Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation

Jiazhuo Cheng; Qiru Wang

doi:10.1515/anona-2023-0133

Open Access Published by De Gruyter February 24, 2024

Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation

Jiazhuo Cheng and Qiru Wang

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2023-0133

Abstract

This article is devoted to the study of the initial boundary value problem for a mixed pseudo-parabolic r ( x ) -Laplacian-type equation. First, by employing the imbedding theorems, the theory of potential wells, and the Galerkin method, we establish the existence and uniqueness of global solutions with subcritical initial energy, critical initial energy, and supercritical initial energy, respectively. Then, we obtain the decay estimate of global solutions with sub-sharp-critical initial energy, sharp-critical initial energy, and supercritical initial energy, respectively. For supercritical initial energy, we also need to analyze the properties of ω -limits of solutions. Finally, we discuss the finite-time blowup of solutions with sub-sharp-critical initial energy and sharp-critical initial energy, respectively.

Keywords: a mixed pseudo-parabolic r(x)-Laplacian equation; global existence; uniqueness and asymptotic behavior; finite-time blowup; the imbedding theorems; the theory of potential wells and the Galerkin method

MSC 2010: 35K20; 35K58; 35K91; 35D30

1 Introduction

Consider the initial boundary value problem of semilinear pseudo-parabolic equation with r ( x ) -Laplacian:

(1.1) u t − div ( ∣ ∇ u ∣ r ( x ) − 2 ∇ u ) − Δ u t = ∣ u ∣ m ( x ) − 1 u − ⨍ Ω ∣ u ∣ m ( x ) − 1 u d x , x ∈ Ω , 0 < t < T , ∂ u ∂ ν ( x , t ) = 0 , x ∈ ∂ Ω , 0 < t < T , u ( x , 0 ) = u 0 ( x ) ≢ 0 , ⨍ Ω u 0 d x = 0 , x ∈ Ω ,

where Ω ⊂ R n ( n ≥ 1 ) is a bounded domain with smooth boundary, T ∈ ( 0 , ∞ ] , u 0 ( x ) ∈ W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) , ⨍ Ω u d x = 1 ∣ Ω ∣ ∫ Ω u d x , and ⨍ Ω ∣ u ∣ m ( x ) − 1 u d x is conserved (at least when you make precise the meaning of the solution), ∂ ∂ ν denotes the differentiation with respect to the outward normal ν on ∂ Ω . The exponents r ( x ) and m ( x ) are two measurable functions satisfying the following conditions:

( a ) ∣ r ( x ) − r ( y ) ∣ + ∣ m ( x ) − m ( y ) ∣ ≤ A log ( e + 1 ∕ ∣ x − y ∣ ) , for all x , y ∈ Ω with x ≠ y ; ( b ) 0 < r − − 1 ≤ r ( x ) − 1 ≤ r + − 1 < m − ≤ m ( x ) ≤ m + < r − ∗ − 1 ; ( c ) m ( x ) > max { r ( x ) − 1 , 1 } ,

where r − ≔ ess inf x ∈ Ω r ( x ) , r + ≔ ess sup x ∈ Ω r ( x ) , m − ≔ ess inf x ∈ Ω m ( x ) , m + ≔ ess sup x ∈ Ω m ( x ) , r − ∗ ≔ n r − n − r − , if r − < n , + ∞ , if r − ≥ n , A > 0 is a constant.

In recent years, more and more research studies devoted to the study of population dynamics and biological sciences where the total mass is conserved or known [5,10,41,43]. These works have enriched the researches of biological and chemical problems where conservation properties dominate. Our Problem (1.1) is also one of these problems.

The pseudo-parabolic equation

(1.2) u t − Δ u t = F ( x , t , ∇ u , Δ u )

has been studied extensively by many authors [3,4,14,38,45,46]. From 2013 to 2018, the authors in [32,52,53,54] studied (1.2) for F ( x , t , ∇ u , Δ u ) = Δ u + u p . In 2021, Wang and Xu [48] expanded the previous studies and studied the following nonlocal semilinear pseudo-parabolic equation subject to the Neumann boundary condition:

u t − Δ u − Δ u t = ∣ u ∣ p − 1 u − ⨍ Ω ∣ u ∣ p − 1 u d x .

Considering the physical significance of practical problems, such as atoms and ions, researchers often replace the term Δ u in the pseudo-parabolic (1.2) with the p -Laplacian term div ( ∣ ∇ u ∣ p − 2 ∇ u ) . In recent years, many researchers have studied the existence of solutions to pseudo-parabolic p -Laplace equations [11,16,27,31,33,44]. Very recently, Cheng and Wang [9] considered the following semilinear pseudo-parabolic p -Laplace equation:

u t − div ( ∣ ∇ u ∣ p − 2 ∇ u ) − Δ u t = ∣ u ∣ q − 1 u − ⨍ Ω ∣ u ∣ q − 1 u d x ,

which proved the existence, uniqueness, and decay estimate of global solutions with subcritical initial energy, critical initial energy, and supercritical initial energy.

The r ( x ) -Laplacian term div ( ∣ ∇ u ∣ r ( x ) − 2 ∇ u ) is the natural generalization of the usual term div ( ∣ ∇ u ∣ p − 2 ∇ u ) . Since r ( x ) is a function, div ( ∣ ∇ u ∣ r ( x ) − 2 ∇ u ) is non-homogeneous and has more complex nonlinear properties, which lead to the emergence of many new properties in the variable index problems [2,24,37]. In 2016, Guo and Gao [20] studied the Neumann boundary value problem of parabolic equation with nonlocal source:

(1.3) u t − div ( ∣ ∇ u ∣ p ( x , t ) − 2 ∇ u ) = ∣ ∇ u ∣ r ( x , t ) − 2 u − 1 ∣ Ω ∣ ∫ Ω ∣ ∇ u ∣ r ( x , t ) − 2 u d x .

They constructed a suitable control function, improved the regularity of the approximate solution, obtained a new energy inequality, and proved that if

max 1 , 2 N N + 2 < p − < N , max { p + , 2 } < r − ≤ r ( x , t ) ≤ 2 N + ( N + 2 ) p − 2 N ,

and the initial value satisfies the appropriate conditions, the solution of Problem (1.3) blows up in finite-time.

In 2017, Di et al. [15] studied the following pseudo-parabolic equation with nonlinearities of variable exponent type:

(1.4) u t − v Δ u t − div ( ∣ ∇ u ∣ m ( x ) − 2 ∇ u ) = ∣ u ∣ p ( x ) − 2 u ,

and proved a blowup result when initial energy is non-positive by means of a differential inequality technique. In 2020, Liao et al. [30] improved and extended the results of Di et al. and obtained a non-global existence result by combining the concavity method [26,29] with some differential inequalities when the initial energy is positive and bounded. In the same year, Zhu et al. [56] further improved the results of Liao et al. and proved the global existence and blowup results of weak solutions with arbitrarily high initial energy by analyzing the properties of ω -limits of solutions. In addition to the aforementioned studies, there are some studies on variable exponential partial differential equations [1,13,21,22].

The potential well method was proposed by Sattinger [42] in 1968 to overcome the difficulties encountered by the Galerkin method in the prior estimation of solutions. Later, Liu et al. [34,35] extended and improved the method by introducing a family of potential wells, which includes the known potential well as a special case. Now, it is one of the most useful methods for proving the global existence and non-existence of solutions, as well as the vacuum isolation of solutions for parabolic equations [7,36,49, 50,51,55].

The study of Problem (1.1) can help us to understand the role of the corresponding conservation properties in the real world and its connection to biological and chemical problems (1.4). In this article, by combining the theory of potential wells with the Galerkin method, we shall establish the global existence, uniqueness, asymptotic behavior, and finite-time blowup of solutions to Problem (1.1). The goal of this study is as follows:

In Section 2, we give some preliminaries and notations. Meanwhile, we state our main results of this article.
In Sections 3, we prove the local existence of solution using the standard Galerkin method.
In Section 4, we establish the existence and uniqueness of global solutions with subcritical initial energy, critical initial energy, and supercritical initial energy, respectively.
In Section 5, we prove the decay estimate of global solutions with sub-sharp-critical initial energy, sharp-critical initial energy, and supercritical initial energy, respectively.
In Section 6, we discuss the global nonexistence of solutions with sub-sharp-critical initial energy and sharp-critical initial energy, respectively.
In Section 7, we make some conclusions and discussions.

