Kaixuan Zhu and Yongqin Xie

Global attractors for a class of semilinear degenerate parabolic equations

Open Access
De Gruyter | Published online: May 4, 2021

Abstract

In this paper, we consider the long-time behavior for a class of semi-linear degenerate parabolic equations with the nonlinearity f satisfying the polynomial growth of arbitrary p 1 order. We establish some new estimates, i.e., asymptotic higher-order integrability for the difference of the solutions near the initial time. As an application, we obtain the ( L 2 ( Ω ) , L p ( Ω ) ) -global attractors immediately; moreover, such an attractor can attract every bounded subset of L 2 ( Ω ) with the L p + δ -norm for any δ [ 0 , + ) .

MSC 2010: 35B40; 35B41; 35K65

1 Introduction

Let Ω R N ( N 3 ) be a bounded domain with smooth boundary, we consider the long-time behavior for the solutions of the following semi-linear degenerate parabolic equation:

(1.1) t u ( x , t ) = Δ λ u ( x , t ) + f ( u ( x , t ) ) + g ( x ) in Ω × ( 0 , ) , u ( x , t ) = 0 on Ω × [ 0 , ) , u ( x , 0 ) = u 0 , x Ω ,
where Δ λ is the degenerate elliptic operator, which will be characterized in Section 2. The external forcing g ( x ) = i = 1 N D i g i + g 0 ( x ) H 1 ( Ω ) , the nonlinearity f C 1 ( R , R ) and satisfies the following classical assumptions (e.g., see [ 1, 2, 3]):
(1.2) f ( u ) l ,
and
(1.3) c 0 c 1 u p f ( u ) u c 0 c 2 u p , p 2
for some positive constants c 0 , c 1 , c 2 and u R .

Let ( u ) = 0 u f ( r ) d r , then there exist constants c ˜ i > 0 ( i = 0 , 1 , 2 ) such that

(1.4) c ˜ 0 c ˜ 1 u p ( u ) c ˜ 0 c ˜ 2 u p , u R .

The long-time behavior for the solutions of semi-linear degenerate parabolic equations has been considered by many researchers, e.g., see [4,5,6] and references therein. For the subcritical growth case, the authors have established the existence of the global attractors in S 0 1 ( Ω ) (see [4] for details) for equation (1.1) involving the Grushin operator. In [5], the authors extended the result for the Δ λ -Laplacian. They considered Δ λ as a self-adjoint operator and showed that Δ λ generates an analytic semigroup in L 2 ( Ω ) . They not only proved the existence of the global attractors in W ˚ λ 1 , 2 ( Ω ) together with its fractal dimension but also showed the convergence of solutions to an equilibrium solution as the time t . In the following, the authors of [6] have considered the case of critical growth nonlinearity and obtained the global attractors in W ˚ λ 1 , 2 ( Ω ) by applying the decomposition technique introduced in [7].

In this paper, similar to the reaction–diffusion equations in [8,9], we consider equation (1.1) with the nonlinearity f satisfying the polynomial growth of arbitrary p 1 order, i.e., f ( u ) u p 2 u ( p 2 ) . For our problem, we will confront two main difficulties when we establish the asymptotic higher-order integrability of solutions and the existence of the ( L 2 ( Ω ) , L p ( Ω ) ) -global attractors. One difficulty is that the external force term g ( x ) = i = 1 N D i g i + g 0 ( x ) with g i ( x ) ( i = 0 , 1 , , N ) belongs only to L 2 ( Ω ) , which leads to the fact that the solutions of equation (1.1) are only bounded in W ˚ λ 1 , 2 ( Ω ) L p ( Ω ) and do not have any higher regularity than the order max p , 2 Q Q 2 (where Q is the homogeneous dimension of R N with a group of dilations corresponding to the Δ λ -operator, see [5] for details). The other difficulty is that the Sobolev embedding theorem is not any longer valid since the growth exponent is arbitrary p 1 ( p 2 ) order.

Based on the aforementioned difficulties and motivated by the idea of [10,11, 12,13], we first decompose equation (1.1) as a stationary equation and an evolutionary equation, then establish some asymptotic higher-order integrability results about the difference of the solutions near the initial time by using the bootstrap method (see Theorems 4.1 and 4.3). As an application, we obtain the ( L 2 ( Ω ) , L p ( Ω ) ) -global attractors immediately (see Corollary 4.4). Moreover, the ( L 2 ( Ω ) , L p ( Ω ) ) -global attractors indeed can attract every bounded subset of L 2 ( Ω ) with the L p + δ -norm for any δ [ 0 , + ) . Finally, we also obtain the ( L 2 ( Ω ) , W ˚ λ 1 , 2 ( Ω ) ) -global attractors (see Theorem 5.3).

