Open Access

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

Boundary Value Problems20132013:164

DOI: 10.1186/1687-2770-2013-164

Received: 11 March 2013

Accepted: 26 June 2013

Published: 10 July 2013

Abstract

The aim of this paper is to study the existence and extinction of weak solutions of the initial and boundary value problem for u t = div ( ( | u | σ ( x , t ) + d 0 ) | 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 L 2 ( Ω ) norm as t .

Keywords

nonlinear parabolic equation nonstandard growth condition extinction p ( x , t ) -Laplace operator

1 Introduction

Let Ω R N ( N 1 ) be a bounded simply connected domain and 0 < T < . Consider the following quasilinear degenerate parabolic problem:
{ u t = div ( a ( x , t , u ) | u | p ( x , t ) 2 u ) + f ( x , t , u ) , ( x , t ) Q T , u ( x , t ) = 0 , ( x , t ) Γ T , u ( x , 0 ) = u 0 ( x ) , x Ω ,
(1.1)
where Q T = Ω × ( 0 , T ] , Γ T denotes the lateral boundary of the cylinder Q T , a ( x , t , u ) = | u | σ ( x , t ) + d 0 with the assumption that d 0 is a positive constant and the nonlinear source f ( x , t , u ) satisfies
f ( x , t , u ) = b ( x , t ) b 0 u ( x , t ) , u ( , + ) , x Ω , t > 0
(1.2)
with b 0 > 0 , b ( x , t ) 0 , ( x , t ) Q T . It will be assumed throughout the paper that the exponents p ( x , t ) , σ ( x , t ) are continuous in Q = Q T ¯ with the logarithmic module of continuity:
1 < p = inf ( x , t ) Q p ( x , t ) p ( x , t ) p + = sup ( x , t ) Q p ( x , t ) < ,
(1.3)
0 < σ = inf ( x , t ) Q σ ( x , t ) σ ( x , t ) σ + = sup ( x , t ) Q σ ( x , t ) < ,
(1.4)
z = ( x , t ) Q T , ξ = ( y , s ) Q T , | z ξ | < 1 , | p ( z ) p ( ξ ) | ω ( | z ξ | ) ,
(1.5)
where
lim sup τ 0 + ω ( τ ) ln 1 τ = C < + .

Model (1.1) may describe some properties of image restoration in space and time. Especially when the nonlinear source f ( x , u ) = b ( x , t ) b 0 u , 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 [13]. 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 [49].

To the best of our knowledge, there are only a few works about parabolic equations with variable exponents of nonlinearity. In [6], Chen, Levine and Rao obtained the existence and uniqueness of weak solutions with the assumption that the exponent σ ( x , t ) 0 , 1 < p < p + < 2 . In [10], we applied the method of parabolic regularization and Galerkin’s method to prove the existence of weak solutions to problem (1.1) with the assumption that σ ( x , t ) constant , f ( x , t , u ) f ( x , t ) . In this paper, we generalize the results in [10]. Especially, unlike [10], we obtain the existence and uniqueness of weak solution not only in the case when σ ( x , t ) ( 2 , 2 p + p + 1 ) , p + 2 , but also in the case when σ ( x , t ) ( 1 , 2 ) , 1 < p < p + 1 + 2 . 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 0 in L 2 ( Ω ) norm as t by some techniques in convex analysis.

The outline of this paper is the following. In Section 2, 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 3 is devoted to the proof of the extinction of the solution obtained in Section 2. In Section 4, we get the long time asymptotic behavior of the solution.

