THE GENERALIZED DIFFUSION PHENOMENON AND
APPLICATIONS
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
Abstract. We study the asymptotic behavior of solutions to dissipative wave
equations involving two non-commuting self-adjoint operators in a Hilbert
space. The main result is that the abstract diffusion phenomenon takes place,
as solutions of such equations approach solutions of diffusion equations at large
times. When the diffusion semigroup has the Markov property and satisfies a
Nash-type inequality, we obtain precise estimates for the consecutive diffusion
approximations and remainder. We present several important applications
including sharp decay estimates for dissipative hyperbolic equations with variable coefficients on an exterior domain. In the nonlocal case we obtain the first
decay estimates for nonlocal wave equations with damping; the decay rates are
sharp.
1. Introduction
Let B : D(B) → H and C : H → H be nonnegative self-adjoint operators on a
real Hilbert space (H, k · k) and consider the initial value problem on (0, ∞)
C∂t2 u + ∂t u + Bu = 0, u(0) = u0 , ∂t u(0) = u1 ,
√
where (u0 , u1 ) ∈ D( B) × H. The purpose of this paper is to study the validity of
(1.1)
u(t) ≈ e−tB (u0 + Cu1 ),
t → ∞,
known as the diffusion phenomenon, and to use this approximation for transferring
decay estimates from the diffusion semigroup {e−tB }t≥0 to the solutions of (1.1).
The diffusion phenomenon was discovered by Hsiao and Liu [19] who observed
that the asymptotic profiles of solutions to hyperbolic conservation laws with damp2
ing were proportional to the heat kernel (4πt)−1/2 e−|x| /(4t) . Consecutively, Li [35],
Nishihara [42], [43] and Han and Milani [17] improved the convergence rate and extended the diffusion phenomenon to quasilinear hyperbolic equations with damping
in dimensions n ≤ 3. These results applied only to the Laplace operator ∆ in Rn ,
as they were obtained by means of the Fourier transform or fundamental solutions.
Orive, Zuazua and Pazoto [46] verified the diffusion phenomenon for damped
wave equations involving uniformly elliptic operators with periodic coefficients.
Their result used Bloch waves decomposition instead of the Fourier transform.
Another case of well understood spectral decomposition was the Laplace operator
with rapidly decaying potential in Rn or exterior domain; see Racke [47].
Date: July 1, 2015.
1991 Mathematics Subject Classification. Primary 35L90, 35L20; Secondary 37B40.
Key words and phrases. Diffusion phenomenon, dissipative wave equation, decay estimates,
nonlocal wave equation.
The first author acknowledges the support through the award NSF - DMS 0908435.
1
2
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
The abstract setting of diffusion phenomenon was introduced by Ikehata [27]
who only considered applications to the Laplace operator in exterior domain. Later
Ikehata and Nishihara [28] gave a completely abstract version involving one selfadjoint operator B : D(B) → H and C = I in problem (1.1):
(1.2)
ku(t) − e−tB (u0 + u1 )k . c(t)(t + 1)−1 (ku0 k + ku1 k),
t ≥ 0,
1/2+ε
for c(t) = ln
(t + 2) with any ε > 0. The latter result was not sharp, but the
method of [28] could also work in the two operator setting of (1.1).
Chill and Haraux [7] removed the logarithmic losses and obtained the sharp
version of abstract diffusion phenomenon (1.2) with c(t) = 1. Their method was
heavily based on the spectral theorem for B. In the same setting, Yamazaki [58]
derived an extension to time-dependent damping a(t)∂t u.
Results of the form (1.2) are, however, unsuitable for decay estimates of dissipative wave equations in high spatial dimensions. The remainder in these estimates
is always O(t−1 ), while the norms ku(t)k are expected to depend on the dimension
and asymptotic behavior of coefficients in (1.1). In fact, Matsumura [36] has shown
that ku(t)k = O(t−n/4 ) for (1.1) with B = −∆ in L2 (Rn ) and C = I. An improvement of (1.2) was found in a recent paper of Radu, Todorova and Yordanov [50]
where the diffusion phenomenon was established in the form
ku(t) − e−tB (u0 + u1 )k
(1.3)
. e−t/16 (ku0 k + ku1 k)
+(t + 1)−1 ke−tB/2 u0 k + ke−tB/2 u1 k ,
t ≥ 0.
The above remainder decays faster than the solutions of both equations, whenever these decay like powers of t. What makes approximation (1.3) useful is the
availability of sharp decay estimates for the diffusion semigroup {e−tB }t≥0 in Lq
spaces with q 6= 2; see Saloff-Coste [53]. In particular, this approach can generalize
Matsumura’s estimates to elliptic operators with variable coefficients in the entire
space or exterior domain when B has the Markov property and satisfies a Nash
type inequality [50]. The proof of (1.3) follows from the representation, for u0 = 0,
p
−t/2 sinh t 1/4 − B
p
u1 , t ≥ 0,
u(t) = e
1/4 − B
p
and the low frequency expansion 1/4 − B = 1/2−B +O(B 2 ), where the functions
of B are defined through operator calculus or spectral theorem. In the paper of
Ikehata, Todorova and Yordanov [29], the same method was used to show a more
complex diffusion phenomenon
for the abstract wave equation with strong damping:
√
if (u, ∂t u) ∈ C(R+ , D( B) × H) is the unique mild solution to ∂t2 u + B∂t u + Bu = 0
and u(0) = u0 , ∂t u(0) = u1 , then
!
√
√
sin
t
B
u(t) ≈ e−tB/2 cos t B u0 + √
u1 , t → ∞.
B
It is difficult to generalize the diffusion approximation (1.3) to non-commuting
operators B and C in problem (1.1) when the spectral theorem is no longer available.
First, there is no explicit formula for u(t) in terms of the diffusion semigroup.
Second, the decomposition into low and high frequencies with respect to B is not
preserved by C. The approach of this paper is straightforward but requires many
inductive steps and leads to long and tedious calculations. We transform (1.1) into
a nonhomogeneous equation C∂t2 u + ∂t u + Bu = f with zero initial data and source
GENERALIZED DIFFUSION PHENOMENON
3
f depending on (u0 , u1 ). Treating C∂t2 u as a perturbation, we iterate the equation
k times to find an approximate solution ukdif (t) = u0 (t) + u1 (t) + · · · + uk (t), where
uj (t) are determined one after another from
∂t u0 + Bu0
1
∂t u + Bu
1
∂t uk + Buk
=
f,
= −C∂t2 u0 ,
..
.
= −C∂t2 uk−1 .
The initial values are uj (0) = 0, j = 0, . . . , k, and k ≥ 1 is arbitrary. We bound the
remainder u(t) − ukdif (t) using an abstract version of the weighted energy method
from [49]; the final estimate is ku(t) − ukdif (t)k = O(t1−k ), since the remainder
has 2k time derivatives. There are additional difficulties related to the diffusion
approximations ukdif (t). We work with mollified u(t) to avoid losses of regularity at
each C∂t2 uj (t), j = 1, . . . , k. To show that uj (t) decay faster than u0 (t), we need to
rely on the maximal regularity of Markov semigroups {e−tB }t≥0 in Lq spaces.
1.1. Assumptions. Let us restate the abstract initial value problem on (0, ∞):
C∂t2 u + ∂t u + Bu = 0,
(1.4)
u(0) = u0 ,
∂t u(0) = u1 ,
√
where (u0 , u1 ) ∈ D( B) × H. The self-adjoint operators B and C satisfy
(H1)
D(B) is dense in H and C is a bounded operator on H;
(H2)
hBu, ui > 0 for u ∈ D(B) and u 6= 0;
(H3)
c1 kuk2 ≥ hCu, ui ≥ c0 kuk2 for u ∈ H, where c1 ≥ c0 > 0.
Recall that we work in a real Hilbert space H with inner product h·, ·i and norm
k · k. The above assumptions not √
only guarantee the existence and uniqueness of
mild solutions (u, ∂t u) ∈ C(R+ , D( B) × H), but also imply the estimate
√
√
√
√
k C∂t u(t)k2 + k Bu(t)k2 ≤ k Cu1 k2 + k Bu0 k2 ,
t ≥ 0.
In Appendix A we outline the proof based on the Lumer-Phillips theorem.
1.2. Results. To state
√ our results we introduce E(s; v) the energy associated with
(v, ∂s v) ∈ C(R+ , D( B) × H):
(1.5)
E(s; v) =
√
1 √
k C∂s v(s)k2 + k Bv(s)k2 ,
2
s ≥ 0.
4
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
√
Theorem 1.1. Assume that (H1)-(H3) hold and let (u, ∂t u) ∈ C(R+ , D( B) × H)
be the unique mild solution of (1.4). For all k ≥ 1,
t
Z
(s + 1)2(k−1) ku(s) − ukdif (s)k2 ds .
(i)
0
√
ku0 k2 + k Bu0 k2 + ku1 k2
Z t
(s + 1)2(k−2) kukdif (s)k2 ds,
+
0
t
Z
(s + 1)2(k−1/2) E(s; u − ukdif )ds .
(ii)
0
√
ku0 k2 + k Bu0 k2 + ku1 k2
Z t
+
(s + 1)2(k−2) kukdif (s)k2 ds,
0
Z
t
2k
(s + 1) k∂s (u(s) −
(iii)
ukdif (s))k2 ds
.
0
√
ku0 k2 + k Bu0 k2 + ku1 k2
Z t
+
(s + 1)2(k−2) kukdif (s)k2 ds,
0
where the diffusion approximation ukdif (t) = u0 (t) + u1 (t) + · · · + uk (t) is determined
as follows: given χ ∈ Cc∞ (R), with supp χ ⊂ (0, 1), and such that
χ ≥ 0,
kχkL1 = 1;
the leading term u0 (t) and corrections uj (t) are solutions to the chain of equations
∂t u0 + Bu0
1
∂t u + Bu
1
∂t uk + Buk
= χ(t)Cu1 + ∂t χ(t)Cu0 + χ(t)u0 ,
= −C∂t2 u0 ,
..