2 Preliminaries

2.1 The Δ λ -operator

As in [5,14, 15,16,17], we consider the degenerate elliptic operator of the form

(2.1) Δ λ i = 1 N x i ( λ i 2 x i ) , λ = ( λ 1 , λ 2 , , λ N ) : R N R N ,
where x i = x i , i = 1 , 2 , , N . The functions λ i : R N R are continuous, strictly positive and of C 1 outside the coordinate hyperplane [1] and satisfy the following properties (see [ 5, 17] for details):
  1. (1)

    λ 1 ( x ) 1 , λ i ( x ) = λ i ( x 1 , x 2 , , x i 1 ) , i = 2 , 3 , , N .

  2. (2)

    For every x R N , the function λ i ( x ) = λ i ( x ) , i = 1 , 2 , , N , where

    x = ( x 1 , x 2 , , x N ) if x = ( x 1 , x 2 , , x N ) .

  3. (3)

    There exists a constant ρ 0 such that

    0 x k x k λ i ( x ) ρ λ i ( x ) , k { 1 , 2 , , i 1 } , i = 2 , 3 , , N ,
    and for every x R + N { ( x 1 , x 2 , , x N ) R N x i 0 , i = 2 , 3 , , N } .

  4. (4)

    There exists a group of dilations ( δ r ) r > 0 which satisfy

    δ r : R N R N , δ r ( x ) = δ r ( x 1 , x 2 , , x N ) = ( r ε 1 x 1 , r ε 2 x 2 , , r ε N x N ) ,
    where 1 ε 1 ε 2 ε N , such that λ i is δ r -homogeneous of degree ε i 1 , i.e.,
    λ i ( δ r ( x ) ) = r ε i 1 λ i ( x ) , x R N , r > 0 , i = 1 , 2 , , N .

This implies that the operator Δ λ is δ r -homogeneous of degree two, i.e.,

Δ λ ( u ( δ r ( x ) ) ) = r 2 ( Δ λ u ) ( δ r ( x ) ) , u C ( R N ) .

We will denote by Q the homogeneous dimension of R N with respect to the group of dilations ( δ r ) r > 0 , that is,

Q = ε 1 + ε 2 + + ε N .

We have another expression of operator Δ λ from the property (1) as (e.g., see [5])

Δ λ = i = 1 N ( λ i x i ) 2 .

2.2 Functional settings

For a function u C 1 , we define

λ u = ( λ 1 x 1 u , λ 2 x 2 u , , λ N x N u ) , λ u 2 i = 1 N λ i x i u 2 , Δ λ u = ( λ 1 2 x 1 2 2 u , λ 2 2 x 2 2 2 u , , λ N 2 x N 2 2 u ) , Δ λ u 2 i = 1 N λ i 2 x i 2 2 u 2 ,
where λ i ( i = 1 , 2 , , N ) comes from ( 2.1). For any p ( 1 , ) , we define that W ˚ λ 1 , p ( Ω ) is the closure of C 0 1 ( Ω ) with respect to the norm
u W ˚ λ 1 , p ( Ω ) Ω λ u p d x 1 p ,
and W λ 2 , p ( Ω ) with respect to the norm
u W λ 2 , p ( Ω ) Ω λ u p d x + Ω Δ λ u p d x 1 p .

Similar to that in [5,17], we have the following embedding theorem.

Lemma 2.1

For any p ( 1 , Q ) , q [ 1 , p λ ] , the embedding

W ˚ λ 1 , p ( Ω ) L q ( Ω )
is continuous for q [ 1 , p λ ] and compact for q [ 1 , p λ ) , where p λ p Q Q p .

In particular, for any u C 0 1 ( Ω ) , there exists a constant C such that

(2.2) u L 2 ( Ω ) C u W ˚ λ 1 , 2 ( Ω ) , u C 0 1 ( Ω ) ,
and the optimal constant of C is 1 μ 1 , where μ 1 > 0 denotes the first eigenvalue of the operator Δ λ on Ω with homogeneous Dirichlet boundary conditions.

For later application, we recall the following results (see [12] for details).

Lemma 2.2

For any ϕ W ˚ λ 1 , 2 ( Ω ) L ( Ω ) and any r > 0 , the following equality holds:

(2.3) Ω λ ϕ λ ( ϕ r ϕ ) d x = ( r + 1 ) 2 r + 2 2 Ω λ ϕ r + 2 2 2 d x ,
where stands for the usual inner product in R N .

Lemma 2.3

Let X , Y , Z be three Banach spaces satisfying Z Y X with continuous embedding and suppose that there exist θ 0 ( 0 , 1 ] and a constant C 0 such that Y C 0 X θ 0 Z 1 θ 0 . Then, for any bounded sets A , B Z , we have

dist Y ( A , B ) C 0 ( A Z + B Z + 1 ) dist X θ 0 ( A , B ) ,
where A Z = sup x A x Z , B Z = sup x B x Z for any A , B Z .

Similar to Theorem 3.2 in [12], we have the following theorem.

