g(ut )φ =
f (u)φ .
(32)
t=0
0
0
QT
QT
This conclusion holds for every test function
φ ∈ C 1 ([0, T ]; span{ek }∞
k=1 ) .
These functions are dense in H 1 (0, T ; D(Ar )) for any r ≥ 0 which yields the desired
variational identity (10). Moreover:
• u ∈ L∞ (0, T ; D(A))
• ut ∈ L∞ (0, T ; H 1 (Ω))
as well as
16
LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV
• u ∈ C([0, T ]; H01 (Ω))
• ut ∈ LΦ (QT ) .
From the first two bullets one infers (e.g. [21, Lemma 8.1, p. 275]) that
(u, ut ) ∈ Cw ([0, T ]; H 2 (Ω) × H 1 (Ω)) .
In addition, these solutions have the following property:
Proposition 4 (Inequality for finite energy). If u, v are two regular solutions,
ũ := u − v and Eũ (t) := E(ũ(t), ũt (t)), then
Z
Z
Eũ (T ) +
(g(ut ) − g(vt ))ũt ≤ Eũ (0) +
(f (u) − f (v))ũt .
(33)
QT
QT
Proof. Use the variational Galerkin identity (26) for the initial data (un0 , un1 ) and
(v0n , v1n ) and take the difference of the resulting equations:
Z T
Z T
Z
Z
n
n
n
n
(ũtt , φ)Ω +
(∇ũ , ∇φ)Ω +
(g(ut ) − g(vt ))φ =
(f (un ) − f (v n ))φ .
0
0
Substitute φ =
Eũn (T ) +
ũnt
QT
QT
and integrate by parts
Z
(g(unt ) − g(vtn ))ũnt = Eũn (0) +
QT
Z
(f (un ) − f (v n ))ũnt .
(34)
QT
Recall from Proposition 3 that ũnt converges strongly in Revery Lq≥1 (0, T ; H 1−ε (Ω)),
2n
hence strongly in every Lρ (QT ), ρ < n−2
. Therefore QT (f (un ) − f (v n ))ũnt will
R
converge on a subsequence to some QT ξut provided f (un ) and f (v n ) are bounded
0
2n
in Lρ (QT ) where ρ0 > min{1, n+2
}. Specifically, in dimensions n ≥ 5 it necessitates
n+2
p < n−4 which is implied by (f -2). Since f (un ) − f (v n ) converges pointwise a.e., it
does so in measure on the bounded domain, hence we can identify ξ = f (u) − f (v).
Next take the limit inferior as n → ∞ in (34), use strong convergence of the energy functionals, and invoke Fatou’s lemma for the damping integral (the integrand
is non-negative for each n), to get the desired result.
5.5. Uniqueness of solutions. In 3 dimensions the uniqueness follows immedi6m
ately if p ≤ m+1
since in this region we have the uniqueness result for weak solutions from [6], and the regular solution described by Theorem 4.1 coincides with
the unique weak solution. Now we extend the uniqueness result for regular solution
in the complementary region of exponents p.
Let u and v be any two regular solutions of (1) defined on [0, TM ). Then for
t ∈ [0, T ] ⊂ [0, TM ) find a constant RT such that for all t ∈ [0, T ]
ku(t)kH 2 (Ω) + kut (t)kH 1 (Ω) ≤ RT .
From (33) by the monotonicity of g we have
Z
Eũ (T ) ≤ Eũ (0) +
(f (u) − f (v))ũt .
QT
Since pointwise |f 0 (u)| . |u|p−1 then
|f (u) − f (v)| ≤ C|u − v| |u|p−1 + |v|p−1 .
(35)
REGULAR SOLUTIONS
17
Therefore, using Hölder estimates with conjugate exponents as in (18) gives for each
t ∈ [0, T ]
Z
Z
Z
(f (u) − f (v))ũt dΩ ≤
|f (u) − f (v)||ũt |dΩ .
|ũ| |u|p−1 + |v|p−1 |ũt |dΩ
Ω
Ω
Ω
p−1
p−1
. kukH 2 (Ω) + kvkH 2 (Ω) Eũ (t) ≤ C(RT )Eũ (t) .
Substitute this result into (35) and use Gronwall’s inequality to conclude that
Eũ (0) = 0 would imply that the solutions u and v coincide.