2 Preliminaries and main results

Throughout this article, let C be a general positive constant that may change from line to line. For 1 ≤ p ≤ ∞ , denote the L p ( Ω ) norm by ∥ ⋅ ∥ p . Define the set

W m , p ( Ω ) = { u ∈ L p ( Ω ) ∣ D α u ∈ L p ( Ω ) , ∀ α ∈ Z + n , ∣ α ∣ ≤ m } , 1 ≤ p ≤ + ∞ ,

where D α u is the α -order weak derivative of u , and H k ( Ω ) = W k , 2 ( Ω ) ( k = 0 , 1 , … ) when p = 2 . Furthermore, we define the following set:

W N m , p ( Ω ) = u ∈ W m , p ( Ω ) ∂ u ∂ ν ∂ Ω = 0 , ∫ Ω u d x = 0 , 1 ≤ p ≤ + ∞ .

Let ∥ ⋅ ∥ H 1 represent the norm of H 1 ( Ω ) and ( ⋅ , ⋅ ) be the inner product in L 2 ( Ω ) , i.e.,

∥ u ∥ H 1 = ( ∥ u ∥ 2 2 + ∥ ∇ u ∥ 2 2 ) 1 2 , ∀ u ∈ H 1 ( Ω )

and

( u , v ) = ∫ Ω u ( x ) v ( x ) d x , ∀ u , v ∈ L 2 ( Ω ) ,

then ∥ ⋅ ∥ H 1 is equivalent to ∥ ∇ ( ⋅ ) ∥ 2 .

Define the Banach space of the Orlicz-Sobolev type as follows:

L r ( x ) ( Ω ) = u ∣ u is a measurable real-valued function , ∫ Ω ∣ u ∣ r ( x ) d x < ∞ ,

equipped with the following Luxemburg norm:

∥ u ∥ r ( x ) = inf λ > 0 ∣ ∫ Ω u λ r ( x ) d x ≤ 1 ,

and the conjugate space is L m ( x ) ( Ω ) , where 1 m ( x ) + 1 r ( x ) = 1 , for all x ∈ Ω .

Correspondingly, the space W 1 , r ( x ) ( Ω ) can be defined as:

W 1 , r ( x ) ( Ω ) = { u ∈ L r ( x ) ( Ω ) ∣ ∣ ∇ u ∣ ∈ L r ( x ) ( Ω ) } ,

and it can be equipped with the following norm:

∥ u ∥ W 1 , r ( x ) = ∥ u ∥ r ( x ) + ∥ ∇ u ∥ r ( x ) .

Integrating (1.1) over Ω × ( 0 , t ) , we obtain ∫ Ω u d x = ∫ Ω u 0 d x = 0 , which means the conservation law for Problem (1.1). Hence, we define

W N 1 , r ( x ) ( Ω ) = u ∈ W 1 , r ( x ) ( Ω ) ∂ u ∂ ν ∂ Ω = 0 , ∫ Ω u d x = 0 .

Lemma 2.1

[25] If r 1 ( x ) ≤ r 2 ( x ) for any x ∈ Ω , then there exists the continuous embedding:

L r 2 ( x ) ( Ω ) ↪ L r 1 ( x ) ( Ω ) ,

where the norm of the embedding operator does not exceed ∣ Ω ∣ + 1 .

Lemma 2.2

[18] For any u ∈ L r ( x ) ( Ω ) ,

∥ u ∥ r ( x ) < 1 ( = 1 ; > 1 ) ⇔ ∫ Ω ∣ u ∣ r ( x ) d x < 1 ( = 1 ; > 1 ) ;
∥ u ∥ r ( x ) < 1 , t h e n ∥ u ∥ r ( x ) r + ≤ ∫ Ω ∣ u ∣ r ( x ) d x ≤ ∥ u ∥ r ( x ) r − ;
∥ u ∥ r ( x ) > 1 , t h e n ∥ u ∥ r ( x ) r − ≤ ∫ Ω ∣ u ∣ r ( x ) d x ≤ ∥ u ∥ r ( x ) r + .

Lemma 2.3

[18] Let r , m ∈ C + ( Ω ¯ ) = { f ∈ C ( Ω ¯ ) : min x ∈ Ω ¯ f ( x ) > 1 } . Assume that

m ( x ) + 1 < r ∗ ( x ) = n r ( x ) n − r ( x ) , i f r ( x ) < n , + ∞ , i f r ( x ) ≥ n .

Then, there is a continuous and compact embedding W N 1 , r ( x ) ( Ω ) ↪ L m ( x ) + 1 ( Ω ) .

Let

1 B = inf u ∈ W N 1 , r ( x ) , u ≠ 0 ∥ ∇ u ∥ r ( x ) ∥ u ∥ m ( x ) + 1

be the imbedding constant for W N 1 , r ( x ) ( Ω ) ↪ L m ( x ) + 1 and B 1 = max { B , 1 } .

Next, the energy functional J ( u ) and the Nehari functional I ( u ) are defined as:

(2.1) J ( u ) = ∫ Ω 1 r ( x ) ∣ ∇ u ∣ r ( x ) − 1 m ( x ) + 1 ∣ u ∣ m ( x ) + 1 d x , I ( u ) = ∫ Ω ( ∣ ∇ u ∣ r ( x ) − ∣ u ∣ m ( x ) + 1 ) d x ,

and the Nehari manifold is as follows:

N = { u ∈ W N 1 , r ( x ) ( Ω ) \ { 0 } ∣ I ( u ) = 0 } .

Based on the definitions of J ( u ) and I ( u ) , we introduce sets

W = { u ∈ W N 1 , r ( x ) ( Ω ) ∣ J ( u ) < d , I ( u ) > 0 } ∪ { 0 } , V = { u ∈ W N 1 , r ( x ) ( Ω ) ∣ J ( u ) < d , I ( u ) < 0 } ,

where d is the depth of the potential well, which can be defined as:

d = inf u ∈ N J ( u ) .

Define a family of potential wells

(2.2) I δ ( u ) = δ ∫ Ω ∣ ∇ u ∣ r ( x ) d x − ∫ Ω ∣ u ∣ m ( x ) + 1 d x , N δ = { u ∈ W N 1 , r ( x ) ( Ω ) \ { 0 } ∣ I δ ( u ) = 0 } , W δ = { u ∈ W N 1 , r ( x ) ( Ω ) ∣ J ( u ) < d ( δ ) , I δ ( u ) > 0 } ∪ { 0 } , V δ = { u ∈ W N 1 , r ( x ) ( Ω ) ∣ J ( u ) < d ( δ ) , I δ ( u ) < 0 } , d ( δ ) = inf u ∈ N δ J ( u ) ,

where δ > 0 .

In the following, let us give some sets and functionals in order to consider the weak solution with high energy level:

N + = { u ∈ W N 1 , r ( x ) ( Ω ) ∣ I ( u ) > 0 } , J α = { u ∈ W N 1 , r ( x ) ( Ω ) ∣ J ( u ) ≤ α } , for all α > d , N α = N ∩ J α = { u ∈ N ∣ J ( u ) ≤ α } , for all α > d , λ α = inf { ∥ u ∥ H 1 2 ∣ u ∈ N α } , for all α > d .

Clearly, we know that λ α is not increasing with respect to α .

As in Evans [17, Page 8], we present the following definition.

Definition 2.1

(Weak solution [17]) Function u ( x , t ) ∈ Ω × ( 0 , T ) is called a weak solution to (1.1), if u ∈ L ∞ ( 0 , T ; W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) ) , u t ∈ L 2 ( 0 , T ; W N 1 , 2 ( Ω ) ) , u ( x , 0 ) = u 0 ( x ) ∈ W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) and satisfies

(2.3) ∫ 0 t ( ( u τ , φ ) + ( ∇ u τ , ∇ φ ) + ( ∣ ∇ u ∣ r ( x ) − 2 ∇ u , ∇ φ ) ) d τ = ∫ 0 t ( ∣ u ∣ m ( x ) − 1 u − ⨍ Ω ∣ u ∣ m ( x ) − 1 u d x , φ ) d τ ,

for any φ ∈ W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) .

Moreover, the following equality

∫ 0 t ∥ u τ ∥ H 1 2 d τ + J ( u ) = J ( u 0 )

holds for t ∈ [ 0 , T ) .

Definition 2.2

[Maximal existence time] Let u ( t ) be a weak solution of Problem (1.1), for maximal existence time T of u ( t ) and we have the following definitions:

If u ( t ) exists for t ∈ [ 0 , ∞ ) , then T = + ∞ ;
If there exists a t 0 ∈ ( 0 , ∞ ) such that u ( t ) exists for t ∈ [ 0 , t 0 ) , but does not exist at t = t 0 , then T = t 0 .