Theorem 2.4

Let X , Y , Z be three Banach spaces satisfying Z Y X with continuous embedding, S ( ) be a semigroup defined on X . Moreover, we assume that

  1. (a)

    S ( ) has a global attractor A in X ;

  2. (b)

    v ( x ) is the solution of the stationary equation Δ λ v + f ( v ) + g ( x ) = 0 for (1.1);

  3. (c)

    there exist a family of operators S 1 ( ) defined on X satisfying

    S ( t ) = v ( x ) + S 1 ( t ) ( v ( x ) ) for any t 0 ,
    and a bounded set B 0 in Z satisfying that for any t 0 and any bounded set B in X there exists a T = T ( B ) such that
    (2.4) S 1 ( t ) ( B v ( x ) ) B 0 for any t T .

Then the following conclusions hold:
  1. (i)

    B 1 = v ( x ) + B 0 is a bounded absorbing set for the semigroup S ( ) ;

  2. (ii)

    dist X ( A , B 1 ) = 0 , i.e.,

    (2.5) dist X ( A , v ( x ) + B 0 ) = dist X ( A v ( x ) , B 0 ) = 0 ;

  3. (iii)

    if B 0 is closed in X , then

    (2.6) A v ( x ) B 0 .

Furthermore, we assume that the space Y satisfies Y C 0 X θ 0 Z 1 θ 0 for constant C 0 and some θ 0 ( 0 , 1 ] . Then,
(2.7) dist Y ( S ( t ) B , A ) 0 as t + .

Proof

For any t , T 0 with t T and any bounded set B in X , from (2.4) we have

S ( t ) B = v ( x ) + S 1 ( t ) ( B v ( x ) ) v ( x ) + B 0 for any t T ,
which implies ( i) immediately.

Given the fact that A B 1 , (ii) is a direct result of the invariant of A and (i):

dist X ( A , v ( x ) + B 0 ) = dist X ( S ( t ) A , v ( x ) + B 0 ) 0 as t + .

In the following, we prove (iii).

At first, (2.6) follows directly from (2.5) and the assumption that B 0 is closed in X .

Then, from the assumption ( c ) we know that

S ( t ) B v ( x ) B 0 as t T ,
and consequently, by applying Lemma 2.3, we obtain that
dist Y ( S ( t ) B , A ) = dist Y ( S ( t ) B v ( x ) , A v ( x ) ) C 0 ( 2 B 0 Z + 1 ) dist X θ 0 ( S ( t ) B v ( x ) , A v ( x ) ) as t T = C 0 ( 2 B 0 Z + 1 ) dist X θ 0 ( S ( t ) B , A ) ,
which implies ( 2.7) immediately.□

2.3 Solutions for equations (1.1)

In this subsection, being similar to the reaction–diffusion equations, we will give the definition of different solutions about equation (1.1) (see [2] for details).

Definition 2.5

(Weak solutions) A function u = u ( x , t ) is said to be a weak solution of equation (1.1) if

u C ( [ 0 , T ] ; L 2 ( Ω ) ) L 2 ( 0 , T ; W ˚ λ 1 , 2 ( Ω ) ) L p ( 0 , T ; L p ( Ω ) ) , u t L 2 ( 0 , T ; H 1 ( Ω ) )
with u ( 0 ) = u 0 L 2 ( Ω ) , g H 1 ( Ω ) and for all φ L 2 ( 0 , T ; W ˚ λ 1 , 2 ( Ω ) ) L p ( 0 , T ; L p ( Ω ) ) , it satisfies
0 T Ω u t φ + λ u λ φ d x d t = 0 T Ω ( f ( u ) φ + g φ ) d x d t .

Definition 2.6

(Strong solutions) A weak solution u = u ( x , t ) of equation (1.1) is said to be a strong solution if

u C ( [ 0 , T ] ; W ˚ λ 1 , 2 ( Ω ) ) L 2 ( 0 , T ; W λ 2 , 2 ( Ω ) ) L ( 0 , T ; L p ( Ω ) ) , u t L 2 ( 0 , T ; L 2 ( Ω ) )
with u ( 0 ) = u 0 W ˚ λ 1 , 2 ( Ω ) L p ( Ω ) and g L 2 ( Ω ) .

3 ( L 2 ( Ω ) , L 2 ( Ω ) ) -global attractors

3.1 Existence and uniqueness of solutions

In this subsection, we first give the existence and uniqueness of weak solutions, which can be obtained by the Fadeo-Galerkin method (similar to Theorem 8.4, p. 221 in [2]). Here we only state the results.

Lemma 3.1

Let f satisfy (1.2)–(1.4), g H 1 ( Ω ) and u 0 L 2 ( Ω ) . Then, for any T > 0 , there exists a unique weak solution u to equation (1.1) such that

u C ( [ 0 , T ] ; L 2 ( Ω ) ) L 2 ( 0 , T ; W ˚ λ 1 , 2 ( Ω ) ) L p ( 0 , T ; L p ( Ω ) ) .

