Large solutions of some parabolic equations
AL SAYED Waad
Laboratoire de Mathématiques et Physique Théorique
(LMPT-TOURS)
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
1/21
Plan
1
On uniqueness of large solutions in nonsmooth domains
2
Solutions of nonlinear parabolic equations with initial blow-up
3
Large solution on lateral boundary and at t = 0
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
2/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
1
On uniqueness of large solutions in nonsmooth domains
2
Solutions of nonlinear parabolic equations with initial blow-up
3
Large solution on lateral boundary and at t = 0
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
3/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
Concept of Large Solution
Large Solution of elliptic equations :
Let Ω be a domain in RN . A function u ∈ C 1 (Ω) is a large solution of
q−1
− ∆u + |u|
u = 0 in Ω,
(1)
if
lim u (x) = ∞,
(2)
ρ(x)→0
where ρ(x) = dist (x, ∂Ω).
Keller and Osserman [1957] proved the existence of a maximal large solution of the
equation (1) when 1 < q < N/(N − 2). In addition, Marcus and Véron [1997]
proved that there exists at most one large solution of the equation (1) when ∂Ω
has the local graph property. In order to establish the uniqueness, Véron [2000]
c
supposed that ∂Ω = ∂Ω .
Purpose : Extend this concept to some parabolic problems.
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
4/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
Large solution on lateral boundary
Let q > 1 and Ω be a bounded domain in RN . We study the problem :
∂t u − ∆u + |u|
q−1
Ω
u = 0 in Q∞
:= Ω × (0, ∞).
(3)
We are interested in positive solutions of (3) which satisfy
lim u (., t) = f (.)
t→0
in L1loc (Ω)
(4)
where f ∈ L1loc + (Ω) and
lim
(x,t)→(y ,s)
u (x, t) = ∞
∀(y , s) ∈ ∂Ω × (0, ∞).
(5)
Procedure :
Construct a maximal solution.
Prove that this maximal solution is a large solution when 1 < q < N/(N − 2).
Search conditions ensuring the uniqueness of this solution.
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
5/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
The Maximal Solution
How to construct the maximal solution ūf of the problem (3)-(4) ?
Ωn increasing sequence of bounded regular domains such that
Ωn ⊂ Ωn+1 ⊂ Ω and ∪Ωn = Ω.
un,f is the increasing limit when k → ∞ de un,k,f solution of

q
Ωn
∂u
− ∆un,k,f + un,k,f
= 0 in Q∞


 t n,k,f
un,k,f (x, t) = k in ∂Ωn × (0, ∞)



un,k,f (x, 0) = f χΩn in Ωn .
(6)
ūf is the limit of the un,f when n → ∞.
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
6/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
Minimal Solution : The minimal solution u f is the limit when n tends to infinity
of the fonction un,0,f solution of

q
Ωn
∂u
− ∆un,0,f + un,0,f
= 0 in Q∞


 t n,0,f
un,0,f (x, t) = 0 in ∂Ωn × (0, ∞)
(7)



un,0,f (x, 0) = f χΩn in Ωn .
Relation between minimal solution and maximal solution :
uf positive solution of (3) satisfying (4).
Construct a solution u0 of (3) satisfying
lim u0 (., t) = 0 in L1loc (Ω),
(8)
0 ≤ uf − u f ≤ u0 ≤ uf ,
(9)
0 ≤ u f − uf ≤ u 0 − u0 .
(10)
t→0
such that
and
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
7/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
Case where 1 < q < N/(N − 2).
We suppose that the domain Ω has a compact boundary.
The maximal solution is large :
RN \{0}
Construct autosimilar solution VN of the equation (3) in Q∞
lim
(x,t)→(y ,0)
.
N
VN (x, t) = 0 ∀y ∈ R \ {0}.
lim VN (x, t) = ∞ locally uniformly in [τ, ∞), for all τ > 0.
|x|→0
√
VN (x, t) = t −1/(q−1) HN (|x|/ t), where H := HN is the unique positive
function satisfying