Lemma 2.4

[Relations between I δ ( u ) and ∫ Ω ∣ ∇ u ∣ r ( x ) d x ] Let u ∈ W N 1 , r ( x ) ( Ω ) .

If 0 < ∫ Ω ∣ ∇ u ∣ r ( x ) d x < h ( δ ) , then I δ ( u ) > 0 ;
If I δ ( u ) < 0 , then ∫ Ω ∣ ∇ u ∣ r ( x ) d x > h ( δ ) ;
If I δ ( u ) = 0 and ∫ Ω ∣ ∇ u ∣ r ( x ) d x ≠ 0 , then ∫ Ω ∣ ∇ u ∣ r ( x ) d x ≥ h ( δ ) , where h ( δ ) is a function of δ and satisfies
h ( δ ) = min δ B 1 m + + 1 r − m + + 1 − r − , δ B 1 m + + 1 r + m − + 1 − r + .

Proof

By 0 < ∫ Ω ∣ ∇ u ∣ r ( x ) d x < h ( δ ) , we obtain
∫ Ω ∣ u ∣ m ( x ) + 1 d x ≤ max { ∥ u ∥ m ( x ) + 1 m + + 1 , ∥ u ∥ m ( x ) + 1 m − + 1 } ≤ max B 1 m + + 1 max ∫ Ω ∣ ∇ u ∣ r ( x ) d x m + + 1 r + , ∫ Ω ∣ ∇ u ∣ r ( x ) d x m + + 1 r − , B 1 m − + 1 max ∫ Ω ∣ ∇ u ∣ r ( x ) d x m − + 1 r + , ∫ Ω ∣ ∇ u ∣ r ( x ) d x m − + 1 r − ≤ B 1 m + + 1 max ∫ Ω ∣ ∇ u ∣ r ( x ) d x m + + 1 r − , ∫ Ω ∣ ∇ u ∣ r ( x ) d x m − + 1 r + = B 1 m + + 1 ∫ Ω ∣ ∇ u ∣ r ( x ) d x max ∫ Ω ∣ ∇ u ∣ r ( x ) d x m + + 1 − r − r − , ∫ Ω ∣ ∇ u ∣ r ( x ) d x m − + 1 − r + r + < B 1 m + + 1 ∫ Ω ∣ ∇ u ∣ r ( x ) d x max ( h ( δ ) ) m + + 1 − r − r − , ( h ( δ ) ) m − + 1 − r + r + = δ ∫ Ω ∣ ∇ u ∣ r ( x ) d x ,
which yields I δ ( u ) > 0 .
From I δ ( u ) < 0 , we know that ∫ Ω ∣ ∇ u ∣ r ( x ) d x ≠ 0 . Thus,
δ ∫ Ω ∣ ∇ u ∣ r ( x ) d x < ∫ Ω ∣ u ∣ m ( x ) + 1 d x ≤ B 1 m + + 1 ∫ Ω ∣ ∇ u ∣ r ( x ) d x max ∫ Ω ∣ ∇ u ∣ r ( x ) d x m + + 1 − r − r − , ∫ Ω ∣ ∇ u ∣ r ( x ) d x m − + 1 − r + r + ,
which gives ∫ Ω ∣ ∇ u ∣ r ( x ) d x > h ( δ ) .
If I δ ( u ) = 0 , ∫ Ω ∣ ∇ u ∣ r ( x ) d x ≠ 0 , we have
δ ∫ Ω ∣ ∇ u ∣ r ( x ) d x = ∫ Ω ∣ u ∣ m ( x ) + 1 d x ≤ B 1 m + + 1 ∫ Ω ∣ ∇ u ∣ r ( x ) d x max ∫ Ω ∣ ∇ u ∣ r ( x ) d x m + + 1 − r − r − , ∫ Ω ∣ ∇ u ∣ r ( x ) d x m − + 1 − r + r + ,
then ∫ Ω ∣ ∇ u ∣ r ( x ) d x ≥ h ( δ ) .□

Lemma 2.5

[Properties of J ( λ u ) ] Assume that u ∈ W N 1 , r ( x ) ( Ω ) and ∫ Ω ∣ u ∣ m ( x ) + 1 d x ≠ 0 . Then, there exist ε ∈ [ r − − 1 , r + − 1 ] and η ∈ [ m − , m + ] , such that

J ( λ u ) is decreasing on λ ∈ [ λ ∗ , ∞ ) and increasing on λ ∈ [ 0 , λ ∗ ] , and λ = λ ∗ is the maximum point of J ( λ u ) . In addition, lim λ → 0 J ( λ u ) = 0 and lim λ → + ∞ J ( λ u ) = − ∞ ;
I ( λ u ) < 0 for λ ∗ < λ < ∞ and I ( λ u ) > 0 for 0 < λ < λ ∗ , and I ( λ ∗ u ) = 0 , where
λ ∗ = ∫ Ω ∣ ∇ u ∣ r ( x ) d x ∫ Ω ∣ u ∣ m ( x ) + 1 d x 1 η − ε .

Proof

From the definition of J ( u ) , we have

J ( λ u ) = ∫ Ω λ r ( x ) r ( x ) ∣ ∇ u ∣ r ( x ) d x − ∫ Ω λ m ( x ) + 1 m ( x ) + 1 ∣ u ∣ m ( x ) + 1 d x ,

which gives lim λ → 0 J ( λ u ) = 0 and lim λ → + ∞ J ( λ u ) = − ∞ . Furthermore, from the mean value theorem of integrals, we have that

(2.4) d J ( λ u ) d λ = ∫ Ω λ r ( x ) − 1 ∣ ∇ u ∣ r ( x ) d x − ∫ Ω λ m ( x ) ∣ u ∣ m ( x ) + 1 d x = λ ε ∫ Ω ∣ ∇ u ∣ r ( x ) d x − λ η ∫ Ω ∣ u ∣ m ( x ) + 1 d x = λ ε ∫ Ω ∣ ∇ u ∣ r ( x ) d x − λ η − ε ∫ Ω ∣ u ∣ m ( x ) + 1 d x ,

where ε ∈ [ r − − 1 , r + − 1 ] , η ∈ [ m − , m + ] . Considering d J ( λ u ) d λ λ = λ ∗ = 0 , we obtain

λ ∗ = ∫ Ω ∣ ∇ u ∣ r ( x ) d x ∫ Ω ∣ u ∣ m ( x ) + 1 d x 1 η − ε .

In addition, by (2.4), we obtain

d J ( λ u ) d λ > 0 , for 0 < λ < λ ∗ ; d J ( λ u ) d λ < 0 , for λ ∗ < λ < ∞ ,

which yields

□ I ( λ u ) = λ d J ( λ u ) d λ = > 0 , 0 < λ < λ ∗ ; = 0 , λ = λ ∗ ; < 0 , λ ∗ < λ < ∞ .

Let M ≔ 1 r + − 1 m − + 1 B 1 − r + ( m + + 1 ) m − + 1 − r + , and we have the following lemma.

Lemma 2.6

[Depth d of potential well] Assume that r ( x ) and m ( x ) satisfy the conditions given in (1.1). Then, the potential well depth

d ≥ M .

Proof

If u ∈ N , then I ( u ) = 0 and u ≠ 0 . According to the definitions of J ( u ) and I ( u ) , we obtain

∫ Ω ∣ ∇ u ∣ r ( x ) d x = ∫ Ω ∣ u ∣ m ( x ) + 1 d x > 0

and

J ( u ) ≥ 1 r + ∫ Ω ∣ ∇ u ∣ r ( x ) d x − 1 m − + 1 ∫ Ω ∣ u ∣ m ( x ) + 1 d x = 1 r + − 1 m − + 1 ∫ Ω ∣ ∇ u ∣ r ( x ) d x .

By Lemma 2.4 (iii), we have

d = inf u ∈ N J ( u ) ≥ inf u ∈ N 1 r + − 1 m − + 1 ∫ Ω ∣ ∇ u ∣ r ( x ) d x ≥ 1 r + − 1 m − + 1 B 1 − r + ( m + + 1 ) m − + 1 − r + ,

which implies

□ d ≥ M .

Remark 2.1

When r ( x ) and m ( x ) are constants, d = M (see [9, Lemma 2.3]). In this case, we call d the critical initial energy and M the sharp-critical initial energy, respectively.

Lemma 2.7

For d ( δ ) in (2.2), we obtain

d ( δ ) ≥ 1 r + ( 1 − δ ) h ( δ ) + m − + 1 − r + r + ( m − + 1 ) δ h ( δ ) , for 0 < δ ≤ 1 . In particular, d ( 1 ) ≥ M ;
there is a constant b , b ∈ 1 , m + + 1 r − such that d ( b ) = 0 , d ( δ ) is decreasing on 1 ≤ δ ≤ b and increasing on 0 < δ ≤ 1 .