By Lemma 3.1, we can define the semigroup { S ( t ) } t 0 in L 2 ( Ω ) as follows:

(3.1) S ( t ) : L 2 ( Ω ) L 2 ( Ω ) , S ( t ) u 0 u ( t ) , t 0 ,
and { S ( t ) } t 0 is Lipschitz continuous in L 2 ( Ω ) .

Moreover, we also have the following lemma about strong solutions, which can be obtained as that in [2] (similar to Theorem 8.5 in p. 227).

Lemma 3.2

Let f satisfy (1.2)–(1.4), g L 2 ( Ω ) and u 0 W ˚ λ 1 , 2 ( Ω ) L p ( Ω ) . Then, for any T > 0 , there exists a unique strong solution u to equation (1.1) such that

u C ( [ 0 , T ] ; W ˚ λ 1 , 2 ( Ω ) ) L 2 ( 0 , T ; W λ 2 , 2 ( Ω ) ) L ( 0 , T ; L p ( Ω ) ) , u t L 2 ( 0 , T ; L 2 ( Ω ) ) .

Similar to [12,13], we need the following lemma about the strong solutions, which guarantees that the test functions that we used in the following are meaningful.

Theorem 3.3

Let f satisfy (1.2)–(1.4). Then for any T > 0 and any initial data ( u 0 , g ) ( W ˚ λ 1 , 2 ( Ω ) L ( Ω ) , L ( 0 , T ; L ( Ω ) ) , the unique strong solution u ( t ) of equation (1.1) belongs to L ( 0 , T ; L ( Ω ) ) .

3.2 Some results

In this subsection, we give some results about the solutions of equation (1.1) under conditions (1.2)–(1.4).

At first, similar to that in [2] (see Proposition 11.1 in p. 286 and Theorem 11.4 in p. 290), we have the following lemmas, which will be used to obtain the bounded absorbing sets and the global attractors in L 2 ( Ω ) .

Lemma 3.4

Let f satisfy (1.2)–(1.3), g H 1 ( Ω ) and u 0 L 2 ( Ω ) . Then the semigroup { S ( t ) } t 0 defined by (3.1) has a ( L 2 ( Ω ) , L 2 ( Ω ) ) -bounded absorbing set, that is, for any bounded subset B in L 2 ( Ω ) , there exists a T = T ( B ) , depending only on the L 2 -norm of B , such that

u ( t ) 2 2 ρ 0 2 for any t T and u 0 B ,
where ρ 0 > 0 is a constant independent of B and u ( t ) .

Lemma 3.5

Let f satisfy (1.2)–(1.4), g H 1 ( Ω ) and u 0 L 2 ( Ω ) . Then the semigroup { S ( t ) } t 0 defined by (3.1) has a ( L 2 ( Ω ) , L 2 ( Ω ) ) -global attractor, which is nonempty, compact, invariant in L 2 ( Ω ) and attracts every bounded subset of L 2 ( Ω ) with respect to the L 2 -norm.

Next, similar to Theorem 3.4 in [11], we have the following lemma, which will be used to obtain the bounded absorbing sets in L p ( Ω ) and W ˚ λ 1 , 2 ( Ω ) .

Lemma 3.6

Let f satisfy (1.2)–(1.4), g H 1 ( Ω ) and u 0 L 2 ( Ω ) . Then the semigroup { S ( t ) } t 0 defined by (3.1) has a ( L 2 ( Ω ) , L p ( Ω ) ) and ( L 2 ( Ω ) , W ˚ λ 1 , 2 ( Ω ) ) -bounded absorbing set, that is, for any bounded subset B in L 2 ( Ω ) , there exists a T = T ( B ) , depending only on the L 2 -norm of B , such that

u ( t ) p p ρ 1 2 for any t T and u 0 B ,
and
λ u ( t ) 2 2 ρ 1 2 for any t T and u 0 B ,
where ρ 1 > 0 is a constant independent of B and u ( t ) .

4 Asymptotic higher-order integrability

In this section, we will establish some asymptotic higher-order integrability for the solutions of equation (1.1).

4.1 Some a priori estimate for the solutions of (4.7)

In this subsection, based on the Alikakos-Moser iteration technique (see [18]), we will establish the following induction estimates about the solutions of (4.7).

For this purpose, we decompose equation (1.1) as a stationary equation

(4.1) Δ λ v + f ( v ) + i = 1 N D i g i + g 0 ( x ) = 0 in Ω , v ( x ) = 0 , on Ω ,
and an evolutionary equation
(4.2) t w ( x , t ) = Δ λ w ( x , t ) + f ( u ( x , t ) ) f ( v ( x ) ) in Ω × ( 0 , ) , w ( x , t ) = 0 on Ω × [ 0 , ) , w ( x , 0 ) = u 0 ( x ) v ( x ) , x Ω .

By [19] we know that equation (4.1) has a weak solution v W ˚ λ 1 , 2 ( Ω ) L p ( Ω ) , and then we cannot expect any higher-order integrability of the solutions of (4.1) than the order max p , 2 Q Q 2 .

