#### Extinction and asymptotic behavior of solutions for nonlinear parabolic equations with variable exponent of nonlinearity

Boundary Value Problems
Extinction and asymptotic behavior of solutions for nonlinear parabolic equations with variable exponent of nonlinearity
Yanchao Gao 0
Wenjie Gao 0
0 School of Mathematics, Jilin University , Changchun, 130012 , P.R. China
The aim of this paper is to study the existence and extinction of weak solutions of the initial and boundary value problem for ut = div((|u|σ (x,t) + d0)|∇u|p(x,t)-2∇u) + f (x, t, u). First, the authors apply the method of parabolic regularization and Galerkin's method to prove the existence of solutions to the problem mentioned and then obtain the comparison principle by arguing by contradiction. Furthermore, the authors prove that the solution vanishes in finite time and approaches 0 in L2( ) norm as t → ∞.
nonlinear parabolic equation; nonstandard growth condition; extinction; p(x; t)-Laplace operator
1 Introduction
following quasilinear degenerate parabolic problem:
u(x, t) = , (x, t) ∈ T ,
⎧⎪⎪ ut = div(a(x, t, u)|∇u|p(x,t)–∇u) + f (x, t, u), (x, t) ∈ QT ,
⎨
where QT =
f (x, t, u) satisfies
f (x, t, u) = b(x, t) – bu(x, t),
u ∈ (–∞, +∞), x ∈ , t >
∀z = (x, t) ∈ QT , ξ = (y, s) ∈ QT , |z – ξ | < ,
liτm→su+p ω(τ ) ln τ = C < +∞.
Model (.) may describe some properties of image restoration in space and time.
Especially when the nonlinear source f (x, u) = b(x, t) – bu, the functions u(x, t), b(x, t)
represent a recovering image and its observed noisy image, respectively. In the case when p(x, t),
σ (x, t) are fixed constants, there have been many results about the existence, uniqueness,
blowing-up and so on; we refer to the bibliography [–]. When p, σ are functions with
respect to the space variable and time variable, this problem arises from elastic mechanics,
electro-rheological fluids dynamics and image processing, etc.; see [–].
To the best of our knowledge, there are only a few works about parabolic equations
with variable exponents of nonlinearity. In [], Chen, Levine and Rao obtained the
existence and uniqueness of weak solutions with the assumption that the exponent σ (x, t) ≡ ,
< p– < p+ < . In [], we applied the method of parabolic regularization and Galerkin’s
method to prove the existence of weak solutions to problem (.) with the assumption
that σ (x, t) ≡ constant, f (x, t, u) ≡ f (x, t). In this paper, we generalize the results in [].
Especially, unlike [], we obtain the existence and uniqueness of weak solution not only
in the case when σ (x, t) ∈ (, p+p–+ ), p+ ≥ , but also in the case when σ (x, t) ∈ (, ),
< p– < p+ ≤ + √. Furthermore, we apply energy estimates and Gronwall’s
inequality to obtain the extinction of solutions when the exponents p– and p+ belong to different
intervals; as we know such results are seldom seen for the problems with variable
exponents. At the end of this paper, we prove that the solution approaches in L( ) norm as
t → ∞ by some techniques in convex analysis.
The outline of this paper is the following. In Section , we introduce the function spaces
of Orlicz-Sobolev type, give the definition of weak solution to the problem and prove the
existence of weak solutions with a method of regularization and the uniqueness of
solutions by arguing by contradiction. Section is devoted to the proof of the extinction of the
solution obtained in Section . In Section , we get the long time asymptotic behavior of
the solution.
2 Existence and uniqueness of weak solutions
u p(·) = inf λ > , Ap(·)(u/λ) ≤ ;
u Vt( ) = u , + ∇u p(.,t) ;
|u|p(x,t) dx dt < ∞ ,
u W(QT ) = u ,QT + ∇u p(x,t),QT
and denote by W (QT ) the dual of W(QT ) with respect to the inner product in L(QT ).