2 Existence and uniqueness of weak solutions

We study the existence of weak solutions in this section. Let us introduce the Banach spaces
L p ( x , t ) ( Q T ) = { u ( x , t ) | u  is measurable in  Q T , A p ( ) ( u ) = Q T | u | p ( x , t ) d x d t < } , u p ( ) = inf { λ > 0 , A p ( ) ( u / λ ) 1 } ; V t ( Ω ) = { u | u L 2 ( Ω ) W 0 1 , 1 ( Ω ) , | u | L p ( x , t ) ( Ω ) } , u V t ( Ω ) = u 2 , Ω + u p ( . , t ) Ω ; W ( Q T ) = { u : [ 0 , T ] V t ( Ω ) | u L 2 ( Q T ) , | u | L p ( x , t ) ( Q T ) , u = 0  on  Γ T } , u W ( Q T ) = u 2 , Q T + u p ( x , t ) , Q T

and denote by W ( Q T ) the dual of W ( Q T ) with respect to the inner product in L 2 ( Q T ) .

Definition 2.1 A function u ( x , t ) W ( Q T ) L ( 0 , T ; L ( Ω ) ) is called a weak solution of problem (1.1) if for every test-function
ξ Z { η ( z ) : η W ( Q T ) L ( 0 , T ; L 2 ( Ω ) ) , η t W ( Q T ) } ,
and every t 1 , t 2 [ 0 , T ] , the following identity holds:
t 1 t 2 Ω [ u ξ t ( | u | σ ( x , t ) + d 0 ) | u | p ( x , t ) 2 u ξ + f ( x , t , u ) ξ ] d x d t = Ω u ξ d x | t 1 t 2 .
(2.1)

Following the line of the proof of Theorem 2.1 in [10, 11], we have the following theorem about the existence of weak solutions.

Theorem 2.1 Let the function f ( x , t , u ) and the exponents p ( x , t ) , σ ( x , t ) satisfy Conditions (1.2)-(1.5). If the following conditions hold:
(H ) 1 max { 1 , 2 N N + 2 } < p , (H ) 1 σ + ( p + 1 ) p + σ < σ + < 2 or 2 σ < σ + < 2 p + p + 1 ; (H ) 2 u 0 , Ω + 0 T b ( x , t ) , Ω d t = K ( T ) < ,

then problem (1.1) has at least one weak solution u satisfying u , Q T u 0 , Ω .

The theorems about the uniqueness of weak solutions are as follows.

Theorem 2.2 Suppose that the conditions in Theorem 2.1 are fulfilled and the following condition is satisfied:
(H ) 3 σ + ( p + 1 ) p + σ < σ + < 2 , 1 < p < p + 1 + 2 .

Then the nonnegative bounded solution of problem (1.1) 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 (1.1) and there is a δ > 0 such that for some 0 < τ T , w = u v > δ on the set Ω δ = Ω { x : w ( x , τ ) > δ } and μ ( Ω δ ) > 0 . Let
F ε ( ξ ) = { 1 1 α ε α 1 1 1 α ξ α 1 if  ξ > ε , 0 if  ξ ε ,

where δ > 2 ε > 0 and α = σ + 2 .

By the definition of weak solutions, pick a test-function ξ = F ε ( w ) Z ,
0 = Q τ [ w t F ε ( w ) + ( v σ ( x , t ) + d 0 ) ( | u | p ( x , t ) 2 u | v | p ( x , t ) 2 v ) F ε ( w ) ] d x d t + Q τ ( u σ ( x , t ) v σ ( x , t ) ) | u | p ( x , t ) 2 u F ε ( w ) d x d t Q τ [ f ( x , t , u ) f ( x , t , v ) ] F ε ( w ) d x d t = Q ε , τ w t F ε ( w ) d x d t + Q ε , τ ( v σ ( x , t ) + d 0 ) w α 2 ( | u | p ( x , t ) 2 u | v | p ( x , t ) 2 v ) w d x d t + Q ε , τ ( u σ ( x , t ) v σ ( x , t ) ) w α 2 | u | p ( x , t ) 2 u w d x d t + Q ε , τ w F ε ( w ) d x d t : = J 1 + J 2 + J 3 + J 4 ,
(2.2)

with Q ε , τ = Q τ { ( x , t ) Q τ : w > ε } .