5.6. Relaxing the condition p < n−2
n−4 in dimensions n ≥ 5. Here we outline
how the above proof of local existence and uniqueness can be modified to accommodate higher values of p in dimensions n ≥ 5 under the extended Assumptions of
Theorem 4.3. To verify this result we must essentially refine the proof of a priori
bounds claimed in Proposition 2.
5.6.1. Adjusting the proof for the bounds un ∈ H 2 (Ω), unt ∈ H 1 (Ω). Because the
sought result is local in time, then without loss of generality (up to a constant
dependent on g(1), |Ω| and T ) we can assume that
|s|m . |g(s)|
holds for all s ∈ R as opposed to just |s| ≥ 1. Let E n denote the finite energy of
the approximate solution (un , unt ). Then plugging test-function unt into (13) gives
Z
Z
n
n n
n
kun km+1
.
E
(t)
+
g(u
)u
=
E
(0)
+
f (un )unt
(36)
t
t
Lm+1 (QT )
QT
QT
n
n
We can estimate
E (0) by higher energy E (0) and add this inequality to (17).
Because f (un )Γ ≡ 0, then from the symmetry of the operator A = −∆ we get:
1,n
kun km+1
(T )
Lm+1 (QT ) + E
Z
Z
. E 1,n (0) −
(∆f (un ))unt +
QT
f (un )unt
QT
(37)
I
=E 1,n (0) −
zZ
}|
II
{
f 00 (un )|∇un |2 unt −
QT
z
Z
III
}|
{
f 0 (un )∆un unt +
QT
z
Z
}|
{
f (un )unt
QT
Recall that we are interested in dimensions n ≥ 5. For the integral I use repeated
Hölder’s inequality with a triple of conjugate exponents
α :=
n
,
n−2
(m + 1)n
,
2(m + 1) − n
m + 1.
Since the assumption on f is |f 00 (s)| ≤ |s|p−2 then for any ε > 0
Z T
|I| .
kukp−2
k∇uk2 2n
kut kLm+1 (Ω)
Lα(p−2) (Ω)
L n−2 (Ω)
0
Z
T
kuk
.Cε
0
(p−2) m+1
m
Lα(p−2) (Ω)
2(m+1)
m
2n
L n−2 (Ω)
k∇uk
dt + εkut km+1
Lm+1 (QT ) .
The term εkut km+1
Lm+1 (QT ) can be absorbed into the left-hand side of (37) for small
enough ε > 0. The rest of the quantities can be expressed in terms of the energy
18
LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV
E 1 (t) as in (19) using Sobolev embeddings of H 2 (Ω) into Lebesgue spaces provided:
α(p − 2) =
(m + 1)n
2n
(p − 2) ≤ 2∗∗ =
.
2(m + 1) − n
n−4
If we are allowed to take the damping exponent m arbitrarily large (depending on
p), then effectively it suffices to have
2n − 4
2n
⇐⇒ p <
.
n−4
n−4
If we express it more precisely, then we require
p−2<
m>
p(n − 4) + 4
(2n − 4) − p(n − 4)
(38)
(39)
For the integral II in (37) use the triple of conjugate exponents
2(m + 1)
, 2,
m−1
Then the same conclusion holds, provided
m + 1.
2n
2(m + 1)
<
m−1
n−4
which leads to the same condition on m. To estimate the integral III using un ∈
2n
H 2 (Ω) and unt ∈ H 1 (Ω) one can appeal to the condition p ≤ 2∗∗ = n−4
, which is
automatically satisfied by (39).
(p − 1)
5.6.2. Proving the bounds on unt in LΦ (QT ). In order to establish that unt was
bounded in LΦ (QT ) we invoked (20) along with the boundedness of the product
Z
f (un )unt
QT
uniformly in n. Observe that by (36) the functions unt are bounded uniformly in
m+1
Lm+1 (QT ). Consequently, it suffices to have f (un ) bounded in L m (QT ) independently of n. Since un us bounded in L∞ (0, T ; H 2 (Ω)), then it remains to guarantee
the embedding
H 2 (Ω) ⊂ Lp
m+1
m
(Ω)
(40)
which requires
2n
p≤
n−4
m
m+1
or m ≥
p(n − 4)
.
2n − (n − 4)p
(41)
The resulting condition is already implied by (39).