.
= −C∂t2 uk−1 ,
where the initial values are uj (0) = 0 for j = 0, 1, . . . , k.
Remark 1.2. The dependance on χ is not essential for the applications to Markov
semigroups considered here.
Remark 1.3. Our approach naturally leads to the diffusion phenomenon in average.
A pointwise version for Markov semigroups is given in Corollary 1.5 below.
We have more precise results under additional assumptions on the Hilbert space
and operators. Let H = L2 (Ω, µ), where (Ω, µ) is a σ-finite measure space.
(H4)
−B generates Markov semigroups {e−tB }t≥0 on Lq (Ω, µ), q ∈ [1, 2].
(H5)
∃ m > 0 such that ke−tB gk2 ≤ cq t−m/2(1/q−1/2) (kgkq + kgk2 )
for g ∈ Lq (Ω, µ) ∩ L2 (Ω, µ), t > 0, q ∈ [1, 2].
(H6)
C is a bounded operator Lq (Ω, µ) → Lq (Ω, µ) for q ∈ [1, 2].
GENERALIZED DIFFUSION PHENOMENON
5
√
Theorem 1.4. Assume that (H1)-(H6) hold and let (u, ∂t u) ∈ C(R+ , D( B) × H)
be the unique mild solution of (1.4). If k ≥ m/2(1/q − 1/2) + 2 and q ∈ (1, 2], then
Z t
(s + 1)2(k−1) ku(s) − v(s)k22 ds
(i)
0
√
. (ku0 k2 + k Bu0 k2 + ku1 k2 + ku0 kq + ku1 kq )2 (t + 1)2(k−3/2)−m(1/q−1/2) ,
Z t
(s + 1)2(k−1/2) E(s; u − v)ds
(ii)
0
√
. (ku0 k2 + k Bu0 k2 + ku1 k2 + ku0 kq + ku1 kq )2 (t + 1)2(k−3/2)−m(1/q−1/2) ,
Z t
(s + 1)2k k∂s (u(s) − v(s))k22 ds
(iii)
0
√
. (ku0 k2 + k Bu0 k2 + ku1 k2 + ku0 kq + ku1 kq )2 (t + 1)2(k−3/2)−m(1/q−1/2) ,
where v(s) = e−sB (u0 + Cu1 ) for s ≥ 0. (We denote the norm of Lq (Ω) by k · kq .)
√
Corollary 1.5. Assume that (H1)-(H6) hold and let (u, ∂t u) ∈ C(R+ , D( B)×H)
be the unique mild solution of (1.4). If q ∈ (1, 2], then
(i)
ku(t) − e−tB (u0 + Cu1 )k2
√
. (ku0 k2 + k Bu0 k2 + ku1 k2 + ku0 kq + ku1 kq )(t + 1)−m/2(1/q−1/2)−1 .
(The diffusion semigroup decays slower: ke−tB gk2 . t−m/2(1/q−1/2) (kgkq + kgk2 ).)
(ii)
(iii)
ku(t)k2 . (ku0 k2 + ku1 k2 + ku0 k1 + ku1 k1 )(t + 1)−m/4
√
+(ku0 k2 + k Bu0 k2 + ku1 k2 + ku0 kq + ku1 kq )(t + 1)−m/2(1/q−1/2)−1 ,
√
E 1/2 (t; u) . (ku0 k2 + k Bu0 k2 + ku1 k2 + ku0 k1 + ku1 k1 )(t + 1)−m/4−1
√
+(ku0 k2 + k Bu0 k2 + ku1 k2 + ku0 kq + ku1 kq )(t + 1)−m/2(1/q−1/2)−3/2 .
(Hence, ku(t)k2 and E 1/2 (t; u) have the decay rates of the diffusion semigroup if
the exponent q is sufficiently close to 1.)
Included in the abstract problem (1.1) are many important examples, such as
damped hyperbolic equations in exterior domain Ω ⊆ Rn with smooth ∂Ω and the
Dirichlet boundary condition:
c(x)∂t2 u − ∇ · b(x)∇u + a(x)∂t u = 0
(1.6)
u=0
u = u0 ,
∂t u = u1
x ∈ Ω, t > 0,
x ∈ ∂Ω, t ≥ 0,
x ∈ Ω, t = 0,
where the coefficients a, c ∈ C(Ω) and bij ∈ C 1 (Ω) satisfy ∀ξ ∈ Rn
a0 ≤ a(x) ≤ a1 ,
(1.7)
β
2
b0 (1 + |x|) |ξ| ≤ b(x)ξ · ξ ≤ b1 (1 + |x|)β |ξ|2 ,
c0 ≤ c(x) ≤ c1 ,
with β ∈ [0, 2) and positive ai , bi , ci for i = 0, 1. In fact, H = L2 (Ω, a(x)dx) and
(1.8)
B=−
1
∇ · b(x)∇,
a(x)
C=
c(x)
,
a(x)
followed by a suitable definition of D(B), will reduce (1.6) to (1.1); see Section 4.1.
As a consequence of the abstract results we obtain
6
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
Theorem 1.6. Assume that (1.7) hold and (u0 , u1 ) ∈ H01 (Ω) × L2 (Ω) are compactly supported. Let (u, ∂t u) ∈ C(R+ , H01 (Ω) × L2 (Ω)) be the unique weak solution
to problem (1.6) and B be the self-adjoint operator defined through (4.1) and the
Dirichlet boundary condition; see Section 4. If m = 2n/(2 − β) and q ∈ (1, 2], then
(1.9)
ku(t) − e−tB (u0 + cu1 /a)k2
. (ku0 k2 + k∇u0 k2 + ku1 k2 + ku0 kq + ku1 kq )(t + 1)−m/2(1/q−1/2)−1 .
As a consequence, u satisfies the Matsumura decay estimate
(1.10)
ku(t)k2 .(ku0 k2 + k∇u0 k2 + ku1 k2 + ku0 kq + ku1 kq )(t + 1)−m/2(1/q−1/2)−1
+ (ku0 k1 + ku1 k1 + ku0 k2 + ku1 k2 )(t + 1)−m/4 .
Corollary 1.7. Under the conditions of Theorem 1.6, the solution u satisfies
ku(t)k2
. (ku0 k2 + k∇u0 k2 + ku1 k2 + ku0 k1 + ku1 k1 )(t + 1)−m/4 .
The abstract results of the paper also apply to the nonlocal setting. Thus, we
obtain the first to our knowledge decay rates for solutions of the Cauchy problem
for nonlocal wave equations with damped terms in the form:
(1.11)
c(x)∂t2 u − L(u) + ∂t u = 0
u = u0 ,
∂t u = u1
x ∈ Rn , t > 0,
x ∈ Rn , t = 0.
In the system above c ∈ L∞ (Rn ), the initial data u0 , u1 ∈ L2 (Rn ), and the operator
Z
(1.12)
L(u)(t, x) =
(u(t, y) − u(t, x))J(y − x) dy
Rn
is a nonlocal Laplacian with J a nonnegative integrable kernel that satisfies some
mild conditions formulated in Section 4.2.
Theorem 1.8 (Decay estimates for the nonlocal damped wave equation). Assume
that u0 , u1 ∈ L2 (Rn )∩Lq (Rn ) for q ∈ (1, 2] and c ∈ L∞ (Rn ) satisfies c(x) ≥ c0 > 0.
Let (u, ∂t u) ∈ C(R+ , L2 (Rn ) × L2 (Rn )) be the unique weak solution of (1.11). Then
(i)
ku(t) − e−tL (u0 + cu1 )k2
√
. (ku0 k2 + k −Lu0 k2 + ku1 k2 + ku0 kq + ku1 kq )(t + 1)−n/2(1/q−1/2)−1 .
(ii)
ku(t)k2 . (ku0 k2 + ku1 k2 + ku0 k1 + ku1 k1 )(t + 1)−n/4
√
+(ku0 k2 + k −Lu0 k2 + ku1 k2 + ku0 kq + ku1 kq )(t + 1)−n/2(1/q−1/2)−1 ,
√
E 1/2 (t; u) . (ku0 k2 + k −Lu0 k2 + ku1 k2 + ku0 k1 + ku1 k1 )(t + 1)−n/4−1
√
+(ku0 k2 + k −Lu0 k2 + ku1 k2 + ku0 kq + ku1 kq )(t + 1)−n/2(1/q−1/2)−3/2 .
(iii)
Hence, ku(t)k2 and E 1/2 (t; u) decay as fast as the corresponding norms of nonlocal diffusion semigroup when the exponent q is sufficiently close to 1.
The paper is organized in the following way. In Section 2 we establish the
generalized diffusion phenomenon in average for two non-commuting self-adjoint
operators satisfying (H1) − (H3). We study the diffusion approximations ukdif (t)
in Section 3 under the additional assumptions (H4) − (H6). In Section 4 we apply
the abstract results to derive sharp decay estimates for
GENERALIZED DIFFUSION PHENOMENON
7
• damped hyperbolic equations with variable coefficients in the entire space
or exterior domain (Section 4.1)
• nonlocal damped wave equations (Section 4.2).
The paper also includes Appendices A, B and C which present useful results concerning well-posedness, decay estimates for dissipative equations and properties of
Markov semigroups.
2. Diffusion Phenomenon in Hilbert Spaces
The common idea behind all proofs of diffusion phenomenon is to treat C∂t2 u as
a decaying source in ∂t u + Bu = −C∂t2 u. Solving for u,
Z t
−tB
(2.1)
u(t) = e
u0 −
e−(t−s)B C∂s2 u(s) ds.
0
However, the suggested approximation u(t) ≈ e−tB u0 is incorrect; notice that the
integral on [0, t/2] has the form
!