Now, let t 0 = inf { t ( 0 , τ ] : w > ε } , then we estimate J 1 , J 2 , J 3 , J 4 as follows:
J 1 = Q ε , τ w t F ε ( w ) d x d t = Ω ( 0 t 0 w t F ε ( w ) d t + t 0 τ w t F ε ( w ) d t ) d x J 1 Ω ε w ( x , τ ) F ε ( s ) d s d x Ω ε ε w ( x , τ ) F ε ( s ) d s d x J 1 Ω δ ( w 2 ε ) F ε ( 2 ε ) d x ( δ 2 ε ) F ε ( 2 ε ) μ ( Ω δ ) ; J 4 = Q ε , τ w F ε ( w ) d x d t 0 .
(2.3)
Let us consider first the case 2 p < p + 1 + 2 . By virtue of the first inequality of Lemma 4.4 in [2], we get
J 2 = Q ε , τ ( v σ ( x , t ) + d 0 ) w α 2 ( | u | p ( x , t ) 2 u | v | p ( x , t ) 2 v ) w d x d t Q ε , τ ( v σ ( x , t ) + d 0 ) w α 2 2 p ( x , t ) | w | p ( x , t ) d x d t 2 p + Q ε , τ ( v σ ( x , t ) + d 0 ) w α 2 | w | p ( x , t ) d x d t 0 .
(2.4)
According to the condition (H3), we have
2 p + 4 p + 1 σ + ( p + 1 ) p + < σ + 0 < ( 4 σ + ) ( p + 1 ) 2 p + < 1 ( 4 σ + ) ( p + 1 ) 2 p + p ( x , t ) p ( x , t ) 1 2 σ + 2 = 2 α ,
and then applying Young’s inequality, we may estimate the integrand of J 3 in the following way:
| ( u σ ( x , t ) v σ ( x , t ) ) w α 2 | u | p ( x , t ) 2 u w | = | σ ( x , t ) w 0 1 ( θ u + ( 1 θ ) v ) σ ( x , t ) 1 d θ w α 2 | u | p ( x , t ) 2 u w | C w 2 α [ v σ ( x , t ) + d 0 C | w | p ( x , t ) + C 1 ( σ ± , d 0 , K ( T ) , p ± ) | w | ( 4 σ + ) ( p + 1 ) 2 p + p ( x , t ) p ( x , t ) 1 | u | p ( x , t ) ] = ( v σ ( x , t ) + d 0 ) 2 p + + 1 w 2 α | w | p ( x , t ) + C 1 ( σ ± , d 0 , K ( T ) , p ± ) | w | ( 4 σ + ) ( p + 1 ) 2 p + p ( x , t ) p ( x , t ) 1 + α 2 | u | p ( x , t ) ( v σ ( x , t ) + d 0 ) 2 p + + 1 w 2 α | w | p ( x , t ) + C 1 ( σ , d 0 , K ( T ) , p ± ) | u | p ( x , t ) .
(2.5)
Substituting (2.5) into J 3 , we get
J 3 1 2 J 2 + C Q ε , τ | u | p ( x , t ) d x d t .
(2.6)
Secondly, we consider the case 1 < p < 2 , 1 < p + 1 + 2 . According to the second inequality of Lemma 4.4 in [2], it is easily seen that the following inequalities hold:
J 2 = Q ε , τ ( v σ ( x , t ) + d 0 ) w α 2 ( | u | p ( x , t ) 2 u | v | p ( x , t ) 2 v ) w d x d t ( p 1 ) Q ε , τ ( v σ ( x , t ) + d 0 ) w α 2 ( | u | + | v | ) p ( x , t ) 2 | w | 2 d x d t 0 .
(2.7)
Using Young’s inequality, we may evaluate integrand of J 3 as follows:
| ( u σ ( x , t ) v σ ( x , t ) ) w α 2 | u | p ( x , t ) 2 u w | = | σ ( x , t ) w 0 1 ( θ u + ( 1 θ ) v ) σ ( x , t ) 1 d θ w α 2 | u | p ( x , t ) 2 u w | ( v σ ( x , t ) + d 0 ) ( p 1 ) 2 w 2 α ( | u | + | v | ) p ( x , t ) 2 | w | 2 + C 1 ( σ ± , d 0 , K ( T ) , p ± ) | w | 4 σ + 2 2 + α ( | u | + | v | ) p ( x , t ) ( v σ + d 0 ) ( p 1 ) 2 w 2 α ( | u | + | v | ) p ( x , t ) 2 | w | 2 + C 1 ( | u | + | v | ) p ( x , t ) .
(2.8)
Plugging (2.8) into J 3 , we get
J 3 1 2 J 2 + C Q ε , τ ( | u | + | v | ) p ( x , t ) d x d t .
(2.9)
Plugging the above estimates (2.3), (2.4), (2.6) and (2.3), (2.7), (2.9) into (2.2) and dropping the nonnegative terms, we arrive at the inequality
( δ 2 ε ) ( 1 2 α 1 ) ε α 1 μ ( Ω δ ) C ˜ ,
(2.10)