Remark 4. This relation between p and m essentially becomes the H 2 analog of
6m
the restriction p ≤ m+1
in 3D (as in [26, 23, 4]) for finite-energy solutions.
5.6.3. Obtaining a regular solution to the problem. When passing to the limit n →
∞ the exponent p also plays a role in identifying the limit of the source term (as in
(28)). In this case it suffices to have p < 2∗∗ = 2n/(n − 4), which follows from (38).
The rest of the argument is analogous to the original proof. Note that in this
scenario the solution ultimately satisfies
ut ∈ LΦ (QT ) ⊂ Lm+1 (QT ) .
(42)
REGULAR SOLUTIONS
19
5.6.4. Adjusting the proof of uniqueness. The uniqueness result relied, first of all, on
the inequality for the
R energy in Proposition 4. That conclusion in turn necessitates
the convergence of QT (f (un ) − f (v n ))(unt − vtn ). From (36) we may assume that
unt and vtn converge weakly in Lm+1 (QT ). Thus we require strong convergence of
m+1
f (un ) and f (v n ) in the dual space L m (QT ) . Note the estimate
m+1
Z
Z Z 1
m
m+1
0
n
n
n
f
(u
s
+
(1
−
s)u)ds(u
−
u)
|f (u ) − f (u)| m ≤
0
QT
QT
Z
m+1
m+1
(43)
.
|un |p−1 + |u|p−1 m |un − u| m
QT
m+1
α
. kun kα
+
kuk
kun − uk m
0
0
αρ
αρ
L
(QT )
L
(QT )
ρ m+1
L
m
(QT )
∞
where α = (p−1)(m+1)/m. As a consequence of their convergence in L (0, T ; H 2−ε ),
∗∗
the functions un converge strongly in every L2 −δ (QT ), δ > 0. Thus we may choose
∗∗
ρ m+1
m arbitrarily close from below to 2 . So we should only ensure that the norm
n
αρ0
of u in L (QT ) is bounded, meaning αρ0 ≤ 2∗∗ . The resulting inequality is
2n
(p − 1)(m + 1) 0
ρ ≤
m
n−4
2mn
. This inwhere ρ0 can be chosen arbitrarily close from above to 2mn−(n−4)(m+1)
equality gives a strict version of (41), which, as it has been mentioned, is already
satisfied via (39). These steps verify the desired strong convergence of f (un ) in
m+1
L m (QT ) (and analogously for f (v n )).
Now with the help of the assumption (12) we get at our disposal the following
inequality for the difference Eũ of two solutions u and v:
Eũ (T ) + ckut − vt km+1
Lm+1 (QT )
Z
(f (u) − f (v))(ut − vt )
. Eũ (0) +
QT
. Eũ (0) + Cε kf (u) − f (v)k
m+1
m
m+1
L m
(QT )
+ εkut − vt km+1
Lm+1 (QT ) .
The parameter ε is chosen small enough to absorb the velocity term into the lefthand side. The source term is estimated as in (43) with powers above 1 factored
into a constant dependent on E(0) and time T . That step ultimately gives
Z
Z T
(f (u) − f (v))(ut − vt ) ≤ C(RT )
Eũ (t)dt
QT
0
as in Section 5.5 wherefrom the uniqueness result follows.
The above modifications to the original proof of existence and uniqueness imply
the result of Theorem 4.3.
5.7. From local regular to global regular solutions. Now we return to the
original Assumption 1 and proceed to verify global existence claimed in Theorem
4.2 by building upon the result of Theorem 4.1. At this stage we will demonstrate
that both the finite (H 1 × L2 ) and higher (H 2 × H 1 ) energies cannot blow up as
t % TM provided p ≤ m (which excludes the scenario with sublinear damping since
p ≥ 1, unless of course the source is entirely absent). Then for any TM < ∞ the
20
LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV
solutions obtained by the Galerkin approximations converge on a neighborhood of
t = TM and therefore can be extended past any finite interval of existence.
Proposition 5 (Non-blowup of finite energy). Suppose (u, ut ) is a local regular
solution defined on the maximal interval of existence [0, TM ). Assume that p ≤ m,
then the finite energy remains uniformly bounded on [0, TM ].