Proof

If u ∈ N δ , i.e., I δ ( u ) = 0 and ∫ Ω ∣ ∇ u ∣ r ( x ) d x ≠ 0 . From Lemma 2.4 (iii), we have
J ( u ) = ∫ Ω 1 r ( x ) ( 1 − δ ) ∣ ∇ u ∣ r ( x ) d x + ∫ Ω δ r ( x ) ∣ ∇ u ∣ r ( x ) d x − ∫ Ω 1 m ( x ) + 1 ∣ u ∣ m ( x ) + 1 d x ≥ 1 r + ( 1 − δ ) ∫ Ω ∣ ∇ u ∣ r ( x ) d x + 1 r + − 1 m − + 1 ∫ Ω ∣ u ∣ m ( x ) + 1 d x ≥ 1 r + ( 1 − δ ) h ( δ ) + m − + 1 − r + r + ( m − + 1 ) δ h ( δ ) ,
which yields d ( 1 ) ≥ M .
Let λ ( δ ) = δ ∫ Ω ∣ ∇ u ∣ r ( x ) d x ∫ Ω ∣ u ∣ m ( x ) + 1 d x 1 m ( x ) + 1 − r ( x ) , then

(2.5) ∣ I δ ( λ ( δ ) u ) ∣ = δ ∫ Ω ∣ ∇ u ∣ r ( x ) λ ( δ ) r ( x ) d x − ∫ Ω ∣ u ∣ m ( x ) + 1 λ ( δ ) m ( x ) + 1 d x = ∫ Ω λ ( δ ) r ( x ) ( δ ∣ ∇ u ∣ r ( x ) − ∣ u ∣ m ( x ) + 1 λ ( δ ) m ( x ) + 1 − r ( x ) ) d x ≤ max δ ∫ Ω ∣ ∇ u ∣ r ( x ) d x ∫ Ω ∣ u ∣ m ( x ) + 1 d x r − m + + 1 − r − , δ ∫ Ω ∣ ∇ u ∣ r ( x ) d x ∫ Ω ∣ u ∣ m ( x ) + 1 d x r + m − + 1 − r + × ∫ Ω ( δ ∣ ∇ u ∣ r ( x ) − ∣ u ∣ m ( x ) + 1 λ ( δ ) m ( x ) + 1 − r ( x ) ) d x = max δ ∫ Ω ∣ ∇ u ∣ r ( x ) d x ∫ Ω ∣ u ∣ m ( x ) + 1 d x r − m + + 1 − r − , δ ∫ Ω ∣ ∇ u ∣ r ( x ) d x ∫ Ω ∣ u ∣ m ( x ) + 1 d x r + m − + 1 − r + × ∫ Ω δ ∣ ∇ u ∣ r ( x ) d x − ∫ Ω ∣ u ∣ m ( x ) + 1 δ ∫ Ω ∣ ∇ u ∣ r ( x ) d x ∫ Ω ∣ u ∣ m ( x ) + 1 d x d x = 0 ,
i.e., I δ ( λ ( δ ) u ) = 0 . For λ ( δ ) u ∈ N δ , we obtain
d ( δ ) ≤ J ( λ ( δ ) u ) ≤ 1 r − ∫ Ω λ ( δ ) r ( x ) ∣ ∇ u ∣ r ( x ) d x − 1 m + + 1 ∫ Ω λ ( δ ) m ( x ) + 1 ∣ u ∣ m ( x ) + 1 d x = 1 r − − δ m + + 1 ∫ Ω λ ( δ ) r ( x ) ∣ ∇ u ∣ r ( x ) d x .
When δ = m + + 1 r − , we have d ( δ ) ≤ 0 . Since d ( δ ) is continuous with respect to δ and d ( 1 ) = d > 0 , there exists a b ∈ 1 , m + + 1 r − that satisfies d ( b ) = 0 .

Now, we need to prove inf N δ ′ J ( u ) = d ( δ ′ ) < d ( δ ″ ) = inf N δ ″ J ( u ) for 0 < δ ′ < δ ″ < 1 and 1 < δ ″ < δ ′ < b , respectively, and if successful, then the lemma is proved.

In fact, by the definition of λ ( δ ) , for 0 < δ ′ < δ ″ < 1 , similar to the process of (2.5), we see that
(2.6) δ ∫ Ω 1 m ( x ) + 1 − r ( x ) ∣ ∇ u ∣ r ( x ) λ ( δ ) r ( x ) d x − ∫ Ω 1 m ( x ) + 1 − r ( x ) ∣ u ∣ m ( x ) + 1 λ ( δ ) m ( x ) + 1 d x = 0 .
Let β ( δ ) = δ ∫ Ω ∣ ∇ u ∣ r ( x ) d x ∫ Ω ∣ u ∣ m ( x ) + 1 d x , taking into account (2.6), we obtain

(2.7) J ( λ ( δ ″ ) u ) − J ( λ ( δ ′ ) u ) = ∫ Ω 1 r ( x ) λ ( δ ″ ) r ( x ) ∣ ∇ u ∣ r ( x ) − 1 r ( x ) λ ( δ ′ ) r ( x ) ∣ ∇ u ∣ r ( x ) d x − ∫ Ω 1 m ( x ) + 1 λ ( δ ″ ) m ( x ) + 1 ∣ u ∣ m ( x ) + 1 − 1 m ( x ) + 1 λ ( δ ′ ) m ( x ) + 1 ∣ u ∣ m ( x ) + 1 d x = ∫ Ω 1 r ( x ) ∣ ∇ u ∣ r ( x ) ∫ β ( δ ′ ) β ( δ ″ ) d d β β ( δ ) r ( x ) m ( x ) + 1 − r ( x ) d β d x − ∫ Ω 1 m ( x ) + 1 ∣ u ∣ m ( x ) + 1 ∫ β ( δ ′ ) β ( δ ″ ) d d β β ( δ ) m ( x ) + 1 m ( x ) + 1 − r ( x ) d β d x = ∫ β ( δ ′ ) β ( δ ″ ) ∫ Ω 1 r ( x ) ∣ ∇ u ∣ r ( x ) r ( x ) m ( x ) + 1 − r ( x ) β ( δ ) r ( x ) m ( x ) + 1 − r ( x ) − 1 d x d β − ∫ β ( δ ′ ) β ( δ ″ ) ∫ Ω 1 m ( x ) + 1 ∣ u ∣ m ( x ) + 1 m ( x ) + 1 m ( x ) + 1 − r ( x ) β ( δ ) m ( x ) + 1 m ( x ) + 1 − r ( x ) − 1 d x d β = ∫ β ( δ ′ ) β ( δ ″ ) β ( δ ) − 1 ( 1 − δ ) ∫ Ω ∣ ∇ u ∣ r ( x ) 1 m ( x ) + 1 − r ( x ) β ( δ ) r ( x ) m ( x ) + 1 − r ( x ) d x d β .

From the definition of β ( δ ) , it is easy to know that β ( δ ) is increasing with respect to δ . Hence, by (2.7), we obtain J ( λ ( δ ″ ) u ) − J ( λ ( δ ′ ) u ) > 0 .

Similarly, we obtain J ( λ ( δ ″ ) u ) − J ( λ ( δ ′ ) u ) > 0 , for 1 < δ ″ < δ ′ < b .□

By Lemma 2.7, define d 0 = lim δ → 0 + d ( δ ) ≥ 0 .

Lemma 2.8

Let d 0 < J ( u ) < d , u ∈ W N 1 , r ( x ) , and δ 1 < 1 < δ 2 , δ 1 and δ 2 are the two roots of equation d ( δ ) = J ( u ) , then the sign of I δ ( u ) is invariable in δ 1 < δ < δ 2 .

Similar to [9, Proposition 2.1] and [6, Proposition 3.7], we have the following proposition.

Proposition 2.1

Assume that u is the weak solution of (1.1), u 0 ∈ W N 1 , r ( x ) and J ( u 0 ) = σ . Then, we have

If 0 < σ ≤ d 0 , there exists a unique δ ¯ ∈ ( 1 , b ) that satisfies d ( δ ¯ ) = σ , where b is the constant in Lemma 2.7(ii). Furthermore, if I ( u 0 ) > 0 , then for any 1 ≤ δ < δ ¯ , there is u ∈ W δ . Otherwise, if I ( u 0 ) < 0 , then for any 1 ≤ δ < δ ¯ , there is u ∈ V δ ;
If d 0 < σ < d , then δ 1 and δ 2 satisfy δ 1 < 1 < δ 2 and d ( δ 1 ) = d ( δ 2 ) = σ . Furthermore, if I ( u 0 ) > 0 , then for any δ 1 < δ < δ 2 , there is u ∈ W δ . Otherwise, if I ( u 0 ) < 0 , then for any δ 1 < δ < δ 2 , there is u ∈ V δ .