with a constant C ˜ independent of ε.

Noticing that lim ε 0 ( δ 2 ε ) ( 1 2 α 1 ) ε α 1 μ ( Ω δ ) = + , we obtain a contradiction. This means μ ( Ω δ ) = 0 and w 0 , a.e. in Q τ . □

In the case when σ > 2 , following the lines of the proof of Theorem 2.2, we have the following theorem.

Theorem 2.3 Suppose that the conditions in Theorem 2.1 are fulfilled and the following condition is satisfied:
(H ) 4 2 < σ < σ + < 2 p + p + 1 , p + 2 .

Then the nonnegative solution of problem (1.1) 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 (1.1). Namely, we study the extinction of the solution. We discuss the extinction of weak solutions in the case of 2 N N + 2 p < p + < 2 and 1 < p < 2 N N + 2 , 1 < p + < N p N p , respectively. Our main results are the following.

Theorem 3.1 Suppose that b ( x , t ) 0 , 2 N N + 2 p < p + < 2 , then any bounded nonnegative solution of problem (1.1) vanishes in finite time for any nonnegative initial data 0 u 0 L ( Ω ) W 1 , p ( x ) ( Ω ) and satisfies the following estimate:
u 2 2 [ u 0 2 2 p + 2 + C 1 b 0 e p + 2 2 b 0 t C 1 b 0 ] 2 2 p + , 0 < t < T 1 ; u 2 = 0 , t [ T 1 , + ) ,

where T 1 = 2 b 0 ( 2 p + ) ln ( 1 + b 0 C 1 u 0 2 2 p + ) , C 1 , b 0 are two positive constants.

Proof In Definition 2.1, we choose u as a test-function to show
1 2 Ω u 2 d x | t 1 t 2 + t 1 t 2 Ω | u | p ( x , t ) d x d t + b 0 t 1 t 2 Ω u 2 d x d t = 0 .
(3.1)
Applying the conditions t 1 t 2 Ω | u | p ( x , t ) d x d t 1 2 u 0 2 2 , the inequalities min { u p ( ) p , u p ( ) p + } t 1 t 2 Ω | u | p ( x , t ) d x d t max { u p ( ) p , u p ( ) p + } and the imbedding theorem W 0 1 , p ( x , t ) ( Ω ) W 0 1 , p ( Ω ) L 2 ( Ω ) , we have
t 1 t 2 Ω | u | p ( x , t ) d x d t min { u p ( ) p , u p ( ) p + } C ( p + , p ) min { 1 , u 0 2 p p + } u p ( ) p + C ( p + , p , u 0 2 ) u p p + C ( p + , p , u 0 2 ) u 2 p + .
(3.2)
By (3.1), (3.2), we have
1 2 Ω u 2 d x | t 1 t 2 + t 1 t 2 ( Ω | u | 2 d x ) p + 2 d t + b 0 t 1 t 2 Ω u 2 d x d t 0 .
(3.3)
In (3.3), let t 1 = t , t 2 = t + h ( 0 < h < T t ), multiply (3.3) by h and apply Lebesgue’s dominated convergence theorem to show that as h 0 ,
d d t ( Ω u 2 ( x , t ) d x ) + 2 C 1 ( Ω | u ( x , t ) | 2 d x ) p + 2 + 2 b 0 Ω u 2 ( x , t ) d x 0 , a.e.  t ( 0 , T ) .
(3.4)
By Gronwall’s inequality, we have
Ω u 2 d x [ ( u 0 2 2 p + + b 0 C 1 ) e ( p + 2 2 ) b 0 t b 0 C 1 ] 2 2 p + .