Z t/2
−tB/2
−(t/2−s)B
2
e
e
C∂s u(s) ds ,
0
so this term behaves like a solution to the diffusion equation at large t. It is also not
evident that the integral on [t/2, t] decays faster; the preliminary estimates of ∂t2 u
from Appendix B show only k∂t2 u(t)k . t−2 ku(t)k on average. The final difficulty
related to (2.1) is the loss of regularity in t; the bound of ku(t)k will involve bounds
of its second-order derivative k∂t2 u(t)k.
We can avoid the shortcomings of direct approximation by repeating step (2.1)
k times, for arbitrary k ≥ 1, and working with mollified solutions. Although the
amount of work increases substantially, the computations remain within elementary
Hilbert space theory. Moreover, we will use only assumptions (H1) − (H3).
2.1. Mollified problem. Let χ ∈ Cc∞ (R), with supp χ ⊂ (0, 1), and such that
χ ≥ 0,
kχkL1 = 1.
We define the convolution of χ and solutions u of (1.4) as
Z t
(2.2)
ũ(t) =
χ(s)u(t − s) ds, t ≥ 0.
0
Clearly u 7→ ũ is a linear operator L1loc (R+ , H) → C ∞ (R+ , H) and ∂ti ũ(0) = 0 for
all i. These two conditions are crucial for the diffusion approximation which is
constructed inductively and requires C ∞ regularity and zero data at each step.
To derive an equation for ũ(t), we differentiate the convolution (2.2) twice and
obtain the following:
g
∂t ũ = ∂
t u + χ(t)u(0),
2
∂t2 ũ = ∂g
t u + χ(t)∂t u(0) + ∂t χ(t)u(0).
Then the initial value problem (1.4) for u(t) transforms into
(2.3)
C∂t2 ũ + ∂t ũ + B ũ = f,
∂tj ũ(0) = 0,
j = 0, 1, . . .
with
(2.4)
f (t) = χ(t)Cu1 + ∂t χ(t)Cu0 + χ(t)u0 .
8
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
Here f ∈ C0∞ ([0, 1], H), so u(t) and ũ(t) satisfy homogeneous equations at t > 1
but different √initial conditions at t = 0.
√ Of course, it remains to be seen that
{u(t), ∂t u(t), Bu(t)} and {ũ(t), ∂t ũ(t), B ũ(t)} have similar long time behaviors.
The preliminary result below uses only the compact support of χ.
Proposition 2.1. Let v ∈ C 1 (R+ , H) and define the mollification ṽ by (2.2). For
all θ ≥ −1 and L ≥ 1, we have
t
Z
θ+1
(s + L)
t
Z
2
(s + L)θ+1 k∂s v(s)k2 ds
kv(s) − ṽ(s)k ds .
0
0
1
Z
kv(s)k2 ds.
+
0
Proof. If s > 1, we use the definition of ṽ and the Newton-Leibniz theorem:
Z s
kv(s) − ṽ(s)k2 .
k∂τ v(τ )k2 dτ.
s−1
Hence
t
Z
θ+1
(s + L)
Z
2
t
kv(s) − ṽ(s)k ds .
1
(τ + L)θ+1 k∂τ v(τ )k2 dτ.
0
If s ≤ 1, we use the definition of ṽ to write
kṽ(s)k2 .
Z
1
kv(τ )k2 dτ.
0
Adding the estimates for s > 1 and s ≤ 1 completes the proof.
We will combine the above result for ũ(t) with the energy estimate for wave
equations to bound ∂t u(t) and rule out the possibility for fast oscillations of u(t).
√
Proposition 2.2. Let (u, ∂t u) ∈ C(R+ , D( B) × H) be the mild solution of (1.4).
For every θ ≥ −1 and L ≥ 1, we have
Z
(i)
t
(s + L)θ+1 ku(s) − ũ(s)k2 ds .
0
√
ku0 k2 + k Bu0 k2 + ku1 k2
Z t
+
(s + L)θ−1 kũ(s)k2 ds,
0
Z
(ii)
t
(s + L)
θ+2
E(s; u − ũ) ds .
0
√
ku0 k2 + k Bu0 k2 + ku1 k2
Z t
+
(s + L)θ−1 kũ(s)k2 ds,
0
Z
(iii)
0
t
(s + L)θ+3 k∂s (u(s) − ũ(s))k2 ds .
√
ku0 k2 + k Bu0 k2 + ku1 k2
Z t
+
(s + L)θ−1 kũ(s)k2 ds.
0
GENERALIZED DIFFUSION PHENOMENON
9
Proof. To prove claim (i) we employ Proposition 2.1 and estimate (B.3) in Proposition B.1 (see Appendix B) and get
√
. ku0 k2 + k Bu0 k2 + ku1 k2
Z t
(2.5) (s + L)θ+1 ku(s) − ũ(s)k2 ds
0
Z
+
t
(s + L)θ−1 ku(s)k2 ds +
1
Z
0
ku(τ )k2 dτ.
0
To estimate the first integral on the RHS of (2.5) in terms of ũ, instead of u, we
apply
ku(s)k2 . kũ(s)k2 + ku(s) − ũ(s)k2 .
Choosing L sufficiently large and applying (s + L)θ−1 ≤ L1 (s + L)θ+1 , we absorb the
Rt
term 0 (s + L)θ−1 ku(s) − ũ(s)k2 ds from the RHS of (2.5) into the LHS to obtain
t
Z
√
(s + L)θ+1 ku(s) − ũ(s)k2 ds . ku0 k2 + k Bu0 k2 + ku1 k2
0
t
Z
(s + L)θ−1 kũ(s)k2 ds +
+
0
Z
1
ku(τ )k2 dτ.
0
The last integral on the RHS above is bounded in terms of the initial data as follows
Z τ
ku(τ )k ≤ ku(0)k +
k∂s u(s)k ds
0
Z τ √
1
k C∂s u(s)k ds
≤ ku(0)k + √
c0 0
. ku(0)k + E 1/2 (0; u)
for τ ∈ [0, 1]; above, c0 is the constant from the assumption (H3). Thus we have
Z
1
√
ku(τ )k2 dτ . ku(0)k2 + E(0; u) . ku(0)k2 + k Bu0 k2 + ku1 k2 ,
0
which completes the proof of (i).
To prove claims (ii) and (iii) we recall that u(t) − ũ(t) solves
C∂t2 (u − ũ) + ∂t (u − ũ) + B(u − ũ) = −f,
(u − ũ)(0) = u0 ,
∂t (u − ũ)(0) = u1 ,
with f (t) = χ(t)Cu1 + ∂t χ(t)Cu0 + χ(t)u0 . We apply (B.1) and (B.3) in Proposition B.1 to u(t) − ũ(t) and derive claims (ii) and (iii) from the claim (i).
It is now clear that we can work with ũ(t) instead of u(t). The next step is to
construct the diffusion approximation of ũ(t).
2.2. Diffusion approximation. We represent ũ(t) as the sum of k + 1 solutions
of diffusion equations and a remainder: ũ(t) = u0 (t) + u1 (t) + · · · + uk (t) + rk (t),
10
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
where the sequence of functions is determined from
∂t u0 + Bu0
1
∂t u + Bu
1
(2.6)
∂t uk + Buk
C∂t2 rk + ∂t rk + Brk
= f,
= −C∂t2 u0 ,
..
.
= −C∂t2 uk−1 ,
= −C∂t2 uk ,
with f defined in (2.4). All functions vanish at t = 0 together with all derivatives:
∂ti uj (0) = ∂ti rk (0) = 0 for j = 0, 1, . . . , k and i = 0, 1, . . .
Adding together equations (2.6), we have the decomposition
ũ(t) = ukdif (t) + rk (t),
ukdif (t) = u0 (t) + u1 (t) + · · · + uk (t).
Our main goal is to show that rk (t) is a true remainder, i.e., rk (t) decays faster
then any power of t when k is sufficiently large. It is helpful to solve equations (2.6)
in terms of the resolvents Rw and Rd defined below.
2.3. Resolvents. Assume that g ∈ C ∞ (R+ , H) and g(t) = 0 for t ≤ 0. The
resolvents g 7→ Rw g and g 7→ Rd g are given by the unique mild solutions of
(2.7)
(wave)
(2.8)
(diffusion)
C∂t2 v + ∂t v + Bv = g,
∂t v + Bv = g,
v(t) = 0 for t ≤ 0,
v(t) = 0 for t ≤ 0,
respectively. The two resolvents commute with all derivatives,
∂ti Rw = Rw ∂ti ,
∂ti Rd = Rd ∂ti ,
i = 1, 2, . . .
when they act on functions vanishing together with all derivatives at t ≤ 0. There
is no simple expression for Rw but {e−tB }t≥0 yields a formula for Rd :
t
Z
e−(t−s)B g(s) ds,
Rd g(t) =
t ≥ 0.
0
This representation of Rd will be used frequently.
We can readily solve the chain of equations (2.6) in terms of Rw and Rd :
u0 (t)
(2.9)
j
u (t)
k
r (t)
= Rd f (t),
=
=
(−∂t )j (∂t Rd C)j u0 (t),
k+1
(−∂t )
j = 1, . . . , k,
(∂t Rw C)(∂t Rd C)k u0 (t).
In fact, we just use the relation uj (t) = −∂t (∂t Rd C)uj−1 (t) with j = 1, . . . , k. To
find rk (t), we substitute (−∂t )k (∂t Rd C)k u0 (t) for uk (t) and obtain
C∂t2 rk + ∂t rk + Brk = (−∂t )k+1 ∂t C(∂t Rd C)k u0 .
Then the third equation in (2.9) follows from the definition of Rw .
GENERALIZED DIFFUSION PHENOMENON
11
2.4. Weighted estimates for high-order time derivatives of Rd and Rw .
The asymptotic behavior and regularity of u0 (t) are not evident from (2.9). These
are provided by the next sequence of claims.
Proposition 2.3. Let u0 (t) =√Rd f (t), where f (t) = χ(t)Cu1 + ∂t χ(t)Cu0 + χ(t)u0
and assume that (u0 , u1 ) ∈ D( B) × H.