Then, we consider the asymptotic higher-order integrability of equation (4.2).

Note that W ˚ λ 1 , 2 ( Ω ) L p ( Ω ) is dense in L 2 ( Ω ) ; then for each g i L 2 ( Ω ) and every initial data u 0 L 2 ( Ω ) , we can find g n i W ˚ λ 1 , 2 ( Ω ) L ( Ω ) and u 0 n W ˚ λ 1 , 2 ( Ω ) L ( Ω ) such that

(4.3) g n i g i in L 2 ( Ω ) , i = 1 , 2 , , N , u 0 n u 0 in L 2 ( Ω ) as n .

Consider the following approximation equations: n = 1 , 2 , ,

(4.4) t u n ( x , t ) = Δ λ u n ( x , t ) + f ( u n ( x , t ) ) + i = 1 N D i g n i + g n 0 ( x ) in Ω × ( 0 , ) , u n ( x , t ) = 0 on Ω × [ 0 , ) , u n ( x , 0 ) = u 0 n , x Ω ,
and
(4.5) Δ λ v n + f ( v n ) + i = 1 N D i g n i + g n 0 ( x ) = 0 in Ω , v n ( x ) = 0 , on Ω .

Then, thanks to Lemma 3.2 and Theorem 3.3, we know that for each n N , equation (4.4) has a unique strong solution u n satisfying

u n L ( 0 , T ; L ( Ω ) ) L ( 0 , T ; W ˚ λ 1 , 2 ( Ω ) ) L 2 ( 0 , T ; W λ 2 , 2 ( Ω ) ) for any T > 0 ,
and by [ 19] we know that equation ( 4.5) has a strong solution v n satisfying
(4.6) v n L ( Ω ) W λ 2 , 2 ( Ω ) .

Now, for each n N , we choose the solutions of (4.5), which satisfies (4.6) and denote it by v n ( x ) .

Set w n = u n v n , then w n satisfies the following equation in distribution sense:

(4.7) t w n ( x , t ) = Δ λ w n ( x , t ) + f ( u n ( x , t ) ) f ( v n ( x , t ) ) in Ω × ( 0 , ) , w n ( x , t ) = 0 on Ω × [ 0 , ) , w n ( x , 0 ) = u 0 n ( x ) v n ( x ) , x Ω .

Moreover, we have

w n ( , t ) W ˚ λ 1 , 2 ( Ω ) L ( Ω ) for a.e. t [ 0 , ) ,
and so for any r 0 ,
(4.8) w n ( , t ) r w n ( , t ) W ˚ λ 1 , 2 ( Ω ) L ( Ω ) for a.e. t [ 0 , ) .

Hence, for any r 0 , we can take w n ( , t ) r w n ( , t ) as a test function for equation (4.7) (this is the main reason for the approximations above).

With the preparation above, we have the following main result of this subsection.

Theorem 4.1

For each k = 0 , 1 , 2 , , there exist two positive constants T k and M k , which depend only on k , p , Q , u 0 n 2 and g n i 2 ( i = 1 , 2 , , N ) , such that for any n = 1 , 2 , , the solutions w n of (4.7) satisfy

(A k′) Ω w n ( t ) 2 Q Q 2 k d x M k for any t T k ,
and
(B k′) t t + 1 Ω w n ( s ) 2 Q Q 2 k + 1 d x Q 2 Q d s M k for any t T k ,
where Q is the homogeneous dimension of R N with a group of dilations corresponding to the Δ λ - operator.

Proof

At first, by taking the L 2 -inner product between equation (4.5) and v n , we know that there exists a positive constant R = R g n i 2 which depends only on the L 2 -bounds of g n i 2 ( i = 0 , 1 , 2 , , N ) such that for every n N , the solution v n ( x ) of (4.5) satisfies

(4.9) v n W ˚ λ 1 , 2 ( Ω ) L p ( Ω ) R g n i 2 .

Second, for the solution of (4.4), similar to the estimate for the solution of equation (1.1) (e.g., Lemma 3.6), we know that for each n N , there are positive constants S g n i 2 (that depends only on the bounds of g n i 2 ) and T u 0 n 2 (that depends only on the bounds of u 0 n 2 ) such that

(4.10) u n ( t ) W ˚ λ 1 , 2 ( Ω ) L p ( Ω ) S g n i 2 for any t T u 0 n 2 .

In the following, we will complete our proof.

(1) For the case k = 0 .

Obviously, (4.9) and (4.10) imply ( A 0 ) directly.

Then, based on (4.9) and (4.10), by applying the embedding W ˚ λ 1 , 2 ( Ω ) L 2 Q Q 2 ( Ω ) (see Lemma 2.1) to w n ( t ) = u n ( t ) v n W ˚ λ 1 , 2 ( Ω ) with w n ( t ) W ˚ λ 1 , 2 ( Ω ) u n ( t ) W ˚ λ 1 , 2 ( Ω ) + v n W ˚ λ 1 , 2 ( Ω ) , we easily obtain ( B 0 ) .