Proof. We need to show that
lim sup E(t) < ∞ .
t%TM
That follows if we prove that Galerkin approximations have energy E n (t) that is
bounded on [0, TM ] independently of n. For brevity we will temporarily suppress
the superscripts:
u := un and ut := unt
for some n ∈ N. Following the argument in [13] introduce a modified energy functional:
1
V (t) := E(t) +
ku(t)kp+1
p+1 .
p+1
Use (20) (which is justified since we’re working with smooth approximations) to
conclude
Z tZ
Z tZ
1
ku(t)kp+1
V (t) +
g(ut )ut = E(0) +
f (u)ut +
p+1 .
p
+
1
0
Ω
0
Ω
Differentiation in time gives
Z
Z
Z
V 0 (t) = −
g(ut (t))ut (t) +
f (u(t))ut (t) +
|u(t)|p−1 u(t)ut (t) .
Ω
Ω
Ω
Using the lower bound on the damping and the upper bound on the source we
obtain
Z
V 0 (t) ≤ − C1 kut (t)km+1
+
C
|u(t)|p |ut (t)|
2
Lm+1
Ω
p
p+1
≤ − C1 kut (t)km+1
Lm+1 + C2 ku(t)kLp+1 kut kL
p+1
p+1
≤ − C1 kut (t)km+1
Lm+1 + C2,ε ku(t)kLp+1 + εkut kLp+1
for any ε > 0. Since 1 ≤ p ≤ m, then for ε ≤ C1 we obtain
V 0 (t) ≤ C2;ε V (t) .
Integrate in time and apply Gronwall’s inequality to conclude (recalling the temporarily omitted dependence on n):
E n (t) ≤ V n (t) ≤ V n (0)eCε t .
The right-hand side is uniformly bounded in n because kun kp+1 is uniformly bounded,
which in dimensions n ≥ 5 is enforced by (f -2). Since the finite energy converges to
that of the true solution as n → ∞ we conclude that E(t) is bounded on [0, TM ].
Proposition 6 (Non-blow up of higher energy). Suppose (u, ut ) is a regular solution
on [0, TM ) such that the finite energy is uniformly bounded in time on [0, TM ]. Then
the higher energy is uniformly bounded on [0, TM ]. In particular, for any T1 ≤ TM
there exists CT1 > 0 dependent only on the H 1 × L2 norms over [0, T1 ], such that
sup ku(t)k2H 2 (Ω) + kut (t)k2H 1 (Ω) . ku0 k2H 2 (Ω) + ku1 k2H 1 (Ω) + CT1 .
(44)
t∈[0,T1 ]
REGULAR SOLUTIONS
21
Proof. What follows is an estimate using a higher-order multiplier −∆ut analogous
to the one used to prove the stability of the Galerkin scheme. We will work with
Galerkin approximations, but for brevity will temporarily suppress the superscripts:
u := un
and ut := unt
for some n ∈ N. Fix any T ≤ T1 ≤ TM . The test function −∆ut gives, as in (17),
Z
kut (T )k2H 1 + ku(T )k2H 2 ≤ ku1 k2H 1 + ku0 k2H 2 −
f (u)∆ut .
(45)
QT
For any t ∈ [0, T1 ] estimate the product involving the source term exactly as in
2n
if n ≥ 3). In dimensions n ≥ 5 we take
(18) (where the exponent q stood for n−2
n−2
advantage of the strict inequality p < n−4 in order to estimate the L2q(p−1)/(q−2) (Ω)
norm via an interpolation of H 2 (Ω) and H 1 (Ω). Specifically, there exists θ0 ∈ (0, 1)
dependent on p such that for all θ ∈ (0, θ0 )
kukL2q(p−1)/(q−2) (Ω) . kukH 2−θ (Ω) . kukθH 2 (Ω) kuk1−θ
H 1 (Ω) .
In lower dimensions this estimate holds a fortiori with the L2q(p−1)/(q−2) (Ω) norm
replaced by the norm in Lr (Ω) for arbitrarily large r. Thus we infer
−(f (u), ∆ut ) =(∇f (u) · ∇ut ) = (f 0 (u)∇u · ∇ut )
p−1
. kukθH 2 kuk1−θ
k∇ukLq k∇ut k
H1
1+θ(p−1)
.kukH 2
(1−θ)(p−1)
kukH 1
kut kH 1 .