Lemma 2.9

[Estimate of nonlinear term ∣ u ∣ m ( x ) − 1 u ] Let m ( x ) satisfy the conditions given by (1.1), ∣ u 1 ∣ + ∣ u 2 ∣ > 0 and u 1 ≠ u 2 for any u 1 ( x , t ) , u 2 ( x , t ) with ( x , t ) ∈ Ω × [ 0 , T ] , then

∣ u 1 ∣ m ( x ) − 1 u 1 − ∣ u 2 ∣ m ( x ) − 1 u 2 ≤ m ( x ) ( ∣ u 1 ∣ + ∣ u 2 ∣ ) m ( x ) − 1 ∣ u 1 − u 2 ∣ .

Proof

Define F ( u ) = ∣ u ∣ m ( x ) − 1 u and κ ( x , t ) = u 1 − u 2 for ( x , t ) ∈ Ω × [ 0 , T ] . Due to the property of G a ˆ teaux derivative, we find

F ( u 1 ) − F ( u 2 ) = ∫ 0 1 d F ( u 2 + s κ ; κ ) d s = ∫ 0 1 lim τ → 0 F ( u 2 + s κ + τ κ ) − F ( u 2 + s κ ) τ d s = ∫ 0 1 d d τ F ( u 2 + s κ ; κ ) τ = 0 d s = ∫ 0 1 d d τ ( ∣ u 2 + s κ + τ κ ∣ m ( x ) − 1 ( u 2 + s κ + τ κ ) ) ∣ τ = 0 d s = ∫ 0 1 m ( x ) ∣ u 2 + s κ + τ κ ∣ m ( x ) − 1 κ ∣ τ = 0 d s = ∫ 0 1 ∣ u 2 + s κ ∣ m ( x ) − 1 m ( x ) κ d s = ∫ 0 1 ∣ s u 1 + ( 1 − s ) u 2 ∣ m ( x ) − 1 m ( x ) ( u 1 − u 2 ) d s ≤ ∫ 0 1 ( ∣ u 1 ∣ + ∣ u 2 ∣ ) m ( x ) − 1 m ( x ) ∣ u 1 − u 2 ∣ d s ≤ ( ∣ u 1 ∣ + ∣ u 2 ∣ ) m ( x ) − 1 m ( x ) ∣ u 1 − u 2 ∣ ,

where s ∈ ( 0 , 1 ) .□

Our main results on local existence and uniqueness as well as global existence, uniqueness, asymptotic behavior, and blowup are stated in the following four theorems.

Theorem 2.1

[Local existence and uniqueness] Let u 0 ∈ W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) and m ( x ) ≤ n + 2 n − 2 , then there exist T > 0 and a unique solution of (1.1) over [ 0 , T ] .

Theorem 2.2

[Global existence and uniqueness] Assume that u 0 ∈ W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) .

If J ( u 0 ) < d and I ( u 0 ) > 0 , then Problem (1.1) has a global weak solution u ∈ L ∞ ( 0 , ∞ ; W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) ) with u t ∈ L 2 ( 0 , ∞ ; W N 1 , 2 ( Ω ) ) ; and the weak solution is unique for m ( x ) ≤ n + r ( x ) n − r ( x ) ;
If J ( u 0 ) = d and I ( u 0 ) ≥ 0 , then Problem (1.1) admits a global weak solution u ∈ L ∞ ( 0 , ∞ ; W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) ) with u t ∈ L 2 ( 0 , ∞ ; W N 1 , 2 ( Ω ) ) ; and the weak solution is unique for m ( x ) ≤ n + r ( x ) n − r ( x ) ;
If J ( u 0 ) is finite and J ( u 0 ) > d , I ( u 0 ) > 0 and ∥ u 0 ∥ H 1 2 ≤ λ J ( u 0 ) , then Problem (1.1) has a global weak solution u ∈ L ∞ ( 0 , ∞ ; W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) ) with u t ∈ L 2 ( 0 , ∞ ; W N 1 , 2 ( Ω ) ) ; and the weak solution is unique for m ( x ) ≤ n + r ( x ) n − r ( x ) .

Theorem 2.3

[Asymptotic behavior] Let u ( x , t ) be the global bounded weak solution in Theorem 2.2.

If J ( u 0 ) < M , I ( u 0 ) > 0 and ∥ ∇ u ∥ 2 2 ≤ C ∫ Ω ∣ ∇ u ∣ r ( x ) d x , then there exists a constant δ > 0 such that ∥ u ∥ H 1 2 < ∥ u 0 ∥ H 1 2 e − 2 ψ t for t ∈ [ 0 , ∞ ) ;
If J ( u 0 ) = M , I ( u 0 ) > 0 and ∥ ∇ u ∥ 2 2 ≤ C ∫ Ω ∣ ∇ u ∣ r ( x ) d x , then there exist constants t 1 > 0 and ς > 0 such that ∥ u ∥ H 1 2 < ∥ u ( t 1 ) ∥ H 1 2 e − 2 ς ( t − t 1 ) for t ∈ ( t 1 , + ∞ ) ;
If J ( u 0 ) is finite and J ( u 0 ) > d , I ( u 0 ) > 0 and ∥ u 0 ∥ H 1 2 ≤ λ J ( u 0 ) , then u ( t ) → 0 as t → + ∞ .

Theorem 2.4

[Blowup] Let u 0 ∈ W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) and m ( x ) ≤ n + 2 n − 2 .

If J ( u 0 ) < M and I ( u 0 ) < 0 , then the weak solution u ( t ) of Problem (1.1) blows up in finite-time;
If J ( u 0 ) = M and I ( u 0 ) < 0 , then the weak solution u ( t ) of Problem (1.1) blows up in finite-time.

Remark 2.2

In Theorem 2.3, we present the following examples to ensure that the assumption ∥ ∇ u ∥ 2 2 ≤ C ∫ Ω ∣ ∇ u ∣ r ( x ) d x holds. For example,

if r − ≥ 2 and ∥ ∇ u ∥ r ( x ) ≥ 1 , by Lemmas 2.1 and 2.2, we have
∥ ∇ u ∥ 2 2 ≤ C ∥ ∇ u ∥ r ( x ) 2 ≤ C ∫ Ω ∣ ∇ u ∣ r ( x ) d x 2 r − ≤ C ∫ Ω ∣ ∇ u ∣ r ( x ) d x ;
if 1 < r ( x ) < 2 and ∣ ∇ u ∣ < 1 , it is easy to know that ∥ ∇ u ∥ 2 2 ≤ C ∫ Ω ∣ ∇ u ∣ r ( x ) d x .

3 Local solutions

In this section, by the standard Galerkin method, we prove the local existence and uniqueness of solutions for Problem (1.1). We refer the interested reader to [9] for similar proof of the existence and uniqueness of weak solutions.

Proof of Theorem 2.1

There are two steps to prove the theorem. The first step is to prove the existence and uniqueness of weak solutions of the following Problem (3.1) corresponding to Problem (1.1) by the Galerkin method. Based on the first step, the second step is to prove the existence and uniqueness of local solutions to Problem (1.1) by the contraction mapping principle.

Step I: Consider space ℋ = C ( [ 0 , T ] ; W N 1 , 2 ( Ω ) ) for every T > 0 , and define the norm on ℋ as follows:

∥ u ∥ ℋ 2 = max t ∈ [ 0 , T ] ∥ ∇ u ∥ 2 2 .

Next, for every T > 0 and u ∈ ℋ , we shall prove that there is a unique v ∈ ℋ satisfying

(3.1) v t − div ( ∣ ∇ v ∣ r ( x ) − 2 ∇ v ) − Δ v t = f ( u ) , in Ω × ( 0 , T ) , ∂ v ∂ ν ( x , t ) = 0 , on ∂ Ω × ( 0 , T ) , v ( x , 0 ) = u 0 ( x ) ≢ 0 , ⨍ Ω v 0 d x = 0 , in Ω ,

where f ( u ) = ∣ u ∣ m ( x ) − 1 u − ⨍ Ω ∣ u ∣ m ( x ) − 1 u d x , m ( x ) ≤ n + 2 n − 2 .