N −1 r
1

′′
 H +
H′ +
+
H − H q = 0 in R+


r
2
q−1
(11)
limr →0 H(r ) = ∞




limr →∞ r 2/(q−1) H(r ) = 0.
Deduce that ūf is a large solution from
√ the inequality
u f (x, t) ≥ t −1/(q−1) HN (ρ(x)/ t).
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
8/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
Uniqueness
Let uf be a large solution.
c
Suppose that ∂Ω = ∂Ω .
Use the constructed solution u0 and the inequality 0 ≤ u f − uf ≤ u 0 − u0 .
Prove that u 0 = u0 := u.
Consequence :
Theorem
Assume q > 1 and Ω is a bounded domain. Then for any f ∈ L1loc + (Ω) there exists
a maximal solution u f to problem (3) satisfying (4). If 1 < q < N/(N − 2), u f
c
satisfies (5). At end, if 1 < q < N/(N − 2) and ∂Ω = ∂Ω , u f is the unique
solution of the problem which satisfies (5).
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
9/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
Local graph property
We suppose that Γ := ∂Ω is compact and has the local graph property, which
means that there exists a finite number of open sets Ωj (j = 1, ..., k) such
that Γ ∩ Ωj is the graph of a continuous function.
Purpose : Find at most one solution of the problem (3) satisfying (4) et (5).
Strategy : Let u be a large solution with zero initial trace. Let
x = (x ′ , xN ) ∈ RN−1 × R the coordinates in RN and Γ ∩ Ωj is the graph of a
continuous positive function φ defined in C = {x ′ ∈ RN−1 : |x ′ | ≤ R}.
For σ > 0, small enough, we consider φσ ∈ C ∞ (C ) satisfying
φ(x ′ ) − σ/2 ≤ φσ (x ′ ) ≤ φ(x ′ ) + σ/2 ∀x ′ ∈ C .
We suppose
φσ (x ′ ) − σ < φσ′ (x ′ ) − σ ′ < φσ′ (x ′ ) + σ ′ < φσ (x ′ ) + σ
∀ 0 < σ′ < σ ∀ x ′ ∈ C .
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
10/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
Localization
GR
Consider the restriction of u to Q∞
:= GR × (0, ∞), thus u is regular in
GR ∪ Γ2 × [0, ∞) and satifies limxN →φ(x ′ ) u(x ′ , xN , t) = ∞ uniformly for
(x ′ , t) ∈ C × [τ, T ], for tout 0 < τ < T .
Ω
Γ
GR
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
11/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
Let vσ be solution of
G
∂t vσ − ∆vσ + vσq = 0 in Q∞σ,R := Gσ,R × (0, ∞).
Gσ,R
Ω
Γ
0
GR
CIMPA-UNESCO-AAST-EGYPTE
0
0
0
AL SAYED Waad
∞
Large solutions of some parabolic equations
12/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
Let wσ be solution of
G′
′
∂t wσ − ∆wσ + wσq = 0 in QT σ,R := Gσ,R
× (0, ∞).
Gσ,R
G′σ,R
Ω
Γ
0
GR
0
0
∞
0
0
∞
∞
0
∞
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
13/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
Establish for σ > 0
GR
vσ (x ′ , xN − 2σ, t) ≤ u(x ′ , xN , t) in Q∞
and
Gσ,R
′
u(x , xN , t) ≤ vσ (x, t) + wσ (x, t) in Q∞ .
Let σ tends to 0 and obtain
GR
v0 ≤ u ≤ v0 + w0 in Q∞
.
Find a relation between w0 et v0 . For ǫ, τ > 0, there exists δǫ > 0 such that, if
Gδ,R ′ = {x = (x ′ , xN ) : |x ′ | < R ′ et φ(x ′ ) − δ ≤ xN < φ(x ′ )}, we have for
√
G ′
R ′ < R/ N − 1, w0 (x, t) ≤ ǫv0 (x, t + τ ) ∀(x, t) ∈ Q∞δ,R .
Obtain then (1 + ǫ)u(., t + τ ) ≥ u 0 (., t) and u ≥ u 0 .
Consequence :
Theorem
Assume q > 1, Ω is a bounded domain and ∂Ω is locally a continuous graph. Then
for any f ∈ L1loc + (Ω) there exists at most one solution to problem (3) satisfying
(4) and (5).
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
14/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
1
On uniqueness of large solutions in nonsmooth domains
2
Solutions of nonlinear parabolic equations with initial blow-up
3
Large solution on lateral boundary and at t = 0
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
15/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
Blow-up of the solution at t = 0
We study the Cauchy-Dirichlet problem P Ω,f