If t ≥ 3, then
u0 (t)
(i)
=
e−tB (Cu1 + u0 )
Z 1
+
χ(s)e−(t−s)B I − e−sB (Cu1 + u0 ) ds
0
Z
−
1
χ(s)e−(t−s)B BCu0 ds
0
and the following estimate holds for all i ≥ 0:
(ii) k∂ti u0 (t) − e−tB (Cu1 + u0 ) k .
t−i−1 ke−tB/2 (Cu1 + u0 )k
+t−i−1 ke−tB/2 Cu0 k.
If t ≥ 0, then
k∂ti u0 (t)k
(iii)
.
for all i ≥ 0. Moreover, (ii) and (iii) imply
Z t
(iv)
(s + 1)2i k∂si+1 u0 (s)k2 ds .
ku0 k + ku1 k
ku0 k2 + ku1 k2 .
0
Rt
Proof. To derive (i), we write e−tB (Cu1 + u0 ) = 0 χ(s)e−(t−s)B e−sB (Cu1 + u0 ) ds
R t −(t−s)B
and subtract this integral from Rd f (t) = 0 e
f (s) ds. Applying
Rd ∂t χ(t)Cu0 = ∂t Rd χ(t)Cu0 = −BRd χ(t)Cu0 ,
we obtain formula (i) for u0 (t).
We can verify (ii) by using (i):
k∂ti u0 (t) − e−tB (Cu1 + u0 ) k
Z 1
.
χ(s)ke−(t/2−s)B B i I − e−sB kop ke−tB/2 (Cu1 + u0 )k ds
Z
0
1
+
χ(s)ke−(t/2−s)B B i+1 kop ke−tB/2 Cu0 k ds,
0
where k · kop means the norm of an operator H → H. The claim follows from
observing that the operator norms satisfy, when s ∈ [0, 1],
ke−(t/2−s)B B i I − e−sB kop . t−i−1 , ke−(t/2−s)B B i+1 kop . t−i−1 .
It is easy to show (iii), since ∂ti Rd f (t) = Rd ∂ti f (t) and the integration in
Rd ∂ti χ(t) takes place over [0, 1].
Finally, we will check (iv). For t < 3, estimate (iii) yields trivially that
Z 3
(s + 1)2i k∂si+1 u0 (s)k2 ds . ku0 k2 + ku1 k2 .
0
12
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
For t ≥ 3, we can use (ii) to obtain
Z
Z t
(s + 1)2i k∂si+1 u0 (s)k2 ds .
t
(s + 1)−2 ds ku0 k2 + ku1 k2 .
3
3
Adding the estimates for [0, 3) and [3, t), we complete the proof of (iv).
It is crucial to prove the fast decay of high-order time derivatives for both Rw
and Rd , in order to bound the remainder rk (t). We will rely on the method of [49].
Lemma 2.4. Let θ ≥ 2l, L ≥ 1 and assume that g(t) = 0 for t ≤ 0. Then
Z t
l Z t
X
θ
l+1
2
(s + L) k∂s Rw g(s)k ds .
(s + L)θ−2l+2j k∂sj g(s)k2 ds
0
0
j=1
t
Z
(s + L)θ−2l k∂s Rw g(s)k2 ds.
+
0
Lemma 2.5. Let θ ≥ 2l, L ≥ 1 and assume that g(t) = 0 for t ≤ 0. Then
Z t
l Z t
X
θ
l+1
2
(s + L) k∂s Rd g(s)k ds .
(s + L)θ−2l+2j k∂sj g(s)k2 ds
0
0
j=1
Z
+
t
(s + L)θ−2l k∂s Rd g(s)k2 ds.
0
The proofs are quite simple and consist of iterating estimates (B.3) and (B.10)
from Appendix B. We omit the details.
2.5. Weighted estimates for high-order time derivatives of (∂t Rd C)i , i ≥ 1.
The diffusion approximation and remainder in (2.9) involve the operators (∂t Rd C)i
with i = 1, . . . , k. We will estimate such powers using induction in i, since each
factor ∂t Rd C is easy to estimate.
Proposition 2.6. Assume that g ∈ C ∞ (R+ , H) and g(t) = 0 for t ≤ 0. Then
Z t
Z t
k(∂s Rd C)i g(s)k2 ds ≤ c2i
kg(s)k2 ds,
1
0
0
for all i ≥ 1. (The constant c1 comes from condition (H3).)
Proof. Let i = 1 and notice that v(t) = Rd Cg(t) solves ∂t v + Bv = Cg, v(0) = 0.
Taking the inner product with ∂t v and integrating on [0, t], we have that
Z t
Z t
Z t
2
2
2
k∂s Rd Cg(s)k ds ≤
kCg(s)k ds ≤ c1
kg(s)k2 ds.
0
0
0
i
i−1
Let i > 1. We write (∂s Rd C) = (∂s Rd C)(∂s Rd C)
and use the estimate for
i = 1 to obtain
Z t
Z t
k(∂s Rd C)i g(s)k2 ds ≤ c21
k(∂s Rd C)i−1 g(s)k2 ds.
0
Now the claim follows from induction in i.
0
The final preliminary result combines the estimates of high order derivatives
with the estimates of high powers for ∂t Rd C.
GENERALIZED DIFFUSION PHENOMENON
13
Proposition 2.7. Assume that g(t) = 0 for t ≤ 0. For l ≥ 1, i ≥ 1 and θ ≥ 2l,
Z
l Z
X
t
(s + 1)
θ
k∂sl (∂s Rd C)i g(s)k2 ds
.
0
j=1
+
t
(s + 1)θ−2l+2j k∂sj g(s)k2 ds
0
i Z
X
t
(s + 1)θ−2l k(∂s Rd C)j g(s)k2 ds.
0
j=1
For θ = 2l, the above estimate simplifies to
Z
l Z
X
t
(s + 1)
2l
k∂sl (∂s Rd C)i g(s)k2 ds
.
0
j=0
t
(s + 1)2j k∂sj g(s)k2 ds.
0
Proof. We already have the estimate for i = 1 as a consequence of Lemma 2.5 and
the fact that C is a bounded operator on H:
Z
l Z
X
t
(s + 1)
θ
k∂sl (∂s Rd C)g(s)k2 ds
.
0
t
(s + 1)θ−2l+2j k∂sj g(s)k2 ds
0
j=1
Z
+
t
(s + 1)θ−2l k(∂s Rd C)g(s)k2 ds.
0
For i > 1, we write (∂s Rd C)i g = (∂s Rd C)(∂s Rd C)i−1 g and apply Lemma 2.5:
Z t
(s + 1)θ k∂sl (∂s Rd C)i g(s)k2 ds
0
.
l Z
X
t
(s + 1)θ−2l+2j k∂sj (∂s Rd C)i−1 g(s)k2 ds
0
j=1
Z
+
t
(s + 1)θ−2l k(∂s Rd C)i g(s)k2 ds.
0
Clearly, the claim is reduced to analogous claims for
Z t
(s + 1)θ−2l+2j k∂sj (∂s Rd C)i−1 g(s)k2 ds,
j = 1, . . . , l.
0
Induction in i completes the proof of the general weighted estimate.
When θ = 2l, the integrals of k(∂t Rd C)j g(s)k2 have no weights and the estimate
in Proposition 2.6 yields
Z t
Z t
k(∂s Rd C)j g(s)k2 ds .
kg(s)k2 ds, j = 1, . . . , i.
0
0
We add the contributions of all j to complete the proof in this case.
2.6. Remainder estimates. The remainder rk (t) = ũ(t) − ukdif (t) is given by
(2.10)
rk (t) = (−∂t )k+1 (∂t Rw C)(∂t Rd C)k u0 (t),
where u0 (t) = Rd f (t). We will show that rk (t) decays fast in average as t → ∞.
14
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
Proposition 2.8. Let k ≥ 1 and rk (t) = ũ(t) − ukdif (t) be defined by (2.10). Then
Z
t
(s + 1)2(k−1) krk (s)k2 ds . ku0 k2 + ku1 k2 ,
(i)
0
Z
t
(s + 1)2(k−1/2) E(s; rk )ds . ku0 k2 + ku1 k2 ,
(ii)
0
Z
(iii)
t
(s + 1)2k k∂s rk (s)k2 ds . ku0 k2 + ku1 k2 ,
0
for all t ≥ 0. (Here u0 and u1 are the initial values in problem (1.4).)
Proof. (i) We have rk (t) = (−1)k+1 ∂tk (∂t Rw C)g k (t) with g k (t) = (∂t Rd C)k ∂t u0 (t)
due to (2.10). Applying Lemma 2.4 with θ = 2(k − 1) and l = k − 1, we derive
t
Z
(s + 1)2(k−1) k∂sk Rw Cg k (s)k2 ds .
0
k−1
XZ t
(s + 1)2j k∂sj Cg k (s)k2 ds
0
j=1
Z
+
t
k∂s Rw Cg k (s)k2 ds.
0
Recall that C is a bounded operator and ∂s Rw Cg k (s) satisfies the energy estimate
Z
t
k
Z
2
k∂s Rw Cg (s)k ds .
0
t
kg k (s)k2 ds.
0
Thus we obtain
Z
t
(s + 1)
2(k−1)
k∂sk Rw Cg k (s)k2 ds
0
.
k−1
XZ t
j=0
The terms on the right hand side are actually
Z t
(s + 1)2j k∂sj (∂t Rd C)k ∂s u0 (s)k2 ds,
(s + 1)2j k∂sj g k (s)k2 ds.
0
j = 0, . . . , k − 1.
0
From Proposition 2.6 (j = 0) and Proposition 2.7 (j ≥ 1), these are bounded by
k−1
XZ t
(2.11)
j=0
(s + 1)2j k∂sj ∂s u0 (s)k2 ds.