Existence. Consider the normalized eigenfunctions { ω j ( x ) } j ∈ N of the Laplace operator in W N 1 , 2 ( Ω ) satisfying

− Δ ω j = λ j ω j , x ∈ Ω , ∂ ω j ∂ ν = 0 , x ∈ ∂ Ω , j = 1 , 2 , … ,

where λ j ( j = 1 , 2 , … ) are the characteristic values. By [47,30,48], we have that ( ω k , ω j ) W N 1 , 2 ( Ω ) = ( 1 + λ j ) ( ω k , ω j ) L 2 ( Ω ) , which yields that { ω j ( x ) } j ∈ N are orthogonal in W N 1 , 2 ( Ω ) and in L 2 ( Ω ) . Construct the following approximation functions:

v m ( x , t ) = ∑ j = 1 m g j m ( t ) ω j ( x ) , m = 1 , 2 , … ,

where g j m ( t ) satisfies the initial value problem of the following equations:

(3.2) ( v m t , ω s ) + ( ∇ v m t , ∇ ω s ) + ( ∣ ∇ v m ∣ r ( x ) − 2 ∇ v m , ∇ ω s ) = ( f ( u ) , ω s ) ,

(3.3) u m ( x , 0 ) = ∑ j = 1 m a j m ω j ( x ) → u 0 ( x ) in W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) ,

for 1 ≤ s ≤ m , in which

a j m = g j m ( 0 ) and f ( u ) = ∣ u ∣ m ( x ) − 1 u − ⨍ Ω ∣ u ∣ m ( x ) − 1 u d x .

From (3.2) and (3.3), we have

(3.4) ( 1 + λ s ) d d t g s m ( t ) = f ∗ ( g 1 m , g 2 m , … , g m m ( t ) , u ( t ) ) , g s m ( 0 ) = ( u 0 , ω s ) ,

where f ∗ ( g 1 m , g 2 m , … , g m m ( t ) , u ( t ) ) = − ( ∣ ∇ v m ∣ r ( x ) − 2 ∇ v m , ∇ ω s ) + ( ∣ u ∣ m ( x ) − 1 u , ω s ) − ⨍ Ω ∣ u ∣ m ( x ) − 1 u d x , ω s . Because of ( ∣ u ∣ m ( x ) − 1 u , ω s ) − ⨍ Ω ∣ u ∣ m ( x ) − 1 u d x , ω s ∈ C ( 0 , T ) and g j m ( t ) ∈ C 1 ( 0 , T ) for all j , by standard existence theory for ordinary differential equations, the initial value Problem (3.4) admits a local solution.

Multiplying the j th equation of (3.2) by g ′ j m ( t ) , summing up with respect to j , we obtain

(3.5) 1 2 d d t ∥ v m ∥ H 1 2 + ∫ Ω ∣ ∇ v ∣ r ( x ) d x − ( ∣ u ∣ m ( x ) − 1 u , v m ) = 0 .

Consider the third term on the left-hand side of the aforementioned equation, we have

(3.6) ( ∣ u ∣ m ( x ) − 1 u , v m ) ≤ ∫ Ω ∣ u ∣ m ( x ) ∣ v m ∣ d x ≤ ∫ Ω ∣ u ∣ 2 n m ( x ) n + 2 d x n + 2 2 n ∫ Ω ∣ v m ∣ 2 n n − 2 d x n − 2 2 n ≤ max ∥ u ∥ 2 n m ( x ) n + 2 2 n m − n + 2 , ∥ u ∥ 2 n m ( x ) n + 2 2 n m + n + 2 n + 2 2 n ∥ v m ∥ 2 n n − 2 ≤ C ∥ ∇ v m ∥ 2 ≤ C + 1 2 ∥ ∇ v m ∥ 2 2 .

By (3.5) and (3.6), we obtain

(3.7) d d t ∥ v m ∥ H 1 2 ≤ C + ∥ ∇ v m ∥ 2 2 .

Then, combining Gronwall’s inequality and (3.7), we have

(3.8) ∥ v m ∥ H 1 2 ≤ C .

Multiplying the j th equation of (3.2) by g j m ′ ( t ) , summing on j , and integrating with respect to the time variable from 0 to t , we obtain

(3.9) ∫ 0 t ∥ v m τ ∥ H 1 2 d τ + ∫ Ω 1 r ( x ) ∣ ∇ v m ∣ r ( x ) d x = ∫ 0 t ( ∣ u ∣ m ( x ) − 1 u , v m τ ) d τ + ∫ Ω 1 r ( x ) ∣ ∇ v m ( 0 ) ∣ r ( x ) d x , 0 ≤ t ≤ T .

Similar to (3.6), we obtain

(3.10) ∫ 0 t ( ∣ u ∣ m ( x ) − 1 u , v m τ ) d τ ≤ C T + 1 2 ∫ 0 t ∥ v m τ ∥ H 1 2 d τ .

Combining (3.9) and (3.10) gives

(3.11) ∫ 0 t ∥ v m τ ∥ H 1 2 d τ + ∫ Ω ∣ ∇ v m ∣ r ( x ) d x ≤ C T .

Hence, from (3.8) and (3.11), there exist a v and a subsequence { v k } of { v m } such that

v k ⇀ v in L ∞ ( [ 0 , T ] ; W N 1 , r ( x ) ∩ H 1 ( Ω ) ) weak star , v k t ⇀ v t in L 2 ( [ 0 , T ] ; H 1 ( Ω ) ) weakly .

Similar to the proof in [6, Theorem 2.3], letting m = k → ∞ in (3.2), we obtain the existence of a weak solution v of (3.1) with the aforementioned regularity. In addition, we have v ∈ C ( [ 0 , T ] ; H 1 ( Ω ) ) by the Aubin-Lions lemma.

Uniqueness. Let v 1 and v 2 be two weak solutions of Problem (3.1) with the same initial value condition. By subtracting the two equations corresponding to v 1 and v 2 , respectively, and testing it with v 1 − v 2 , we obtain

∫ 0 t ∫ Ω [ ( v 1 τ − v 2 τ ) ( v 1 − v 2 ) + ∇ ( v 1 τ − v 2 τ ) ∇ ( v 1 − v 2 ) ] d x d τ + ∫ 0 t ∫ Ω ( ∣ ∇ v 1 ∣ r ( x ) − 2 ∇ v 1 − ∣ ∇ v 2 ∣ r ( x ) − 2 ∇ v 2 ) ∇ ( v 1 − v 2 ) d x d τ = 0 ,

which implies

(3.12) 1 2 ∥ v 1 − v 2 ∥ H 1 2 + ∫ 0 t ∫ Ω ( r ( x ) − 1 ) ∣ ∇ ( θ 1 v 1 + ( 1 − θ 1 ) v 2 ) ∣ r ( x ) − 2 ∣ ∇ ( v 1 − v 2 ) ∣ 2 d x d τ = 0 ,

where θ 1 ∈ [ 0 , 1 ] . From (3.12), it is easy to know that v 1 ≡ v 2 .

Step II: Let R 2 = 2 ∥ ∇ u 0 ∥ 2 2 , for any T > 0 , consider the set:

X T = { u ∈ ℋ ∣ ∥ u ∥ ℋ ≤ R } .

According to Step I, for any u ∈ X T and the unique solution v ∈ ℋ of Problem (3.1), we can define v = Λ ( u ) . We claim that Λ ( X T ) ⊂ X T is a contractive map.