Ω
= Ω × (0, ∞)
∂ u − ∆u + |u|q−1 u = 0 in Q∞


 t
u = f on ∂Ω × (0, ∞)



limt→0 u(x, t) = ∞
∀x ∈ Ω.
(12)
If no assumption of regularity is made on ∂Ω, the boundary data u = f cannot
be prescribed in sense of continuous functions. However, the case f = 0 can
be treated if the vanishing condition on ∂Ω × (0, ∞) is understood in the H01
local sense thanks to Brezis [1973] results of contractions semigroups
generated by subdifferential of proper convex functions in Hilbert spaces.
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
16/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
In the elliptic case, Labutin [2003] proved the existence of a large solution by
using Wiener [1924] criterion : An open set Ω of RN satisfies Wiener criterion
if, for all σ ∈ ∂Ω,
Z 1
C1,2 (Ωc ∩ B (σ, r )) dr
= +∞ for all x ∈ Ωc ,
N−2
r
r
0
où C1,2 is the electrostatic capacity.
For all functions φ ∈ C (∂Ω) and ψ ∈ L∞
loc Ω , a weak solution of
−∆w = ψ in
Ω
w = φ on ∂Ω
is continuous on ∂Ω.
The parabolic Wiener criterion was established by Ziemer [1978] using the
following capacity for any set of Rn+1 :
Z
ZZ
2
2
Γ(K ) = inf sup u(x, t) dx +
|∇x u(x, t)| dx dt.
(13)
t
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
17/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
Method of construction
Construct a positive solution u Ω of (12) with f = 0 belonging to
C (0, ∞; H01 (Ω) ∩ Lq+1 (Ω)) thanks to Brezis results.
Consider an internal increasing approximation of Ω by smooth bounded
domains Ωn such that Ω = ∪n Ωn
For each of these domains, prove that there exists a maximal solution u Ωn of
n
problem P Ω ,0 .
The sequence {u Ωn } is increasing. The limit function uΩ := limn→∞ u Ωn is the
natural candidate to be the minimal positive solution of a solution of P Ω,0
Prove that u Ω = uΩ
If ∂Ω satisfies the parabolic Wiener criterion, prove there truly exist solutions
of P Ω,0 .
Construct a maximal solution u Ω of this problem.
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
18/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
Consequences
Theorem
If ∂Ω is compact and satisfies the parabolic Wiener criterion, there holds
uΩ = uΩ.
Theorem
If ∂Ω is compact and satisfies the parabolic Wiener criterion, and if
f ∈ C (0, ∞; ∂Ω) is nonnegative, u Ω,f is the only positive solution to problem P Ω,f .
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
19/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
1
On uniqueness of large solutions in nonsmooth domains
2
Solutions of nonlinear parabolic equations with initial blow-up
3
Large solution on lateral boundary and at t = 0
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
20/21
On uniqueness of large solutions in nonsmooth domains
Solutions of nonlinear parabolic equations with initial blow-up
Large solution on lateral boundary and at t = 0
Large solution on lateral boundary and at t = 0
We study the problem :
∂t u − ∆u + |u|
q−1
u=0
in QΩ∞ := Ω × (0, ∞).
(14)
We are interested in positive solutions of (14) satisfying
lim u (., t) = ∞
t→0
in Ω
(15)
and
lim
(x,t)→(y ,s)
u (x, t) = ∞
∀(y , s) ∈ ∂Ω × (0, ∞).
(16)
We prove the main result :
Theorem
Suppose that q > 1 and Ω is a bounded domain. Then there exists a maximal
solution ū of the problem (14). If 1 < q < N/ (N − 2), then ū satisfies (15) et
(16). In addition, 1 < q < N/ (N − 2) and ∂Ω = ∂ Ω̄c , then ū is the unique
solution of the problem (14-15-16).
CIMPA-UNESCO-AAST-EGYPTE
AL SAYED Waad
Large solutions of some parabolic equations
21/21