0
We recall Proposition 2.3 (iv) to find that the sum (2.11) is . ku0 k2 +ku1 k2 . Thus,
Z
t
(s + 1)2(k−1) krk (s)k2 ds . ku0 k2 + ku1 k2 .
0
The above is claim (i). Since rk (t) solves a dissipative wave equation, claims (ii)
and (iii) follow from combining (i) with estimates (B.1) and (B.3), respectively.
The proof of diffusion phenomenon is complete for the mollified problem (2.3). GENERALIZED DIFFUSION PHENOMENON
15
2.7. Proof of Theorem 1.1. From Proposition 2.2 (i) with L = 1,
Z t
√
(s + 1)2(k−1) ku(s) − ũ(s)k2 ds . ku0 k2 + k Bu0 k2 + ku1 k2
0
Z t
(s + 1)2(k−2) kũ(s)k2 ds.
+
0
ukdif
ukdif )
Writing ũ =
+ (ũ −
on both sides and applying Proposition 2.8 (i), we
obtain version (i) of the diffusion phenomenon in average:
Z t
√
(s + 1)2(k−1) ku(s) − ukdif (s)k2 ds . ku0 k2 + k Bu0 k2 + ku1 k2
0
Z t
+
(s + 1)2(k−2) kukdif (s)k2 ds.
0
Similarly, we derive (ii) and (iii) from Proposition 2.2 (ii) and (iii) with L = 1. 3. Diffusion Phenomenon for Markov Operators
Let the Hilbert space be H = L2 (Ω, µ), for a σ-finite measure space (Ω, µ), and
assume that (H4)–(H6) hold. The diffusion approximation in Theorem 1.1 is
ukdif (t)
(3.1)
=
k
X
ul (t),
l=0
0
l
l
where u (t) = Rd f (t) and u (t) = (−∂t ) (∂t Rd C)l u0 (t) for l = 1, . . . , k. We expect
that u0 (t) is the leading term of ukdif (t) and e−tB (u0 + Cu1 ) is the leading term
of u0 (t) itself. These conjectures can be verified using the maximal regularity of
Markov semigroups {e−tB }t≥0 in Lq spaces.
It follows from (3.1) that there are three types of terms to consider:
Z t
Ik,l =
(s + 1)2(k−1) kul (s)k22 ds,
0
Z t
(1)
Ik,l =
(3.2)
(s + 1)2(k−1/2) E(s; ul )ds,
0
Z t
(2)
Ik,l =
(s + 1)2k k∂s ul (s)k22 ds,
0
with l = 1, . . . , k. In fact, it will be sufficient to estimate just Ik,l .
(1)
(2)
Proposition 3.1. Let Ik,l , Ik,l and Ik,l be defined in (3.2), where k ≥ 1 and
l = 1, . . . , k. For all t ≥ 0,
Z t
(1)
(2)
Ik,l + Ik,l . Ik−1,l +
(s + 1)2k k∂s2 ul−1 (s)k2 ds.
0
Proof. Recall that ul are solutions to ∂t ul + Bul = −C∂t2 ul−1 . The claim follows
from applying estimates (B.1) and (B.3) in Proposition B.2 to ul .
(1)
(2)
The next preliminary estimate of Ik,l , Ik,l and Ik,l uses only (H1) − (H3).
16
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
(1)
(2)
Proposition 3.2. Let Ik,l , Ik,l and Ik,l be defined in (3.2), where k ≥ 1 and
l = 1, . . . , k. For all t ≥ 0,
Z t
(1)
(2)
Ik,l + Ik,l + Ik,l . ku0 k22 + ku1 k22 +
(s + 1)2k−4 ku0 (s)k22 ds
0
+
l Z
X
j=1
t
(s + 1)2k−4 k(∂s Rd C)j u0 (s)k22 ds.
0
Proof. We begin with Ik,l . Applying Proposition 2.7, we obtain
l Z t
X
Ik,l .
(s + 1)2k−2−2l+2j k∂sj u0 (s)k22 ds
j=1
+
0
l Z t
X
j=1
(s + 1)2k−2−2l k(∂s Rd C)j u0 (s)k22 ds.
0
Lemma 2.5 implies that, for l ≥ 1,
l Z t
X
(s + 1)2k−2−2l+2j k∂sj u0 (s)k22 ds . ku0 k22 + ku1 k22
j=1
0
Z
+
t
(s + 1)2k−2−2l ku0 (s)k22 ds.
0
This establishes the upper bound on Ik,l . Proposition 3.1 shows that the estimates
(1)
(2)
of Ik,l and Ik,l will be similar, as the new term will be
Z
t
(s + 1)
2k
k∂s2 ul−1 (s)k2 ds
0
.
l Z
X
j=1
+
t
(s + 1)2k−2l+2j k∂sj+1 u0 (s)k22 ds
0
l−1 Z
X
j=1
t
(s + 1)2k−2l k∂s (∂s Rd C)j u0 (s)k22 ds.
0
Then another application of Proposition 2.7, followed by Lemma 2.5, completes the
(1)
(2)
upper bounds on Ik,l and Ik,l .
It is clear from Proposition 3.2 and (H5) that we only need estimates for
Z
t
(s + 1)2k−4 k(∂s Rd C)j u0 (s)k22 ds
0
with j = 1, . . . , l. This is our next task, where properties (H4)–(H6) are essential.
3.1. Maximal regularity. The preliminary result holds for all generators of Markov
semigroups independently of condition (H5).
Proposition 3.3. Assume that (H1)–(H4) and (H6) hold. For q ∈ (1, 2] and i ≥ 1,
k(∂s Rd C)i gkL2 ([0,t),Lq ) ≤ ci,q kgkL2 ([0,t),Lq ) .
Proof. This follows from the maximal regularity of Markov operators B; see Lemma C.5.
GENERALIZED DIFFUSION PHENOMENON
17
Proposition 3.4. Assume (H1)–(H6). For q ∈ (1, 2] and θ ≥ m(1/q − 1/2),
Z t
Z t
θ
2
(s + 1) k(∂s Rd C)g(s)k2 ds .
(s + 1)θ+2 k∂s g(s)k22 ds
0
0
X Z t
kg(s)k2r ds,
+(t + 1)θ−m(1/q−1/2)
r=q,2
0
where g satisfies g(s) = 0 for all s ≤ 0.
Proof. It is sufficient to consider t ≥ 3, since (∂t Rd C)g(t) is trivial to estimate in
terms of ∂t g(t) for bounded t. We split the integral for (∂t Rd C)g over [0, t/2]∪[t/2, t]
and integrate by parts on [0, t/2]:
Z t
∂t Rd Cg(t) = e−tB/2 g(t/2) +
e−(t−s)B C∂s g(s)ds
t/2
−e
−tB/2
!
t/2
Z
e
−(t/2−s)B
BCg(s)ds .
0
The third term is estimated my means of the Lq ∩ L2 − L2 estimate from (H5):
Z t
X
−m/2(1/q−1/2)
k∂t Rd Cg(t)k2 . t
kg(t/2)kr +
k∂s g(s)k2 ds
t/2
r=q,2
+t−m/2(1/q−1/2)
Z
t/2
X −(t/2−s)B
e
BCg(s)ds .
0
r=q,2
r
Substituting this upper bound into the weighted integral, we obtain
Z t
(s + 1)θ k(∂s Rd C)g(s)k22 ds
0
Z t
X
.
(s + 1)θ−m(1/q−1/2)
kg(s/2)k2r ds
0
r=q,2
t
Z
+
(s + 1)
θ
0
Z
!2
s
k∂τ g(τ )k2 dτ
t
Z
ds
s/2
+
(s + 1)
θ−m(1/q−1/2)
0
Z
2
s/2
X −(s/2−τ )B
e
BCg(τ )dτ ds.
0
r=q,2
r
The Cauchy inequality and some trivial estimates yield
Z t
(s + 1)θ k(∂s Rd C)g(s)k22 ds
0
X Z t
θ−m(1/q−1/2)
. (t + 1)
kg(s/2)k2r ds
r=q,2
Z
(3.3)
+
t
(s + 1)
0
+(t + 1)
−1
Z
0
!
s
(τ + 1)
θ+2
k∂τ g(τ )k22 dτ
ds
s/2
θ−m(1/q−1/2)
Z
2
s/2
X Z t
−(s/2−τ )B
e
BCg(τ )dτ ds.
0
0
r=q,2
r
18
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
It is easy to bound the double integral involving k∂τ g(τ )k22 : we change the order of
R 2τ
integration and use τ (s + 1)−1 ds ≤ 1. However, the integral of BCg(τ ) requires
the maximal regularity of B and Proposition 3.3:
2
Z
s/2
X Z t
X Z t
−(s/2−τ )B
kg(τ )k2r dτ.
ds
.
e
BCg(τ
)dτ
0
0
0
r=q,2
r=q,2
r
We substitute this estimate in (3.3) and complete the proof.
Corollary 3.5. Assume (H1)–(H6). For q ∈ (1, 2] and θ ≥ m(1/q − 1/2),
Z t
Z t
θ
i
2
(s + 1) k(∂s Rd C) g(s)k2 ds .
(s + 1)θ+2 k∂s g(s)k22 ds
0
0
X Z t
+(t + 1)θ−m(1/q−1/2)
kg(s)k2r ds,
0
r=q,2
where i ≥ 1 and g satisfies ∂sj g(s) = 0 for all s ≤ 0 and j = 0, . . . , i − 1.
Proof. It is convenient to use induction in i. If i = 1, the claim is already verified
in Proposition 3.4. Let i > 1 and apply Proposition 3.4 to (∂s Rd C)i−1 g(s):
Z t
(s + 1)θ k(∂s Rd C)i g(s)k22 ds
0
Z t
.
(s + 1)θ+2 k∂s (∂s Rd C)i−1 g(s)k22 ds
0
X Z t
+(t + 1)θ−m(1/q−1/2)
k(∂s Rd C)i−1 g(s)k2q ds.