 □

Theorem 3.2 Suppose that b ( x , t ) 0 , 1 < p < 2 N N + 2 , 1 < p + < N p N p , then any bounded nonnegative solution of problem (1.1) vanishes in finite time for any nonnegative initial data 0 u 0 L ( Ω ) W 1 , p ( x ) ( Ω ) and satisfies the following estimate:
u r r [ ( u 0 r r r p + ( N p ) N p + C 2 b 0 ) e ( p + ( N p ) N p 1 ) b 0 t C 2 b 0 ] N p N p p + ( N p ) , 0 < t < T 2 ; u r = 0 , t [ T 2 , + ) ,

where r = N ( 2 p ) p , T 2 = N p b 0 ( N p p + ( N p ) ) ln ( 1 + b 0 C 2 u 0 r r r p + ( N p ) N p ) , C 2 , b 0 are two positive constants.

Proof In Definition 2.1, we choose u α ( α = 2 N p ( N + 1 ) p > 1 ) as a test-function to show
1 r Ω u r d x | t 1 t 2 + C ( α , u 0 , N , p ± ) t 1 t 2 Ω | u β | p ( x , t ) d x d t + b 0 t 1 t 2 Ω u r d x d t 0 ,
(3.5)
where r = N ( 2 p ) p > 2 , β = ( 2 p ) ( N p ) p p > 1 . The conditions t 1 t 2 Ω | u | p ( x , t ) d x d t 1 2 u 0 2 2 , u , Q T u 0 , Ω , the inequalities min { u p ( ) p , u p ( ) p + } t 1 t 2 Ω | u | p ( x , t ) d x d t and the imbedding theorem W 0 1 , p ( x , t ) ( Ω ) W 0 1 , p ( Ω ) L N p N p ( Ω ) show that the following inequalities hold:
t 1 t 2 Ω | u β | p ( x , t ) d x d t min { u p ( ) p , u β p ( ) p + } C ( p + , p ) min { 1 , u 0 2 p p + } u p ( ) p + C ( p + , p , u 0 2 ) u β p p + C ( p + , p , u 0 2 ) u β N p N p p + .
(3.6)
A similar argument as above gives that there exists a T 2 > 0 such that u r satisfies that
u r r [ ( u 0 r r r p + ( N p ) N p + C 2 b 0 ) e ( p + ( N p ) N p 1 ) b 0 t C 2 b 0 ] N p N p p + ( N p ) , 0 < t < T 2 ; u r = 0 , t [ T 2 , + ) .

 □

Remark 3.1 In the case when 1 < p < 2 N N + 2 , 2 > p + > N p N p , it is not clear whether any bounded nonnegative solution of problem (1.1) vanishes in finite time.

4 Asymptotic behavior of weak solutions

In this section, we study the asymptotic properties of the weak solution to problem (1.1). Namely, we study the long time asymptotic behavior of the solution, our main result is as follows.

Theorem 4.1 Suppose that b ( x , t ) 0 , 2 p < p + . If the following condition is satisfied

(H4) there exists a positive continuous function g ( t ) such that the following inequality holds:
Ω | u ( x , t ) | p ( x , t ) d x M ( u 0 ) g ( t ) for t > 0 ,

where g ( t ) satisfies 0 + g 2 ( p p + ) p + p ( t ) d t = + .