(2) Assume that ( A k ) and ( B k ) hold for k 0 .

(3) We need to prove that ( A k + 1 ) and ( B k + 1 ) hold.

By (4.8), multiplying (4.7) by w n 2 Q Q 2 k + 1 2 w n and integrating over x Ω , then we obtain that

1 2 Q 2 Q k + 1 d d t Ω w n 2 Q Q 2 k + 1 d x + Ω λ w n λ ( w n 2 Q Q 2 k + 1 2 w n ) d x = Ω ( f ( u n ) f ( v n ) ) w n 2 Q Q 2 k + 1 2 w n d x .
Thanks to ( 1.2) and Lemma 2.2 with r = 2 Q Q 2 k + 1 2 > 0 , we can deduce that
(4.11) d d t Ω w n 2 Q Q 2 k + 1 d x + 2 2 Q Q 2 k + 1 1 Q 2 Q k + 1 Ω λ w n Q Q 2 k + 1 2 d x 2 l Q Q 2 k + 1 Ω w n 2 Q Q 2 k + 1 d x .
Applying the uniform Gronwall lemma (see p. 91, Lemma 1.1 in [ 3]) to ( 4.11) and ( B k ) , we obtain that
Ω w n ( t ) 2 Q Q 2 k + 1 d x C M k , l , Q , k for any t T k + 1 ,
that is, ( A k + 1 ) holds.

Let t T k + 1 , we integrate (4.11) over [ t , t + 1 ] and obtain that

(4.12) t t + 1 Ω λ w n Q Q 2 k + 1 d x d s C M k , l , Q , k .
From ( 4.8) we know that
(4.13) w n ( , s ) Q Q 2 k + 1 W ˚ λ 1 , 2 ( Ω ) for a.e. s [ 0 , ) .

By (4.13), then applying the embedding W ˚ λ 1 , 2 ( Ω ) L 2 Q Q 2 ( Ω ) (see Lemma 2.1) to w n ( , s ) Q Q 2 k + 1 , we can deduce ( B k + 1 ) immediately from (4.12).□

Moreover, similar to Lemmas 3.2 and 3.3 in [10], we have the following results.

Lemma 4.2

Under assumption (4.3), let v n ( n = 1 , 2 , ) be the solutions of (4.5). Then the sequence { v n } n = 1 has a subsequence { v n j } j = 1 satisfying

v n j ( x ) v ( x ) for a.e. x Ω as j .
Moreover, v ( x ) is a solution of equation ( 4.1).

On the other hand, let u n ( n = 1 , 2 , ) be the unique solutions of (4.4) and u be the unique solutions of (1.1). Then we have the following continuity (w.r.t. initial data and forcing term):

u n ( t ) u ( t ) 2 2 e 2 l t 1 + 1 2 l u 0 n u 0 2 2 + i = 1 N g n i g i 2 2 + 1 μ 1 g n 0 g 0 2 2
for all t [ 0 , + ) , where μ 1 > 0 is the first eigenvalue of the operator Δ λ on Ω with homogeneous Dirichlet boundary conditions ( see ( 2.2) in Lemma 2.1).

Moreover, for any t [ 0 , + ) , there is a subsequence { u n j } j = 1 of { u n } n = 1 such that

u n j ( x , t ) u ( x , t ) for a.e. x Ω as j .

4.2 ( L 2 ( Ω ) , L p ( Ω ) ) -global attractors

In the following, we will establish the main results of this paper as follows.

Theorem 4.3

Under the assumptions of Lemma 3.4, suppose that { S ( t ) } t 0 is the semigroup defined by (3.1). Then, for any δ [ 0 , ) , there exists a bounded subset B δ in W ˚ λ 1 , 2 ( Ω ) L p ( Ω ) satisfying the following properties:

B δ = { w W ˚ λ 1 , 2 ( Ω ) L p + δ ( Ω ) : w W ˚ λ 1 , 2 ( Ω ) L p ( Ω ) + w L p + δ ( Ω ) Λ p , Q , δ < } ,
and for any L 2 - bounded subset B L 2 ( Ω ) , there exists a T = T ( B , δ ) such that
(4.14) S ( t ) B v ( x ) + B δ for any t T .
Here, the constant Λ p , Q , δ depends only on p , Q , δ , and v ( x ) is one of the fixed ( independent of δ ) solutions of ( 4.1).

Proof

Note that the homogeneous dimension Q 3 , we have Q Q 2 > 1 and then

Q Q 2 k as k .

Therefore, for any δ [ 0 , ) , we can take k large enough such that

2 p + δ 2 Q Q 2 k .