0
r=q,2
Proposition 2.7 with l = 1 implies
Z t
Z t
(s + 1)θ+2 k∂s (∂s Rd C)i−1 g(s)k22 ds .
(s + 1)θ+2 k∂s g(s)k22 ds
0
0
+
i−1 Z
X
j=1
t
(s + 1)θ k(∂s Rd C)j g(s)k22 ds,
0
while Proposition 3.3 yields
X Z t
X Z t
k(∂s Rd C)i−1 g(s)k2r ds .
kg(s)k2r ds.
r=q,2
0
Adding these estimates, we have that
Z t
(s + 1)θ k(∂s Rd C)i g(s)k22 ds .
r=q,2
0
(t + 1)θ−m(1/q−1/2)
0
X Z
r=q,2
Z
+
+
t
kg(s)k2r ds
0
t
(s + 1)θ+2 k∂s g(s)k22 ds
0
i−1 Z t
X
j=1
0
(s + 1)θ k(∂s Rd C)j g(s)k22 ds.
GENERALIZED DIFFUSION PHENOMENON
19
The above inequality provides a reduction to the corresponding inequalities for
Z t
(s + 1)θ k(∂s Rd C)j g(s)k22 ds, j = 1, . . . , i − 1.
0
Hence, the estimate holds by induction for all i ≥ 1.
(1)
Ik,l
(2)
Ik,l
Proposition 3.6. Let Ik,l ,
and
be defined in (3.2) for all k ≥ 1 and
l = 1, . . . , k. If (H1)–(H6) hold, q ∈ (1, 2] and 2k − 4 ≥ m(1/q − 1/2), then
(1)
(2)
. (ku0 k2 + ku1 k2 + ku0 kq + ku1 kq )2 (t + 1)2k−3−m(1/q−1/2) ,
Ik,l + Ik,l + Ik,l
for all t ≥ 0.
(1)
(2)
Proof. We return to the upper bound of Ik,l + Ik,l + Ik,l in Proposition 3.2. Using
Proposition 2.3 and condition (H5), we obtain
Z t
(3.4)
(s + 1)2k−4 ku0 (s)k22 ds
0
. (ku0 kq + ku1 kq + ku0 k2 + ku1 k2 )2 (t + 1)2k−3−m(1/q−1/2) .
The other key estimate is provided by Corollary 3.5, with θ = 2k − 4 and i = j:
Z t
(s + 1)2k−4 k(∂s Rd C)j u0 (s)k22 ds
0
Z t
.
(s + 1)2k−2 k∂s u0 (s)k22 ds
0
X Z t
+(t + 1)2k−4−m(1/q−1/2)
ku0 (s)k2r ds.
r=q,2
0
We combine again Proposition 2.3 with condition (H5) to show that
Z t
(s + 1)2k−4 k(∂s Rd C)j u0 (s)k22 ds
0
. (ku0 kq + ku1 kq + ku0 k2 + ku1 k2 )2 (t + 1)2k−3−m(1/q−1/2) ,
for j = 1, . . . , l. Adding the above inequalities to (3.4) completes the proof.
3.2. Proof of Theorem 1.4. Combine Propositions 3.6, (2.3) and condition (H5).
3.3. Proof of Corollary 1.5. We write
s=t
(s + 1)2k−1 ku(s) − v(s)k22 s=0
Z t
=
∂s (s + 1)2k−1 ku(s) − v(s)k22 ds,
0
where v(s) = e
−sB
(Cu0 + u1 ). Differentiating the integrand we have that
(t + 1)2k−1 ku(t) − v(t)k22
Z t
√
2
2
2
. ku0 k2 + k Bu0 k2 + ku1 k2 +
(s + 1)2k−2 ku(s) − v(s)k22 ds
0
Z t
+
(s + 1)2k−1 k∂s (u(s) − v(s))k2 ku(s) − v(s)k2 ds.
0
20
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
An application of the Cauchy inequality to the second integral yields
√
(t + 1)2k−1 ku(t) − v(t)k22 . ku0 k22 + k Bu0 k22 + ku1 k22
Z t
+
(s + 1)2k−2 ku(s) − v(s)k22 ds
0
Z t
(s + 1)2k k∂s (u(s) − v(s))k22 ds.
+
0
We apply the average decay estimates in Theorem 1.4 to complete the proof.
4. Applications
4.1. Local damped wave equations with variable coefficients. The energy
decay of solutions of (1.6) in an exterior domain for the Laplacian and variable
damping coefficient has been studied extensively by using the multiplier method:
Mochizuki and Nakazawa [38], Mochizuki and Nakao [37], Nakao [40, 41], Ikehata
[25, 26] and the references therein. The energy decay has been found to be of the
order of (1 + t)−1 , except for [26] where Ikehata shows the faster energy decay
(1 + t)−2 for some types of weighted initial data, which may be sharp only for
n = 2. We remark that our assumptions on the operator B are quite general, so we
do not require it to stabilize to the Laplacian for large |x|, in which case classical
arguments can be used; see Racke [47] where the generalized Fourier transform
was used, and Dan and Shibata [11] for a low-frequency resolvent expansion. More
references for results concerning decay rates of solutions of damped wave equations
can be found in the introductions of the papers [48, 49, 50].
In our first application we consider a wave equation on an exterior domain with
variable coefficients. Thus we consider the system (1.6) where the coefficients satisfy
(1.7) and we chose the operators in the abstract setting to be given by
(4.1)
B=−
1
∇ · b(x)∇,
a(x)
C=
c(x)
a(x)
with H = L2 (Ω, a(x)dx). The diffusion phenomenon for this type of system has
been proved in [50], where the coefficients c(x) and a(x) were considered proportional. Note that in (1.6) we deal with non-proportional coefficients c(x) and a(x),
so the above operators B and C are non-commuting.
Checking conditions (H1) − (H6) is a simple exercise and we omit it. Thus,
by Corollary 1.5 we have that the diffusion phenomenon (1.9) from Theorem 1.6
holds for the system (1.6) under conditions (1.7). In order to establish the decay
rates for solutions of the system (1.6) we will transfer the decay from the solution
e−tB (u0 + cu1 /a). This decay rate was obtained in Corollary 4.4 in [50] by using
a weighted Nash inequality (Proposition 4.2 of [50]). As a consequence, (1.10) and
Corollary 1.7 are proven.
4.2. Nonlocal damped wave equations. In this section we apply the results
from Corollary 1.5 to a nonlocal setting, where the operator B is of integral type.
Nonlocal operators have recently been successfully employed in dynamic fracture
through the recent theory of peridynamics [52], models for biological aggregation
where communications between organisms often take place at large distances [20,
55]. The concept of nonlocality has also been used in image processing [14, 15],
diffusion [3, 4], sandpile formation [2], and other population density models [5].
GENERALIZED DIFFUSION PHENOMENON
21
Previously, Ignat, Rossi, and their collaborators developed a solid framework
(see for example [2, 21, 22]) for decay rates for corresponding nonlocal diffusion
equations.
Here we consider the Cauchy initial value problem (1.11) associated with a nonlocal damped wave equation. We take c ∈ L∞ (Rn ), u0 , u1 ∈ L2 (Rn ) and the kernel
J : Rn → R nonnegative. In addition, we will impose that J satisfies
J ∈ L1 (Rn )
ˆ ∈ L1 (Rn ), ∂ξ J(ξ)
ˆ ∈ L2 (Rn )
J(ξ)
ˆ − 1 + |ξ|2 ∼ |ξ|3 , for ξ close to 0,
J(ξ)
ˆ
where Jˆ is the Fourier transform of J. If J(ξ)
= 1 − |ξ|2 then the operator L
becomes the classical Laplacian; thus we have that the nonlocal diffusion operator
L is an approximation of the classical Laplacian.
Under the above assumptions on the kernel the literature (e.g. [2, 21]) provides
us with many results for decay estimates of solutions to nonlocal diffusion equations.
However, to establish the diffusion phenomenon we need a particular decay rate.
Thus we derive the following estimate:
(J1)
(J2)
(J3)
Proposition 4.1. Let p ∈ [1, ∞]. For any w0 ∈ L1 (Rn ) ∩ L∞ (Rn ) and J ∈ L1 (Rn )
that satisfies (J1 − J3), the solution w(t, x) of the linear problem
(
∂t w(t, x) = (J ∗ w − w)(t, x), t > 0, x ∈ Rn ,
(4.2)
w(0, x) = w0 (x), x ∈ Rn ,
satisfies the decay estimate
(4.3)
kw(t)k2 ≤ cq t−n/2(1/q−1/2) (kw0 kq + kw0 k2 )
for w0 ∈ Lq (Ω, µ) ∩ L2 (Ω, µ), t > 0, q ∈ [1, 2]. The constant cq is independent of
the initial data.
Proof. The ideas for the proof of estimate (4.3) are in [21], however, their estimate
contains a constant which depends implicitly on the initial data, so it is not suitable
for obtaining (H5) which is a key estimate for our diffusion result. In order to
explicitly exhibit the dependence on the initial data and derive (4.3) we will make
some adjustments in the proof.
We start with a decomposition given in Lemmas 2.1 in [21] for the fundamental
solution S of the system (4.2) with w0 (x) = δ0 (x) which can be written as
S(t, x) = e−t δ0 (x) + Kt (x)
(4.4)
with Kt ≥ 0 smooth. Thus from (4.4) we have
(4.5)
ˆ
ˆ
ŵ(ξ, t) = eJ(ξ)t−t ŵ0 (ξ) = e−t ŵ0 (ξ) + eJ(ξ)t−t ŵ0 (ξ) − e−t ŵ0 (ξ)
As in the proof of Lemma 2.2 of [21] choose R > 0, such that
(4.6)
ˆ
|J(ξ)|
≤ 1 − |ξ|2 ,
for all |ξ| ≤ R.
With this choice for R find δ with 0 < δ = δ(R) < 1, such that
(4.7)
ˆ
|J(ξ)|
≤ 1 − δ,
for all |ξ| ≥ R.