Then, for all 0 < σ , the solution to problem (1.1) satisfies
lim t + u ( x , t ) L 2 ( Ω ) = 0 .
Proof Step 1. Let
F ( u ) = 1 2 Ω | u | 2 d x ,

then it is easy to prove that F ( u ) is a convex functional on L 2 ( Ω ) .

For any τ ( 0 , T ) and h > 0 , by δ F ( u ) δ u = u (δ representing Gâteaux differential) and the convexity of F ( u ) , we have
F ( u ( τ + h ) ) F ( u ( τ ) ) Ω ( u ( x , τ + h ) u ( x , τ ) ) u ( x , τ ) d x
(4.1)
for any fixed t 1 , t 2 [ 0 , T ] , t 1 < t 2 . Integrating inequality (4.1) with respect to τ over ( t 1 , t 2 ) , we have
t 1 t 2 F ( u ( τ + h ) ) d τ t 1 t 2 F ( u ( τ ) ) d τ = t 1 + h t 2 + h F ( u ( τ ) ) d τ t 1 t 2 F ( u ( τ ) ) d τ = t 2 t 2 + h F ( u ( τ ) ) d τ t 1 t 1 + h F ( u ( τ ) ) d τ t 1 t 2 Ω ( u ( x , τ + h ) u ( x , τ ) ) u ( x , τ ) d x d τ .
(4.2)
Multiplying both sides of (4.2) by 1 h , and letting h 0 + , we obtain
F ( u ( t 2 ) ) F ( u ( t 1 ) ) t 1 t 2 Ω u t u d x d τ .
Similarly, we have
F ( u ( τ ) ) F ( u ( τ h ) ) Ω ( u ( x , τ ) u ( x , τ h ) ) u ( x , τ ) d x .
Thus,
F ( u ( t 2 ) ) F ( u ( t 1 ) ) t 1 t 2 Ω u t u d x d τ
and hence
F ( u ( t 2 ) ) F ( u ( t 1 ) ) = t 1 t 2 Ω u t u d x d τ .
Choosing t 1 = 0 , t 1 = t , we get from the definition of solutions that
F ( u ( t ) ) F ( u ( 0 ) ) = 0 t Ω ( | u | σ + d 0 ) | u | p ( x ) d x d τ b 0 0 t Ω | u | 2 d x d τ .
(4.3)
Step 2. We prove u ( x , t ) C ( 0 , T ; L 2 ( Ω ) ) . According to Theorem 2.1 and (H4), we have the following conclusions:
{ u ε W ( Q T ) L ( 0 , T ; W 0 1 , p ( Ω ) ) , u ε t W ( Q T ) L p + p + 1 ( 0 , T ; V + ( Ω ) ) , W 0 1 , p ( Ω ) compact L 2 ( Ω ) V + ( Ω ) .

Applying Theorem 5 in [12], we get that the set { u ε ( x , t ) } is relatively compact in C ( 0 , T ; L 2 ( Ω ) ) , so u ( x , t ) C ( 0 , T ; L 2 ( Ω ) ) .

Step 3. Let G ( t ) = 1 2 Ω | u ( x , t ) | 2 d x . Noting that u ( x , t ) C ( 0 , T ; L 2 ( Ω ) ) , then it is easy to prove that G ( t ) is continuous in ( 0 , T ) , so we have G ( t ) = Ω ( | u | σ + d 0 ) | u | p ( x ) d x b 0 Ω | u | 2 d x d τ 0 .