The estimate with p = 1 (f 0 ∼ const) is much simpler so we focus on p > 1. Let
1
1
1
+ + = 1. Then two applications of
α, β, γ > 1 be a conjugate triple, i.e.
α
β
γ
Young’s inequality give for any ε > 0 and any t ∈ [0, T1 ]
α(1+θ(p−1))
−(f (u), ∆ut ) ≤ εkukH 2 (Ω)
γ(1−θ)(p−1)
+ εkut kβH 1 (Ω) + Cε kukH 1 (Ω)
.
1
2
Choose β = 2, θ < min{θ0 , p−1
}, α = 1+θ(p−1)
, and γ −1 = 12 − α−1 . Note that the
right-most term depends on the finite energy only hence by hypothesis there exists
some constant Cε,T1 such that
γ(1−θ)(p−1)
Cε kukH 1 (Ω)
≤ Cε,T1
for all
t ∈ [0, T1 ] .
Then we get from (45) (using the fact that T1 ≥ T )
kut (T )k2H 1 + ku(T )k2H 2
Z
≤ku1 k2H 1 + ku0 k2H 2 + ε
0
≤ku1 k2H 1 + ku0 k2H 2 + εT1
T1
(ku(t)k2H 2 + kut (t)k2H 1 )dt + T1 Cε,T1
sup ku(t)k2H 2 + kut (t)k2H 1 + T1 Cε,T1 .
t∈[0,T1 ]
This estimates holds for all T ≤ T1 , so take the supremum on the left over all
T ∈ [0, T1 ] obtaining
(1 − εT1 ) sup ku(t)k2H 2 + kut (t)k2H 1 ≤ ku1 k2H 1 + ku0 k2H 2 + T1 Cε,T1 .
t∈[0,T1 ]
Choose ε < 1/T1 to get the desired estimate.
22
LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV
Now recall that we have been in fact working with Galerkin approximations.
Thus what we have proven reads
sup kun (t)k2H 2 (Ω) + kunt (t)k2H 1 (Ω)
t∈[0,T1 ]
≤kun0 k2H 2 (Ω) + kun1 k2H 1 (Ω) + CT1 {kun0 kH 1 (Ω) , kun1 kL2 (Ω) }
for n ∈ N .
Since the right-hand side is bounded uniformly in n, then (un , unt ) are confined to
a closed ball in L∞ (0, T ; H 2 (Ω) × H 1 (Ω)). Hence so is the weak* limit (u, ut ) .
We have demonstrated that the higher energy of solutions does not blow-up
near any finite terminal time TM . It means that a priori bounds of Proposition 2
and therefore existence of regular solutions can be established over any finite time
interval. This step completes the proof of global existence stated in Theorem 4.2.
5.8. Semigroup approach to local regular solutions for linear damping.
We conclude the exposition by mentioning a simpler approach to regular solutions
when the damping g is linear and the source is twice-differentiable:
Assumption 2. Suppose
g(s) = γs,
γ>0
and
f ∈ C 2 (R)
with
|f 00 (s)| . |s|p−2
for p as in (f -2) .
The linearity of g is required for the monotonicity of the evolution generator on
the higher-energy space. The stronger assumption on f ensures that its associated
Nemytski operator is locally Lipschitz H 2 (Ω) → H 1 (Ω).
Let A be the Laplace operator as in (9). Based on it we define the evolution
generator
A : D(A ) ⊂ D(A) × D(A1/2 ) → D(A) × D(A1/2 )
0 −I
u
−ut
A =
or A
=
A g
ut
Au + g(ut )
n
o
with domain D(A ) = (u, v) ∈ D(A) × D(A) | Au + g(v) ∈ D(A1/2 ) .
Remark 5. This definition of D(A ) can be used even if g is nonlinear. In the case
when g is linear or linearly bounded at infinity D(A ) is a subset of H 3 (Ω) × H 2 (Ω).
Then system (1) is equivalent to:
u
u
0
∂t
+A
−
= 0.
ut
ut
f (u)
(46)
Here we notice that the operator
u
0
B
=
ut
f (u)
is locally Lipschitz continuous on D(A) × D(A1/2 ). This conclusion follows from the
fact that u 7→ f (u) is locally Lipschitz H 2 (Ω) → H 1 (Ω) via Assumption 2.