Definition . A function u(x, t) ∈ W(QT ) ∩ L∞(, T ; L∞( )) is called a weak solution of
problem (.) if for every test-function
where δ > ε > and α = σ+ .
Following the line of the proof of Theorem . in [, ], we have the following theorem
about the existence of weak solutions.
Theorem . Let the function f (x, t, u) and the exponents p(x, t), σ (x, t) satisfy Conditions
(.)-(.). If the following conditions hold:
≤ σ – < σ + < or ≤ σ – < σ + <
u ∞, +
b(x, t) ∞, dt = K (T ) < ∞,
The theorems about the uniqueness of weak solutions are as follows.
Theorem . Suppose that the conditions in Theorem . are fulfilled and the following
condition is satisfied:
≤ σ – < σ + < , < p– < p+ ≤ + √.
Then the nonnegative bounded solution of problem (.) is unique within the class of all
nonnegative bounded weak solutions.
Proof We argue by contradiction. Suppose that u(x, t) and v(x, t) are two nonnegative weak
solutions of problem (.) and there is a δ > such that for some < τ ≤ T , w = u – v > δ
on the set δ = ∩ {x : w(x, τ ) > δ} and μ( δ) > . Let
=
uσ (x,t) – vσ (x,t) |∇u|p(x,t)–∇u∇Fε(w) dx dt
uσ (x,t) – vσ (x,t) wα–|∇u|p(x,t)–∇u∇w dx dt
J =
J =
J =
with Qε,τ = Qτ ∩ {(x, t) ∈ Qτ : w > ε}.
. By virtue of the first inequality of
Lemma . in [], we get
vσ (x,t) + d wα––p(x,t)|∇w|p(x,t) dx dt
≥ –p+
vσ (x,t) + d wα–|∇w|p(x,t) dx dt ≥ .
According to the condition (H), we have
p+ – ≤
p(x, t) –
≥ –
and then applying Young’s inequality, we may estimate the integrand of J in the following
way:
uσ (x,t) – vσ (x,t) wα–|∇u|p(x,t)–∇u∇w
θ u + ( – θ )v σ (x,t)–dθ wα–|∇u|p(x,t)–∇u∇w
|∇w|p(x,t) + C σ ±, d, K (T ), p± |w|
Plugging the above estimates (.), (.), (.) and (.), (.), (.) into (.) and
dropping the nonnegative terms, we arrive at the inequality
(δ – ε) – α– εα–μ( δ) ≤ C,
(vσ (x,t) + d)
= p++w–α |∇w|p(x,t) + C σ ±, d, K (T ), p± |w|
(vσ (x,t) + d)
≤ p++w–α |∇w|p(x,t) + C σ , d, K (T ), p± |∇u|p(x,t).
(–σ +p)(+p+–) pp(x(,xt,)t–) +α–|∇u|p(x,t)
Substituting (.) into J, we get
|∇u|p(x,t) dx dt.
Secondly, we consider the case < p– < , < p+ ≤ + √. According to the second
inequality of Lemma . in [], it is easily seen that the following inequalities hold:
vσ (x,t) + d wα– |∇u| + |∇v| p(x,t)–|∇w| dx dt ≥ .
J =
≥ p– –
Using Young’s inequality, we may evaluate integrand of J as follows:
uσ (x,t) – vσ (x,t) wα–|∇u|p(x,t)–∇u∇w
θ u + ( – θ )v σ (x,t)–dθ wα–|∇u|p(x,t)–∇u∇w
|∇u| + |∇v| p(x,t)–|∇w|
+ C σ ±, d, K (T ), p± |w| –σ+ –+α |∇u| + |∇v| p(x,t)
Plugging (.) into J, we get
|∇u| + |∇v| p(x,t) dx dt.
Theorem . Suppose that the conditions in Theorem . are fulfilled and the following
condition is satisfied:
(H) < σ – < σ + < p+p–+ , p+ ≥ .