Now, let u n , v n be the strong solution of (4.4) and (4.5), respectively, then for any t T k , by Lemma 4.2, we can find n j such that u n j ( x , t ) u ( x , t ) , v n j ( x ) v ( x ) for a.e. x Ω as j . Then, by applying the Fatou lemma to ( A k ) in Theorem 4.1, we obtain that

(4.15) Ω u ( x , t ) v ( x ) 2 Q Q 2 k d x lim inf j Ω u n j ( x , t ) v n j ( x ) 2 Q Q 2 k d x M k < .

Therefore, combining with (4.15) and the W ˚ λ 1 , 2 L p -dissipation (see Lemma 3.6), applying the interpolation inequality, we can define B δ as follows:

B δ = { w W ˚ λ 1 , 2 L p + δ : w W λ 1 , 2 L p + w L p + δ C M k , M , ρ 0 , δ , v W ˚ λ 1 , 2 L p < } ,
and ( 4.14) follows from ( 4.15) immediately.□

As a directly application of Theorem 4.3, we have the following corollary.

Corollary 4.4

Under the assumptions of Lemma 3.4, the semigroup { S ( t ) } t 0 defined by (3.1) has a ( L 2 ( Ω ) , L p ( Ω ) ) -global attractor A . Moreover, A can attract every bounded subset of L 2 ( Ω ) with L p + δ -norm for any δ [ 0 , ) , and A allows the decomposition A = v ( x ) + B ; here B is bounded in L p + δ ( Ω ) for any δ [ 0 , ) , v ( x ) is a fixed solution of (4.1).

Proof

From Lemma 3.5, we know that the semigroup { S ( t ) } t 0 is ( L 2 ( Ω ) , L 2 ( Ω ) ) -asymptotically compact. Then by (4.15) with k large enough such that p + 1 2 Q Q 2 k , using the following interpolation estimates:

u 1 ( x , t ) u 2 ( x , t ) L p = u 1 ( x , t ) v ( x ) ( u 2 ( x , t ) v ( x ) ) L p u 1 ( x , t ) v ( x ) ( u 2 ( x , t ) v ( x ) ) L p + 1 1 θ u 1 ( x , t ) v ( x ) ( u 2 ( x , t ) v ( x ) ) L 2 θ C M k , θ u 1 ( x , t ) u 2 ( x , t ) L 2 θ ,
where u i ( t ) = S ( t ) u i 0 ( i = 1 , 2 ) and θ ( 0 , 1 ) , we can deduce that the semigroup { S ( t ) } t 0 is ( L 2 ( Ω ) , L p ( Ω ) ) -asymptotically compact.

Hence, combining with Lemmas 3.5 and 3.6, and by the theory of dynamical system (e.g., see [1,2,3, 20,21]), we obtain the existence of the ( L 2 ( Ω ) , L p ( Ω ) ) -global attractors.

Moreover, by applying the interpolation inequality and (4.15), we can get that

(4.16) u 1 ( x , t ) u 2 ( x , t ) L p + δ = u 1 ( x , t ) v ( x ) ( u 2 ( x , t ) v ( x ) ) L p + δ u 1 ( x , t ) v ( x ) ( u 2 ( x , t ) v ( x ) ) L 2 ( Q Q 2 ) k 1 1 θ 1 u 1 ( x , t ) v ( x ) ( u 2 ( x , t ) v ( x ) ) L 2 θ 1 C M k 1 , θ 1 u 1 ( x , t ) u 2 ( x , t ) L 2 θ 1 ,
where θ 1 ( 0 , 1 ) , and for any δ [ 0 , ) , we can take k 1 = log Q Q 2 p + δ 2 + 1 such that
2 Q Q 2 k 1 > p + δ .

On the other hand, from ( A k ) in Theorem 4.1, there are positive constants T k 1 and M k 1 such that for any ε > 0 , we have

(4.17) dist L 2 ( Ω ) ( S ( t ) B v ( x ) , A v ( x ) ) = dist L 2 ( Ω ) ( S ( t ) B , A ) ε C M k 1 , θ 1 1 θ 1
for all t T k 1 , where we have used ( 2.5) in Theorem 2.4 and the attraction of ( L 2 ( Ω ) , L 2 ( Ω ) ) -global attractors, B is a L 2 -bounded set.

Combining with (4.15)–(4.17), we obtain that

dist L p + δ ( Ω ) ( S ( t ) B , A ) ε for any t T k 1 .

Finally, similar to (4.16) and applying (4.15) again, we can complete the proof of the residual part.□

5 ( L 2 ( Ω ) , W ˚ λ 1 , 2 ( Ω ) ) -global attractors

In this section, we will prove the existence of the ( L 2 ( Ω ) , W ˚ λ 1 , 2 ( Ω ) ) -global attractors.

At first, we need the following lemma.

Lemma 5.1

Under the assumptions of Lemma 3.4, then for any bounded subset B in L 2 ( Ω ) , there exists a T = T ( B ) , depending only on the L 2 -norm of B , such that

(5.1) t u ( s ) 2 2 ρ 2 2 for any u 0 B and s T ,
where t u ( s ) = d d t ( S ( t ) u 0 ) t = s , { S ( t ) } t 0 is the semigroup defined by ( 3.1) and ρ 2 > 0 is a constant independent of B .