22
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
By Plancherel’s formula we have
Z
kw(t)k22 = kŵ(t)k22
ˆ
|eJ(|ξ|)t−t − e−t |2 |wˆ0 (ξ)|2 dξ
=
|ξ|<R
Z
ˆ
|eJ(ξ)t−t − e−t |2 |wˆ0 (ξ)|2 dξ.
+
|ξ|>R
For the high frequency integral we use |ea − eb | ≤ |a − b| max{ea , eb } and (4.7):
Z
ˆ
|eJ(ξ)t−t − e−t |2 |ŵ0 (ξ)|2 dξ ≤ t2 e−2δt (1 − δ)2 kŵ0 k22 = t2 e−2δt (1 − δ)2 kw0 k22 .
|ξ|>R
The low frequency integral employs (4.6) to estimate by Hölder’s inequality
Z
ˆ
|eJ(ξ)t−t − e−t |2 |ŵ0 (ξ)|2 dξ
|ξ|<R
!2/q∗
Z
q∗
≤
|wˆ0 (ξ)| dξ
·
|e
|ξ|<R
≤
!1/q1
Z
−t|ξ|2
−t q1
− e | dξ
|ξ|<R
Ckŵ0 k2q∗
!1/q1
Z
·
e
−t|ξ|2 q1
dξ + e
−tq1
|B(R)|
|ξ|<R
where
q∗
q
q
≥ 2, q1 = ∗
=
,
q−1
q −2
2−q
and |B(R)| denotes the volume of the ball of radius R. By the Hausdorff-Young
inequality we conclude that the low frequency integral is bounded by
Z
ˆ
|eJ(ξ)t−t − e−t |2 |wˆ0 (ξ)|2 dξ
q∗ =
|ξ|<R
≤
Ckw0 k2q
!1/q1
Z
e
−t|ξ|2 q1
dξ + e
−tq1
|B(R)|
|ξ|<R
n (2−q)
q
≤ C(q, J)kw0 k2q t− 2 ·
.
By adding together the estimates for the low and high frequency we showed that
(H5) is satisfied for the nonlocal problem, namely we have
kw(t)k2 ≤ cq,J t−n/2(1/q−1/2) (kw0 kq + kw0 k2 ), for w0 ∈ Lq (Ω, µ) ∩ L2 (Ω, µ)
for t > 0, q ∈ [1, 2].
Proof of Theorem 1.8. The result follows as an immediate consequence of
Colrollary 1.5 by taking
Bu := −Lu, Cu(x) := c(x)u, c ∈ L∞ (Rn ).
√
with D(L) = D( L) = H = L2 (Rn ) and m from Corollary 1.5 is taken to be n
according to Proposition 4.1. Thus (H1), (H3) and (H6) are immediately satisfied.
The non-negativity of the operator B as stated in (H2) is a consequence of an
“integration by parts” result for nonlocal operators. We have that
Z
h−Lu, ui = (u(y) − u(x))2 J(y − x) dydx ≥ 0,
GENERALIZED DIFFUSION PHENOMENON
23
which is simply a consequence of the fact that the kernel J is symmetric (see [2] for
a more detailed introduction to nonlocal operators). The fact that the operator −L
generates a Markov semigroup follows according to Definition C.1 from the fact that
the solutions of the nonlocal diffusion problem that start with non-negative initial
data remain non-negative (consequence of the decomposition (4.4)), and from the
fact that the L1 norm is non-increasing (see Lemma 2.2 in [21]).
Finally, (H5) follows from Proposition 4.1 which concludes the proof.
Remark 4.2. Our abstract results allow us to show the diffusion phenomenon in
the nonlocal setting with variable coefficients a, c ∈ L∞ (Rn ), i.e. for the equation
c(x)∂t2 u − L(u) + a(x)∂t u = 0,
x ∈ Rn , t > 0.
Therefore, obtaining the decay of solutions to the nonlocal damped wave equation
requires only the decay estimate of the associated nonlocal diffusion problem.
Remark 4.3. Similar results follow for the nonlocal version of the nonlocal fractional
Laplacian using the decay rates obtained by Ignat and Rossi in [22] (Theorem 1.1).
The assumption (J3) on the kernel is replaced by
ˆ − 1 ∼ |ξ|s
J(ξ)
for ξ close to 0,
for s > 0 given. This operator is shown to converge to the nonlocal Laplacian with
appropriate scaling.
Appendix A. Existence and Uniqueness of Solutions
The existence and uniqueness of mild solutions to abstract dissipative wave equations is a standard result from semigroup theory. We will give a short proof below.
Let us recall the main initial value problem:
C∂t2 u + ∂t u + Bu = 0,
(A.1)
t > 0,
(A.2)
u = u0 , ∂t u = u1 , t = 0.
√
Here (u0 , u1 ) ∈ D( B) × H. Moreover, the self-adjoint operators B and C satisfy
(A.3)
(1)
D(B) is dense in H and C is a bounded operator on H;
(2)
hBu, ui > 0 for u ∈ D(B) and u 6= 0;
c1 kuk2 ≥ hCu, ui ≥ c0 kuk2 for u ∈ H, where c1 ≥ c0 > 0.
√
√
√
It follows from (2) that D( B) is a Hilbert space with inner product h B ·, B ·i.
Condition (3) also shows that C −1 exists as a bounded operator
H → H.
√
We introduce the auxiliary Hilbert space H∗ = D( B) × H with the inner
product
√
√
√
√
(A.4)
hw1 , w2 i∗ = h Bu1 , Bu2 i + h Cv1 , Cv2 i,
(3)
for wi = (ui , wi ) and i = 1, 2. If v = ∂t u, problem (A.1), (A.2) can be written as
(A.5)
∂t w + Aw = 0,
t > 0,
(A.6)
w = w0 ,
t = 0,
where
w(t) =
u(t)
v(t)
,
w0 =
u0
u1
24
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
and the operator A is defined by
0
−I
(A.7)
A=
,
C −1 B C −1
D(A) = {(u, v) ∈ H∗ : u ∈ D(B)}.
Proposition A.1. Assume that conditions (A.3) hold. If A is defined in (A.7),
then −A generates a contraction semigroup on H∗ . Consecutively,
problem (A.5),
√
(A.6) admits a unique mild solution (u, v) ∈ C(R+ , D( B) × H) satisfying
k(u(t), v(t))k∗ ≤ k(u(0), v(0))k∗ ,
t ≥ 0.
Proof. The Lumer–Phillips theorem states that −A generates a contraction semigroup on H∗ if and only if the following hold:
(a) D(A) is dense in H∗ and A is closed,
(b) I + A is surjective,
(c) hAw, wi∗ ≥ 0 for all w ∈ D(A).
The first part of condition (a) is evidently true, since D(B) is dense in H. To show
the second part, let A(un , vn ) → (f, g) and (un , vn ) → (u, v) in H∗ . Then
vn → v in H,
C −1 Bun + C −1 vn → g in H.
This implies C −1 Bun → g − C −1 v and Bun → Cg − v. Since B is closed, we
conclude that u ∈ D(B). Thus, (u, v) ∈ D(A). For condition (b), we consider the
system (I + A)w = h or
u−v
= f,
v + C −1 Bu + C −1 v
= g,
with w = (u, v) ∈ D(A) and h = (f, g) ∈ H∗ . Let us apply C + I to the first
equation and C to the second equation:
(C + I)u − (C + I)v
Cv + Bu + v
= (C + I)f,
= Cg.
Adding these, we obtain (C + I + B)u = (C + I)f + Cg. Hence,
u =
v
(C + I + B)−1 (C + I)f + (C + I + B)−1 Cg,
= −(C + I + B)−1 Bf + (C + I + B)−1 Cg
gives a solution for arbitrary (f, g) ∈ H∗ . Is is important that C +I +B is a positive
self-adjoint operator with D(C + I + B) = D(B). The final condition (c) readily
follows from definitions (A.4) and (A.7).
Corollary A.2. Assume that conditions (A.3) hold.
√ Then problem (A.1), (A.2)
admits a unique mild solution (u, ∂t u) ∈ C(R+ , D( B) × H) satisfying
√
√
√
√
k C∂t u(t)k2 + k Bu(t)k2 ≤ k Cu1 k2 + k Bu0 k2 , t ≥ 0.
Appendix B. Weighted estimates for dissipative equations
We will present abstract versions of the weighted estimates for damped wave
equations from [49] in the case of bounded C. These estimates naturally extend to
diffusion equations as kCkop → 0. Such results show that, in average,
k∂t v(t)k2 . t−2 kv(t)k2 ,
E(t; v) . t−1 kv(t)k2
for all dissipative wave and diffusion equations.
GENERALIZED DIFFUSION PHENOMENON
25
√
Proposition B.1. Assume that (v, vt ) ∈ C(R+ , D( B) × H) and v satisfies
C∂t2 v + ∂t v + Bv = g,
t > 0.
For every θ ≥ 0 and L ≥ 1, the following estimates hold uniformly in L:
Z t
Z t
θ
2
(s + L)θ+1 kg(s)k2 ds
(s + L) E(s; v) ds . E(0; v) + kv(0)k +
0
0
Z t
(B.1)
+
(s + L)θ−1 kv(s)k2 ds,
0
Z
t
θ+1
(s + L)
0
(B.2)
Z
t
(s + L)θ+1 kg(s)k2 ds
k∂s v(s)k ds . E(0; v) +
0
Z t
+
(s + L)θ E(v; s) ds,
2
0
t
Z t
(s + L)θ+1 k∂s v(s)k2 ds . E(0; v) + kv(0)k2 +
(s + L)θ+1 kg(s)k2 ds
0
0
Z t
θ−1
(B.3)
+
(s + L) kv(s)k2 ds.
Z
0
Here E(s; v) is the energy associated with v(s):
√
1 √
E(s; v) =
k C∂s v(s)k2 + k Bv(s)k2 ,
2
s ≥ 0.