For given u ∈ X T , similar to (3.5), we have
(3.13) 1 2 d d t ‖ v ‖ H 1 2 + ∫ Ω ∣ ∇ v ∣ r ( x ) d x − ( ∣ u ∣ m ( x ) − 1 u , v ) = 0 .
Similar to (3.6), we obtain
(3.14) ( ∣ u ∣ m ( x ) − 1 u , v ) ≤ C max R 2 n m − n + 2 , R 2 n m + n + 2 n + 2 n + 1 2 ‖ ∇ v ‖ 2 2 .
Combining (3.13) and (3.14), from Gronwall’s inequality, we have
∥ v ∥ ℋ 2 ≤ e T 1 2 R 2 + C T max R 2 n m − n + 2 , R 2 n m + n + 2 n + 2 n .
Choose a sufficiently small T such that ∥ v ∥ ℋ ≤ R , then Λ ( X T ) ⊂ X T .
Now, we need to prove that such map is contractive. Taking z 1 , z 2 ∈ X T to be the known functions in the right-hand term of (3.1), respectively, and subtracting the two equations in form of (3.1) for v 1 = Λ ( z 1 ) and v 2 = Λ ( z 2 ) , respectively, from (i), we know v 1 , v 2 ∈ X T . Setting v ^ = v 1 − v 2 and testing the both sides by v ^ , we have
1 2 ∥ v ^ ∥ H 1 2 + ∫ 0 t ∫ Ω ( ∣ ∇ v 1 ∣ r ( x ) − 2 ∇ v 1 − ∣ ∇ v 2 ∣ r ( x ) − 2 ∇ v 2 ) ∇ ( v 1 − v 2 ) d x d τ = ∫ 0 t ( ∣ z 1 ∣ m ( x ) − 1 z 1 − ∣ z 2 ∣ m ( x ) − 1 z 2 , v ^ ) d τ ,
which implies
(3.15) 1 2 ∥ v ^ ∥ H 1 2 + ∫ 0 t ∫ Ω ( r ( x ) − 1 ) ∣ ∇ ( θ 1 v 1 + ( 1 − θ 1 ) v 2 ) ∣ r ( x ) − 2 ∣ ∇ ( v 1 − v 2 ) ∣ 2 d x d τ = ∫ 0 t ( ∣ z 1 ∣ m ( x ) − 1 z 1 − ∣ z 2 ∣ m ( x ) − 1 z 2 , v ^ ) d τ ,
where θ 1 ∈ [ 0 , 1 ] . From Lemma 2.9, we obtain
(3.16) ∫ 0 t ∫ Ω ( ∣ z 1 ∣ m ( x ) − 1 z 1 − ∣ z 2 ∣ m ( x ) − 1 z 2 ) v ^ d x d τ ≤ ∫ 0 t ∫ Ω m ( x ) ( ∣ z 1 ∣ + ∣ z 2 ∣ ) m ( x ) − 1 ∣ z 1 − z 2 ∣ ∣ v ^ ∣ d x d τ ≤ C ∫ 0 t ∫ Ω ( ∣ z 1 ∣ + ∣ z 2 ∣ ) ( m ( x ) − 1 ) A 1 d x A 2 1 A 1 ∥ z 1 − z 2 ∥ ∥ v ^ ∥ A 3 d τ ≤ C ∫ 0 t ( max { ∥ ∣ z 1 ∣ + ∣ z 2 ∣ ∥ ( m ( x ) − 1 ) A 1 ( m − − 1 ) A 1 , ∥ ∣ z 1 ∣ + ∣ z 2 ∣ ∥ ( m ( x ) − 1 ) A 1 ( m + − 1 ) A 1 } ) A 2 1 A 1 ∥ z 1 − z 2 ∥ ∥ v ^ ∥ A 3 d τ ≤ C ( max { R ( m − − 1 ) A 1 , R ( m + − 1 ) A 1 } ) 1 A 1 ∫ 0 t ∥ ∇ ( z 1 − z 2 ) ∥ 2 ∥ ∇ v ^ ∥ 2 d τ ≤ C ( max { R ( m − − 1 ) A 1 , R ( m + − 1 ) A 1 } ) 2 A 1 ∫ 0 t ∥ ∇ ( z 1 − z 2 ) ∥ 2 2 d τ + 1 2 ∫ 0 t ∥ ∇ v ^ ∥ 2 2 d τ ,
where A 1 = n 2 , A 2 = A 3 = 2 n n − 2 by m ( x ) ≤ n + 2 n − 2 .

Considering (3.15) and (3.16), we obtain
∥ v ^ ∥ H 1 2 ≤ C ( max { R ( m − − 1 ) A 1 , R ( m + − 1 ) A 1 } ) 2 A 1 ∫ 0 t ∥ ∇ ( z 1 − z 2 ) ∥ 2 2 d τ + ∫ 0 t ∥ ∇ v ^ ∥ 2 2 d τ ,
which gives
∥ Λ ( z 1 ) − Λ ( z 2 ) ∥ ℋ 2 = ∥ v ^ ∥ ℋ 2 = max t ∈ [ 0 , T ] ∥ ∇ v ^ ∥ 2 2 ≤ max t ∈ [ 0 , T ] ∥ v ^ ∥ H 1 2 ≤ C ( max { R ( m − − 1 ) A 1 , R ( m + − 1 ) A 1 } ) 2 A 1 ∫ 0 t ∥ ∇ ( z 1 − z 2 ) ∥ 2 2 d τ + ∫ 0 t ∥ ∇ v ^ ∥ 2 2 d τ ≤ C ( max { R ( m − − 1 ) A 1 , R ( m + − 1 ) A 1 } ) 2 A 1 T max t ∈ [ 0 , T ] ∥ ∇ ( z 1 − z 2 ) ∥ 2 2 + T max t ∈ [ 0 , T ] ∥ ∇ v ^ ∥ 2 2 ,
i.e.,
∥ Λ ( z 1 ) − Λ ( z 2 ) ∥ ℋ 2 ≤ C ( max { R ( m − − 1 ) A 1 , R ( m + − 1 ) A 1 } ) 2 A 1 T 1 − T max t ∈ [ 0 , T ] ∥ ∇ ( z 1 − z 2 ) ∥ 2 2 = △ δ T ∥ z 1 − z 2 ∥ ℋ 2 ,
for some δ T < 1 as long as T is sufficiently small. Then, the map v = Λ ( u ) is contractive. Therefore, there exists a unique local weak solution to (1.1) defined on [ 0 , T ] by the contraction mapping principle. Theorem 2.1 is proved.□

4 Global existence and uniqueness

In this section, we shall consider the global existence and uniqueness of solutions to Problem (1.1) for the subcritical initial energy, critical initial energy, and supercritical initial energy.

Proof of Theorem 2.2

(i) Existence. Similar to the proof process of Theorem 2.1, Let { ω j ( x ) } be the orthogonal basis of W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) . Denote

u m ( x , t ) = ∑ j = 1 m g j m ( t ) ω j ( x ) , m = 1 , 2 , … ,

where g j m ( t ) satisfies the initial value problem of the following equations:

(4.1) ( u m t , ω s ) + ( ∇ u m t , ∇ ω s ) + ( ∣ ∇ u m ∣ r ( x ) − 2 ∇ u m , ∇ ω s ) = ( f ( u m ) , ω s ) ,

(4.2) u m ( x , 0 ) = ∑ j = 1 m a j m ω j ( x ) → u 0 ( x ) in W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) ,

for 1 ≤ s ≤ m , in which

a j m = g j m ( 0 ) and f ( u ) = ∣ u ∣ m ( x ) − 1 u − ⨍ Ω ∣ u ∣ m ( x ) − 1 u d x .

Multiplying the j th equation of (4.1) by g j m ( t ) and summing up with respect to j , we obtain

(4.3) 1 2 d d t ‖ u m ‖ H 1 2 + I ( u m ) = 0 .

By Proposition 2.1, we obtain u m ( t ) ∈ W , i.e., I ( u m ) > 0 . Then, from (4.3), we have

1 2 d d t ‖ u m ‖ H 1 2 < 0 ,

which, in combination with Gronwall’s inequality, implies that

‖ u m ‖ H 1 2 < C , 0 ≤ t < ∞ .

Multiplying the j th equation of (4.1) by g ′ ( t ) , summing on j , and integrating with respect to the time variable from 0 to t , we have

(4.4) ∫ 0 t ∥ u m τ ∥ H 1 2 d τ + J ( u m ) = J ( u m ( 0 ) ) , 0 ≤ t < ∞ .

By (4.2), we obtain J ( u m ( 0 ) ) → J ( u 0 ) < d and I ( u m ( 0 ) ) → I ( u 0 ) > 0 , which, in combination with (4.4), implies that

(4.5) ∫ 0 t ∥ u m τ ∥ H 1 2 d τ + J ( u m ) < d , 0 ≤ t < ∞ .

Considering (4.5) and

J ( u m ) ≥ 1 r + − 1 m − + 1 ∫ Ω ∣ ∇ u m ∣ r ( x ) d x + 1 m − + 1 I ( u m ) ,

we have

∫ 0 t ∥ u m τ ∥ H 1 2 d τ + 1 r + − 1 m − + 1 ∫ Ω ∣ ∇ u m ∣ r ( x ) d x < d , 0 ≤ t < ∞ ,

which implies that

∫ 0 t ∥ u m τ ∥ H 1 2 d τ < d , 0 ≤ t < ∞ ,

∫ Ω ∣ ∇ u m ∣ r ( x ) d x < r + ( m − + 1 ) m − + 1 − r + d , 0 ≤ t < ∞ ,

∫ Ω ∣ ∣ u m ∣ m ( x ) − 1 u m ∣ m ( x ) + 1 m ( x ) d x = ∫ Ω ∣ u m ∣ m ( x ) + 1 d x ≤ B 1 m + + 1 max r + ( m − + 1 ) m − + 1 − r + d m + + 1 r − , r + ( m − + 1 ) m − + 1 − r + d m − + 1 r + , 0 ≤ t < ∞ .