Then the nonnegative solution of problem (.) is unique within the class of all nonnegative
weak solutions.
3 Localization of weak solutions
In this section, we study the localization of the weak solution to problem (.). Namely,
we study the extinction of the solution. We discuss the extinction of weak solutions in the
case of NN+ ≤ p– < p+ < and < p– < NN+ , < p+ < NN–pp–– , respectively. Our main results
are the following.
Theorem . Suppose that b(x, t) ≡ , NN+ ≤ p– < p+ < , then any bounded nonnegative
solution of problem (.) vanishes in finite time for any nonnegative initial data ≡ u ∈
L∞( ) ∩ W ,p(x)( ) and satisfies the following estimate:
u ≤
u –p+ + Cb e p+– bt – Cb –p+
u = , t ∈ [T*, +∞),
Proof In Definition ., we choose u as a test-function to show
|∇u|p(x,t) dx dt + b
u dx dt = .
≥ C p+, p– min , u p––p+
u dx dt ≤ .
In (.), let t = t, t = t + h ( < h < T – t), multiply (.) by h and apply Lebesgue’s
dominated convergence theorem to show that as h → ,
u(x, t) dx + C
u(x, t) dx
By (.), (.), we have
≤ , a.e. t ∈ (, T ).
By Gronwall’s inequality, we have
u dx ≤
u –p+ + Cb e( p+– )bt – Cb –p+
Theorem . Suppose that b(x, t) ≡ , < p– < NN+ , < p+ < NN–pp–– , then any bounded
nonnegative solution of problem (.) vanishes in finite time for any nonnegative initial data
≡ u ∈ L∞( ) ∩ W ,p(x)( ) and satisfies the following estimate:
u rr ≤
u r = , t ∈ [T*, +∞),
), C, b are two positive
conProof In Definition ., we choose uα (α = N–p–(N+) > ) as a test-function to show
p–
t
t
ur dx dt ≤ ,
≥ C p+, p– min , u p––p+
A similar argument as above gives that there exists a T* > such that u r satisfies that
u rr ≤
u r = , t ∈ [T*, +∞).
Np–
+ C e( p+(NNp––p–) –)bt – C Np––p+(N–p–)
b b
Remark . In the case when < p– < NN+ , > p+ > NN–pp–– , it is not clear whether any
bounded nonnegative solution of problem (.) vanishes in finite time.
4 Asymptotic behavior of weak solutions
(H) there exists a positive continuous function g(t) such that the following inequality holds:
∇u(x, t) p(x,t) dx ≤ M(u)g(t) for t > ,
Proof Step . Let
t →lim+∞ u(x, t) L( ) = .
F(u) =
F u(τ + h) – F u(τ ) ≥
Multiplying both sides of (.) by h , and letting h → +, we obtain
F u(τ ) – F u(τ – h) ≤
F u(t) – F u(t) ≥
Similarly, we have
F u(t) – F u(t) ≤
F u(t) – F u(t) =
F u(t) – F u() = –
|u|σ + d |∇u|p(x) dx dτ – b
⎧
⎪⎨⎪⎪ uε ∈ W (QT ) ⊆ L∞(p+, T ; W,p– ( )),
uεt ∈ W (QT ) ⊆ L p+– (, T ; V+( )),
⎪⎪⎪⎩ W,p– ( ) com→pact L( ) → V+( ).
|u| dx ≤ C u ,p(x) ≤ C ∇u p(x)
≤ C | |, p± max
|∇u|p(x) dx
≤ C | |, p±, M(u), d
≤ C | |, p±, M(u), d
|∇u|p(x) dx
|∇u|p(x) dx
, and hence
Ci > , i = , .
This completes the proof of Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors collaborated in all the steps concerning the research and achievements presented in the final manuscript.
Acknowledgements
The work was supported by the Natural Science Foundation of China (11271154) and by the 985 program of Jilin
University. We are very grateful to the anonymous referees for their valuable suggestions that improved the article.
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