Proof. The proof is very simple for C 2 solutions. We take the inner products with
∂t v(t) and v(t) to obtain
(i)
(ii)
∂t E(t; v) + k∂t v(t)k2 = hg(t), ∂t v(t)i ,
√
√
1
∂t hC∂t v(t), v(t)i + kv(t)k2 + k Bv(t)k2 − k C∂t v(t)k2
2
= hg(t), v(t)i.
Adding (2c1 ) × (i) and (ii), we have
1
2
∂t 2c1 E(t; v) + hC∂t v(t), v(t)i + kv(t)k + 2E(t; v)
2
≤ 2c1 hg(t), ∂t v(t)i + hg(t), v(t)i.
Since
2c1 |hg(t), ∂t v(t)i|
≤ (2c21 /c0 )kg(t)k2 + (c0 /2)k∂t v(t)k2 ,
|hg(t), v(t)i|
≤ (t + L)kg(t)k2 + (t + L)−1 kv(t)k2 ,
the combined energy inequality simplifies to
1
2
(B.4)
∂t 2c1 E(t; v) + hC∂t v(t), v(t)i + kv(t)k + E(t; v)
2
≤ (t + L + 2c21 /c0 )kg(t)k2 + (t + L)−1 kv(t)k2 ,
where L ≥ 1 will be chosen later. It is also clear that the energy identity (i) implies
(B.5)
1
∂t E(t; v) + k∂t v(t)k2
2
≤
1
kg(t)k2 .
2
26
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
Finally, we have the lower and upper bounds, respectively,
1
2c1 E(t; v) + hC∂t v(t), v(t)i + kv(t)k2 ≥ 0,
(B.6)
2
1
(B.7)
2c1 E(t; v) + hC∂t v(t), v(t)i + kv(t)k2 ≤ 3c1 E(t; v) + kv(t)k2 .
2
The above combination of terms is important, since it appears after ∂t in (B.4).
We can now complete the proof using only multiplication with (t + L)θ and
integration by parts in t. Estimate (B.1) is derived from (B.4) by using (B.6),
(B.7) and choosing a large L to ensure that
Z
Z t
1 t
(s + L)θ E(s; v) ds.
3c1 θ
(s + L)θ−1 E(s; v) ds ≤
2
0
0
Then we can change the implicit constant after . and extend (B.1) to all L ≥ 1.
Estimate (B.2) readily follows from multiplying (B.5) by (t + L)θ+1 , integrating
by parts in t and applying (B.1). The final estimate (B.3) is only a corollary of
the first two. Hence, the three claims are verified for C 2 solutions. A standard
regularization procedure allows us to show the claims in full generality.
Abstract diffusion equations admit similar estimates which are stated below.
√
Proposition B.2. Assume that (v, vt ) ∈ C(R+ , D( B) × H) and v satisfies
∂t v + Bv = g,
t > 0.
For every θ ≥ 0 and L ≥ 1, the following estimates hold uniformly in L:
Z t
Z t
√
√
(s + L)θ+1 kg(s)k2 ds
(s + L)θ k Bv(s)k2 ds . k Bv(0)k2 + kv(0)k2 +
0
0
Z t
(B.8)
+
(s + L)θ−1 kv(s)k2 ds,
0
Z
t
θ+1
(s + L)
0
(B.9)
2
k∂s v(s)k ds
Z t
√
2
2
. k Bv(0)k + kv(0)k +
(s + L)θ+1 kg(s)k2 ds
0
Z t
√
+
(s + L)θ k Bv(s)k2 ds,
0
Z t
√
(s + L)θ+1 k∂s v(s)k2 ds . k Bv(0)k2 + kv(0)k2 +
(s + L)θ+1 kg(s)k2 ds
0
0
Z t
θ−1
2
(B.10)
+
(s + L) kv(s)k ds.
Z
t
0
Proof. We combine the identities
√
1
(i)
∂t k Bv(t)k2 + k∂t v(t)k2 = hg(t), ∂t v(t)i ,
2
√
1
(ii)
∂t kv(t)k2 + k Bv(t)k2 = hg(t), v(t)i
2
to obtain
√
1 √
∂t k Bv(t)k2 + kv(t)k2 + k∂t v(t)k2 + k Bv(t)k2
2
≤ hg(t), ∂t v(t)i + hg(t), v(t)i.
GENERALIZED DIFFUSION PHENOMENON
The remaining calculations repeat the proof of Proposition B.1.
27
Appendix C. Markov Semigroups and Maximal Regularity
The class of Markov operators, or generators of Markov semigroups, is very
convenient for the Lp − Lq decay estimates used in this paper. Here we present
some basic definitions and results.
Definition C.1. Let (Ω, µ) be a σ-finite measure space and {e−tB }t≥0 be a strongly
continuous semigroup on L1 (Ω, µ) generated by −B. The semigroup is Markov if
g ∈ L1 and g ≥ 0 ⇒ e−tB g ≥ 0 and ke−tB gk1 ≤ kgk1
for all t > 0, where k · kq is the norm in Lq (Ω, µ), q ∈ [1, ∞]. Here g ≥ 0 is a.e.
with respect to µ.
In Section 3, we work with H = L2 (Ω, µ), for a σ-finite measure space (Ω, µ),
and assume that B ≥ 0 is a self-adjoint operator with the Markov property. Then
{e−tB }t≥0 is a symmetric semigroup of contractions in L2 (Ω, µ). Interpolation
between ke−tB gk2 ≤ kgk2 and ke−tB gk1 ≤ kgk1 , combined with a standard duality
argument, allows us to extend this semigroup to all Lq (Ω, µ) with q ∈ [1, ∞], so
that ke−tB gkq ≤ kgkq . (We denote by B all generators of semigroups on Lq (Ω, µ),
q ∈ [1, ∞], which coincide with {e−tB }t≥0 on L2 (Ω, µ) ∩ Lq (Ω, µ).) Thus we obtain
a symmetric Markov semigroup {e−tB }t≥0 on each Lq (Ω, µ) for q ∈ [1, ∞].
Lemma C.2. Let {e−tB }t≥0 be a symmetric Markov semigroup on Lq (Ω, µ) for
q ∈ [1, ∞]. If m > 0, the following conditions are equivalent:
(C.1)
ke−tB gk∞ ≤ C1 t−m/2 kgk1
(C.2)
kgk2
2+4/m
∀g ∈ L1 (Ω, µ) and ∀t > 0,
4/m
≤ C2 hBg, gikgk1
∀g ∈ D(B) ∩ L1 (Ω, µ).
(Here k · kq is the norm in Lq (Ω, µ) while D(B) ⊂ L2 (Ω, µ) is the domain of B.)
The above result and its generalizations are presented in [9]. Condition (C.2),
known as the Nash inequality, is easy to verify for second-order elliptic operators
in exterior domain with various boundary conditions.
Lemma C.3. Let {e−tB }t≥0 be a symmetric Markov semigroup on Lq (Ω, µ) for
q ∈ [1, ∞]. If condition (C.1) or (C.2) holds, then
(i)
(ii)
ke−tB gkp ≤ Cp,q t−m/2(1/q−1/p) kgkq ,
kgkp ≤
t > 0,
1−m/k(1/2−1/p)
m/k(1/2−1/p)
kB k/2 gk2
,
Cp kgk2
for p ∈ [2, ∞], q ∈ [1, p] and k > m(1/2 − 1/p).
The proofs of these inequalities can be found in [10] and [8].
Another crucial fact is that symmetric Markov semigroups have Lp −Lq maximal
regularity. The latter property is actually defined for a wider class of operators.
Definition C.4. Let (Ω, µ) be a σ-finite measure space and assume that −B generates a bounded holomorphic semigroup on Lq (Ω, µ), where q ∈ (1, ∞). The operator
B has maximal regularity of type Lp −Lq on [0, T ) if, for any g ∈ Lp ([0, T ), Lq (Ω, µ))
the unique mild solution of
∂t v(t) + Bv(t) = g(t), t ∈ [0, T ),
v(0) = 0,
28
PETRONELA RADU, GROZDENA TODOROVA AND BORISLAV YORDANOV
satisfies ∂t v, Bv ∈ Lp ([0, T ), Lq (Ω, µ)). This condition is equivalent to
k∂t vkLp ([0,T ),Lq ) + kBvkLp ([0,T ),Lq ) ≤ CkgkLp ([0,T ),Lq )
and independent of T ∈ (0, ∞).
It is well known that the maximal regularity of type Lp − Lq is independent of
p ∈ (1, ∞), [39]. Hence, we can just say that B has maximal regularity in Lq (Ω, µ).
Expressing the solution v(t) in terms of the semigroup {e−tB }t≥0 , we see that the
maximal regularity in Lq (Ω, µ) means
k∂t Rd gkLp ([0,T ),Lq ) ≤ CkgkLp ([0,T ),Lq ) ,
where Rd is given by definition (2.7), i.e.,
Z t
Rd g(t) =
e−(t−s)B g(s)ds,
t > 0.
0
The following is a useful test for maximal regularity [32].
Lemma C.5. Let (Ω, µ) be a σ-finite measure space and assume that −B generates
a holomorphic semigroup of contractions in L2 (Ω, µ). Assume further that
ke−tB gkq ≤ kgkq ,
t > 0,
g ∈ L2 (Ω, µ) ∩ Lq (Ω, µ),
for all q ∈ [1, ∞]. Then the operator −B has maximal regularity of type Lp − Lq
for all p, q ∈ (1, ∞).
A symmetric Markov semigroup extends to a bounded holomorphic semigroup
of angle arccos |1 − 2/q| in Lq (Ω, µ) for q ∈ (1, ∞); see [33]. Hence, Lemma C.5
applies to such semigroups and yields the key estimate in Proposition 3.3.
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UN-Lincoln, UT-Knoxville, UT-Knoxville and Institute of Mathematics and Informatics, Bulgarian Academy of Sciences
E-mail address: [email protected], [email protected], [email protected]