By the imbedding W 1 , p ( x ) ( Ω ) W 1 , p ( Ω ) L 2 ( Ω ) , Lemmas (2.1)-(2.3) in [5] and (H4), we have
Ω | u | 2 d x C u 1 , p ( x ) 2 C u p ( x ) 2 C ( | Ω | , p ± ) max { ( Ω | u | p ( x ) d x ) 2 p , ( Ω | u | p ( x ) d x ) 2 p + } C ( | Ω | , p ± , M ( u 0 ) , d 0 ) max { g 2 ( p + p ) p + p ( t ) , 1 } ( Ω | u | p ( x ) d x ) 2 p + C ( | Ω | , p ± , M ( u 0 ) , d 0 ) max { g 2 ( p + p ) p + p ( t ) , 1 } ( Ω | u | σ + d 0 d 0 | u | p ( x ) d x ) 2 p + C ( | Ω | , p ± , M ( u 0 ) , d 0 , α ) max { g 2 ( p + p ) p + p ( t ) , 1 } | G ( t ) | 2 p + ,

that is, G ( t ) C max { g 2 ( p + p ) p + p ( t ) , 1 } | G ( t ) | 2 p + .

Noting that G ( t ) 0 , G ( t ) 0 , we have G ( t ) C G p + 2 ( t ) 1 + g 2 ( p + p ) p + p ( t ) , and hence
Ω | u ( x , T ) | 2 d x 1 [ C 1 0 T ( 1 + g 2 ( p + p ) p + p ( t ) ) 1 d t + C 2 ] δ , δ = 2 p + 2 , C i > 0 , i = 1 , 2 .

This completes the proof of Theorem 4.1. □

Declarations

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.

Authors’ Affiliations

(1)
School of Mathematics, Jilin University

References

  1. Wei DM: Existence, uniqueness and numerical analysis of solutions of a quasilinear parabolic problem. SIAM J. Numer. Anal. 1992, 29(2):484–497. 10.1137/0729029MathSciNetView ArticleGoogle Scholar
  2. Dibenedetto E: Degenerate Parabolic Equations. Springer, New York; 1993.View ArticleGoogle Scholar
  3. Zhao JN: On the Cauchy problem and initial traces for the evolution p -Laplacian equations with strongly nonlinear sources. J. Differ. Equ. 1995, 121: 329–383. 10.1006/jdeq.1995.1132View ArticleGoogle Scholar
  4. Ruzicka M Lecture Notes in Math. 1748. In Electrorheological Fluids: Modelling and Mathematical Theory. Springer, Berlin; 2000.View ArticleGoogle Scholar
  5. Antontsev SN, Shmarev SI: Anisotropic parabolic equations with variable nonlinearity. Publ. Math. 2009, 53: 355–399.MathSciNetView ArticleGoogle Scholar
  6. Chen Y, Levine S, Rao M: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 2006, 66: 1383–1406. 10.1137/050624522MathSciNetView ArticleGoogle Scholar
  7. Acerbi E, Mingione G, Seregin GA: Regularity results for parabolic systems related to a class of non newtonian fluids. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2004, 21: 25–60.MathSciNetGoogle Scholar
  8. Acerbi E, Mingione G: Regularity results for a class of functionals with nonstandard growth. Arch. Ration. Mech. Anal. 2001, 156(1):121–140.MathSciNetView ArticleGoogle Scholar
  9. Diening L, Harjulehto P, Hästö P, Rûžička M Lecture Notes in Mathematics 2017. In Lebesgue and Sobolev Spaces with Variable Exponents. Springer, Heidelberg; 2011.View ArticleGoogle Scholar
  10. Guo B, Gao WJ: Study of weak solutions for parabolic equations with nonstandard growth conditions. J. Math. Anal. Appl. 2011, 374(2):374–384. 10.1016/j.jmaa.2010.09.039MathSciNetView ArticleGoogle Scholar
  11. Guo B, Gao WJ: Existence and asymptotic behavior of solutions for nonlinear parabolic equations with variable exponent of nonlinearity. Acta Math. Sci. 2012, 32(3):1053–1062.MathSciNetView ArticleGoogle Scholar
  12. Simon J:Compact sets in the space L p ( 0 , T ; B ) . Ann. Math. Pures Appl. 1987, 4(146):65–96.Google Scholar

Copyright

© Gao and Gao; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.