Our goal is to show that A is maximal monotone on the Hilbert space
H = D(A) × D(A1/2 )
REGULAR SOLUTIONS
23
identified with its dual. To verify the monotonicity let U = (u, v), Û = (û, v̂) ∈
D(A ). We must prove
(A U − A Û , U − Û )H ≥ 0 .
Using the definition of A we have
A U − A Û , U − Û
H
v̂ − v
u − û
=
,
Au − Aû + g(v) − g(v̂)
v − v̂ H
=(v̂ − v, u − û)D(A) + (Au − Aû, v − v̂)D(A1/2 ) + (g(v) − g(v̂), v − v̂)D(A1/2 )
=(Av̂ − Av, Au − Aû)Ω + (Au − Aû, Av − Av̂)Ω + (g(v) − g(v̂), Av − Av̂)Ω
= A1/2 g(v) − A1/2 g(v̂), A1/2 v − A1/2 v̂
Ω
which is non-negative since g is linear. This step verifies the monotonicity. For the
maximality property it suffices to prove that A + I is onto H. Let a ∈ D(A) and
b ∈ D(A1/2 ). We need to establish the existence of (u, v) ∈ D(A ) such that
−v + u
a
=
.
Au + g(v)
b
That amounts to solving
A(v + a) + g(v) + v = b
⇔
A(v) + g(v) + v = (b − A(a)) ∈ L2 (Ω) .
Hence it suffices to prove that the operator
T v = A(v) + g(v) + v
is surjective D(A) ⊂ L2 (Ω) → L2 (Ω), which is a direct consequence of the elliptic
theory for the Laplace operator since g here is just an increasing linear function.
Going back to (46), we have a locally Lipschitz perturbation of a maximal monotone flow on H. Now define “truncations” of the source term:
(
f (u)
kukH 2 (Ω) ≤ k
Fk (u) :=
u
kukH 2 (Ω) > k .
f k kuk 2
H
Then for k ∈ N the operator Fk is globally Lipschitz H 2 (Ω) → H 1 (Ω). Replacing
f (u) by Fk (u) in (46) we obtain a “truncated” system with a globally Lipschitz
nonlinearity. It generates a quasi-contractive semigroup [2, Prop. 1.2, p. 230] on
the space H. The energy of solutions in H grows at most exponentially with respect
to the size of the initial data. Thus for a given initial condition (u0 , u1 ) ∈ H if we
choose k > ku0 kH 2 (Ω) then f (u(t)) and Fk (u(t)) will coincide for all sufficiently
small times. Consequently, the semigroup solution to the truncated equation will
also provide a solution to (46) over some time interval [0, T ] where T may depend
on the norm k(u0 , u1 )kH . This solution belongs to the space
C([0, T ], H 2 (Ω) ∩ H01 (Ω)) ∩ C 1 ([0, T ], H01 (Ω))
and satisfies the properties of the regular solutions described in Definition 3.
6. Acknowledgements. The authors are grateful to Dr. Irena Lasiecka for many
insightful and fruitful discussions, and to Dr. Anna Kaminska for providing valuable
references and helping us navigate the extensive theory of Orlicz function spaces.
24
LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV
REFERENCES
[1] R. A. Adams and J. J. F. Fournier. Sobolev spaces, volume 140 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam, second edition, 2003.
[2] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press,
Inc. 1993.
[3] V. Barbu, I. Lasiecka, and M. Rammaha. On nonlinear wave equations with degenerate damping and source terms. Trans. Amer. Math. Soc., 357(7):2571–2611 (electronic), 2005.
[4] L. Bociu, Local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping, Nonlinear Analysis
A: Theory, Methods and Applications, Vol. 71, Issue 12 (2009), e560–e575.
[5] L. Bociu and I. Lasiecka, Blow-up of weak solutions for the semilinear wave equations
with nonlinear boundary and interior sources and damping Applicationes Mathematicae, 35
(2008), 281–304.
[6] L. Bociu and I. Lasiecka, Uniqueness of Weak Solutions for the Semilinear Wave Equations
with Supercritical Boundary/Interior Sources and Damping, Discrete Contin. Dyn. Syst., 22
(2008), 835-860.
[7] L. Bociu and I. Lasiecka. Local Hadamard well-posedness for nonlinear wave equations with
supercritical sources and damping. J. Differential Equations, 249(3):654–683, 2010.