Proof

We give below only formal derivation of the estimate (5.1), which can be justified by the Galerkin approximation method.

By differentiating (1.1) with respect to time t and denoting z = t u , we can get

(5.2) t z ( x , t ) = Δ λ z ( x , t ) + f ( u ( x , t ) ) z ( x , t ) .

Multiplying (5.2) by z and integrating over Ω , then by using (1.2), we obtain that

1 2 d d t z 2 2 + λ z 2 2 l z 2 2 .

Furthermore,

(5.3) d d t z 2 2 2 l z 2 2 .

On the other hand, multiplying (1.1) by t u , we obtain that

(5.4) t u 2 2 + d d t 1 2 λ u 2 2 Ω ( u ) Ω g u = 0 ,
where ( u ) satisfies ( 1.4).

Integrating (5.4) with respect to time t from t to t + 1 , and combining with Lemmas 3.4 and 3.6, we can deduce that

(5.5) t t + 1 t u 2 2 C ( ρ 0 , ρ 1 , Ω , g ˜ 2 2 , g 0 2 2 )
as t is large enough.

Combining with (5.3) and (5.5), and using the uniform Gronwall lemma (see p. 91, Lemma 1.1 in [3]), we have

t u 2 2 C ( ρ 0 , ρ 1 , Ω , g ˜ 2 2 , g 0 2 2 )
as t is large enough.□

Theorem 5.2

Under the assumptions of Lemma 3.4, the semigroup { S ( t ) } t 0 defined by (3.1) is ( L 2 ( Ω ) , W ˚ λ 1 , 2 ( Ω ) ) -asymptotically compact.

Proof

By Lemma 3.6, we know the semigroup { S ( t ) } t 0 has a ( L 2 ( Ω ) , W ˚ λ 1 , 2 ( Ω ) ) -bounded absorbing set B 1 . Let u 1 , u 2 be the solution of equation (1.1) with initial data u 01 , u 02 B 1 . From Theorem 4.3, we know the solutions u 1 , u 2 can be decomposed as

u i = S ( t ) u 0 = w i + v , i = 1 , 2 ,
where w i B δ , v ( x ) is a fixed solution of ( 4.1).

Set ω = u 1 u 2 , then ω satisfies the following equation in distribution sense:

(5.6) t ω ( x , t ) = Δ λ ω ( x , t ) + f ( u 1 ( x , t ) ) f ( u 2 ( x , t ) ) in Ω × ( 0 , ) , ω ( x , t ) = 0 on Ω × [ 0 , ) , ω ( x , 0 ) = u 01 ( x ) u 02 ( x ) , x Ω .

Taking the L 2 -inner product between (5.6) and ω , we have

λ ω 2 2 = ( t ω , ω ) + ( f ( u 1 ) f ( u 2 ) , ω ) .

Thanks to (1.2) and (5.1) in Lemma 5.1, we can deduce that

(5.7) λ ω 2 2 t ω 2 ω 2 + l ω 2 2 C ρ 2 + l ω 2 ,
where the constant C ρ 2 is related to ρ 2 , ρ 2 comes from ( 5.1).

By Lemma 3.5, we know the semigroup { S ( t ) } t 0 is ( L 2 ( Ω ) , L 2 ( Ω ) ) -asymptotically compact. Then by virtue of (5.7), we know the semigroup { S ( t ) } t 0 is ( L 2 ( Ω ) , W ˚ λ 1 , 2 ( Ω ) ) -asymptotically compact.□

Combining with Lemma 3.6 and Theorem 5.2, by the theory of dynamical system (e.g., see [1,2,3, 20,21]), we immediately obtain the following conclusion.

Theorem 5.3

Under the assumptions of Lemma 3.4, the semigroup { S ( t ) } t 0 defined by (3.1) has a ( L 2 ( Ω ) , W ˚ λ 1 , 2 ( Ω ) ) -global attractor, which is nonempty, compact, invariant in W ˚ λ 1 , 2 ( Ω ) and attracts every bounded subset of W ˚ λ 1 , 2 ( Ω ) with respect to the W ˚ λ 1 , 2 -norm.

Acknowledgments

The authors would like to thank the referee for his/her helpful comments and suggestions.

    Funding information: This work was supported by the Natural Science Foundation of Hunan Province (Grants No. 2018JJ2416), the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 20C1263) and the Science and Technology Innovation Team of Hunan University of Arts and Science (Numerical calculation and stochastic process with their applications).

    Conflict of interest: Authors state no conflict of interest.

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Received: 2020-02-21
Revised: 2020-11-22
Accepted: 2021-01-21
Published Online: 2021-05-04

© 2021 Kaixuan Zhu and Yongqin Xie, published by DeGruyter

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