[8] L. Bociu and P. Radu. Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping. Discrete Contin. Dyn.
Syst., (Dynamical Systems, Differential Equations and Applications. 7th AIMS Conference,
suppl.):60–71, 2009.
[9] L. Bociu, M. Rammaha, and D. Toundykov. On a wave equation with supercritical interior
and boundary sources and damping terms. Math. Nachr., 284(16):2032–2064, 2011.
[10] M. Cavalcanti, V.N. Cavalcanti and P. Martinez, Existence and Decay Rates for the Wave
Equation with Nonlinear Boundary Damping and Source Term, J. Differential Equations,
203 (2004), 119–158.
[11] I. Chueshov, M. Eller and I. Lasiecka, On the Attractor for a Semilinear Wave Equation with
Critical Exponent and Nonlinear Boundary Dissipation, Communications in Partial Differential Equations, 27 (2002), 1901–1951.
[12] E. Fereisl, Global attractors for semilinear damped wave equations with supercritical exponent,
Journal of Differential Equations, 116 (1995), 431–447.
[13] V. Georgiev and G. Todorova, Existence of a Solution of the Wave Equation with Nonlinear
Damping and Source Terms, Journal of Differential Equations, 109 (1994), 295–308.
[14] W. Gong and Z. Shi. Drop properties and approximative compactness in Orlicz-Bochner
function spaces. J. Math. Anal. Appl., 344(2):748–756, 2008.
[15] M. A. Krasnosel0 skiı̆ and J. B. Rutickiı̆. Convex functions and Orlicz spaces. Translated from
the first Russian edition by Leo F. Boron. P. Noordhoff Ltd., Groningen, 1961.
[16] I. Lasiecka and J. Ong, Global Solvability and Uniform Decays of Solutions to Quasilinear Equation with Nonlinear Boundary Dissipation, Communications in Partial Differential
Equations, 24 (1999), 2069–2107.
[17] I. Lasiecka and D. Tataru. Uniform Boundary Stabilization of Semilinear Wave Equations
with Nonlinear Boundary Damping, Differential and Integral Equations, 6 (1993), 507–533.
[18] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous
and Approximation Theories, Cambridge, University Press, 2000.
[19] V. Lakshmikantham and S. Leela. Differential and integral inequalities: Theory and applications. Vol. I: Ordinary differential equations. Academic Press, New York, 1969. Mathematics
in Science and Engineering, Vol. 55-I.
[20] P.-K. Lin. Köthe-Bochner function spaces. Birkhäuser Boston Inc., Boston, MA, 2004.
[21] J.-L. Lions and E. Magenes. Problèmes aux limites non homogènes et applications. Vol. 1.
Dunod, Paris, 1968.
[22] L. Payne and D. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,
Israel J. Math 22 (1975), 273 – 303.
[23] P. Radu, Weak Solutions to the Cauchy Problem of a Semilinear Wave Equation with Damping and Source Terms, Advances in Differential Equations, vol 10, No. 11 (2005), 1281–1300.
[24] P. Radu, Weak Solutions to the Initial Boundary Value Problem of a Semilinear Wave Equation with Damping and Source Terms, Applicationae Mathematica (Warsaw), vol 35, no. 3,
(2008), 355–378.
REGULAR SOLUTIONS
25
[25] P. Radu, Strong solutions for semilinear wave equations with damping and source terms, to
appear in Applicable Analysis.
[26] J. Serrin, G. Todorova and E. Vitillaro, Existence for a Nonlinear Wave Equation with Damping and Source Terms, Differential Integral Equations, 16 (2003), 13–50.
[27] J. Simon, Compact sets in the space Lp (0, T ; B), Annali di Mat. Pura et Applicate (4) 146
(1987), 65–96.
[28] G. Todorova. Cauchy problem for a nonlinear wave equation with nonlinear damping and
source terms. Nonlinear Anal., 41(7-8, Ser. A: Theory Methods):891–905, 2000.
[29] G. Todorova and E. Vitillaro. Blow-up for nonlinear dissipative wave equations in Rn . J.
Math. Anal. Appl., 303(1):242–257, 2005.
[30] E. Vitillaro, Global existence of the wave equation with nonlinear boundary damping and
source terms, Journal of Differential Equations, 186 (2002), 259–298.
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