Then, by Lemma 2.2, there exists a subsequence of { u m } , denoted by the same symbol satisfying

u m ⇀ u in L ∞ ( 0 , ∞ ; W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) ) weak star and a.e. in Q = Ω × ( 0 , ∞ ) , u m t ⇀ u t in L 2 ( 0 , ∞ ; W N 1 , 2 ( Ω ) ) weak , ∣ ∇ u m ∣ r ( x ) − 2 ∇ u m ⇀ χ in L ∞ ( 0 , ∞ ; L r ( x ) r ( x ) − 1 ( Ω ) ) weak star , ∣ u m ∣ m ( x ) − 1 u m ⇀ ∣ u ∣ m ( x ) − 1 u in L ∞ ( 0 , ∞ ; L m ( x ) + 1 m ( x ) ( Ω ) ) weak star .

As in the proof of [6, Theorem 2.3], we can prove χ = ∣ ∇ u ∣ r ( x ) − 2 ∇ u . Let m → ∞ and fix s in (4.1), and we conclude that

( u t , ω s ) + ( ∇ u t , ∇ ω s ) + ( ∣ ∇ u ∣ r ( x ) − 2 ∇ u , ∇ ω s ) = ( f ( u ) , ω s ) , for all s

and

( u t , v ) + ( ∇ u t , ∇ v ) + ( ∣ ∇ u ∣ r ( x ) − 2 ∇ u , ∇ v ) = ( f ( u ) , v )

for all v ∈ W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) , t ∈ ( 0 , ∞ ) . In addition, from u m ∈ L ∞ ( 0 , ∞ ; W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) ) and u m t ∈ L 2 ( 0 , ∞ ; W N 1 , 2 ( Ω ) ) , we obtain u ∈ C ( 0 , ∞ ; W N 1 , 2 ( Ω ) ) . With the help of (4.2), we have u ( x , 0 ) = u 0 ( x ) in W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) .

Uniqueness. Assuming that u 1 and u 2 are two weak solutions of (1.1) with the same initial data. Denote v = u 1 − u 2 , and we have

(4.6) v τ + Δ v τ + div ( ∣ ∇ u 1 ∣ r ( x ) − 2 ∇ u 1 ) − div ( ∣ ∇ u 2 ∣ r ( x ) − 2 ∇ u 2 ) = f ( u 1 ) − f ( u 2 ) ,

u 1 , u 2 , v ∈ L ∞ ( 0 , ∞ ; W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) ) , u 1 t , u 2 t , v t ∈ L 2 ( 0 , ∞ ; W N 1 , 2 ( Ω ) ) ,

where f ( u ) = ∣ u ∣ m ( x ) − 1 u − ⨍ Ω ∣ u ∣ m ( x ) − 1 u d x . Multiplying (4.6) by v , and integrating over ( 0 , t ) × Ω , we obtain

(4.7) ∫ 0 t ∫ Ω ( v τ v + ∇ v τ ∇ v + ( ∣ ∇ u 1 ∣ r ( x ) − 2 ∇ u 1 − ∣ ∇ u 2 ∣ r ( x ) − 2 ∇ u 2 ) ∇ v ) d x d τ = ∫ 0 t ∫ Ω ( ∣ u 1 ∣ m ( x ) − 1 u 1 − ∣ u 2 ∣ m ( x ) − 1 u 2 ) v d x d τ .

By Lemma 2.9, we have

(4.8) ∫ 0 t ∫ Ω ( ∣ u 1 ∣ m ( x ) − 1 u 1 − ∣ u 2 ∣ m ( x ) − 1 u 2 ) v d x d τ ≤ ∫ 0 t ∫ Ω m ( x ) ( ∣ u 1 ∣ + ∣ u 2 ∣ ) m ( x ) − 1 ∣ u 1 − u 2 ∣ ∣ v ∣ d x d τ ≤ C ∫ 0 t ∫ Ω ( ∣ u 1 ∣ + ∣ u 2 ∣ ) ( m ( x ) − 1 ) C 1 d x 1 C 1 ∥ z 1 − z 2 ∥ C 2 ∥ v ∥ C 3 d τ ≤ C ∫ 0 t ( max { ∥ ∣ u 1 ∣ + ∣ u 2 ∣ ∥ ( m ( x ) − 1 ) C 1 ( m − − 1 ) C 1 , ∥ ∣ u 1 ∣ + ∣ u 2 ∣ ∥ ( m ( x ) − 1 ) C 1 ( m + − 1 ) C 1 } ) 1 C 1 ∥ z 1 − z 2 ∥ C 2 ∥ v ∥ C 3 d τ ≤ C ∫ 0 t ∥ u 1 − u 2 ∥ H 1 ∥ v ∥ H 1 d τ ≤ C ∫ 0 t ∥ v ∥ H 1 2 d τ ,

where C 1 = n 2 , C 2 = C 3 = 2 n n − 2 by m ( x ) ≤ n + r ( x ) n − r ( x ) . As v ( x , 0 ) = 0 , we obtain

(4.9) ∫ 0 t ∫ Ω ( v τ v + ∇ v τ ∇ v ) d x d τ = 1 2 ∫ Ω ( v 2 ( τ ) ∣ v ( 0 ) v ( t ) + ∣ ∇ v ( τ ) ∣ 2 ∣ v ( 0 ) v ( t ) ) d x = 1 2 ∫ Ω ( v 2 ( t ) + ∣ ∇ v ( t ) ∣ 2 ) d x = 1 2 ∥ v ( t ) ∥ H 1 2 .

Let v = θ 1 u 1 + ( 1 − θ 1 ) u 2 , θ 1 ∈ [ 0 , 1 ] ; similar to the proof of Lemma 2.9, we have

(4.10) ∫ Ω ( ∣ ∇ u 1 ∣ r ( x ) − 2 ∇ u 1 − ∣ ∇ u 2 ∣ r ( x ) − 2 ∇ u 2 ) ∇ v d x = ∫ Ω ( r ( x ) − 1 ) ∣ ∇ v ∣ 2 ∫ 0 1 ∣ ∇ v − ∣ r ( x ) − 2 d θ 1 d x .

Then, it follows from (4.7)–(4.10) that

∥ v ∥ H 1 2 ≤ C ∫ 0 t ∥ v ∥ H 1 2 d τ .

From Gronwall’s inequality, we have ∥ v ∥ H 1 2 ≤ 0 , i.e. ∥ v ∥ H 1 2 = ∥ u 1 − u 2 ∥ H 1 2 = 0 . Then, u 1 = u 2 = 0 a.e. in Ω × ( 0 , ∞ ) .

(ii) From the condition J ( u 0 ) = d , we obtain ∥ u 0 ∥ W 1 , r ( x ) ≠ 0 . Denote μ s = 1 − 1 s and u s 0 = μ s u 0 for s = 2 , 3 , … . Consider Problem (1.1) with the condition:

(4.11) u ( x , 0 ) = u s 0 ( x ) , s = 2 , 3 , … .

By Lemma 2.5 and the initial data I ( u 0 ) ≥ 0 , we obtain λ ∗ ≥ 1 . Furthermore, we obtain I ( u s 0 ) = I ( μ s u 0 ) > 0 and J ( u s 0 ) = J ( μ s u 0 ) < J ( u 0 ) = d . Due to (i), Problem (1.1) with the initial condition (4.11) has a unique global solution u s ( t ) ∈ L ∞ ( 0 , ∞ ; W N 1 , r ( x ) ( Ω ) ∩ H 1 ( Ω ) ) with u s t ( t ) ∈ L 2 ( 0 , ∞ ; W N 1 , 2 ( Ω ) ) for each s = 2 , 3 , … . From Proposition 2.1, we know that u s ( x , t ) ∈ W . Similar to the proof process of (i), we obtain

∥ u s ∥ H 1 2 < C , 0 ≤ t < ∞ ,

∫ 0 t ∥ u s τ ∥ H 1 2 d τ < d , 0 ≤ t < ∞ ,

∫ Ω ∣ ∇ u s ∣ r ( x ) d x < r + ( m − + 1 ) m − + 1 − r + d , 0 ≤ t < ∞ ,

and

∫ Ω ∣ ∣ u s ∣ m ( x ) − 1 u s ∣ m ( x ) + 1 m ( x ) d x = ∫ Ω ∣ u s ∣ m ( x ) + 1 d x ≤ B 1 m + + 1 max r + ( m − + 1 ) m − + 1 − r + d m + + 1 r − , r + ( m − + 1 ) m − + 1 − r + d m − + 1 r + , 0 ≤ t < ∞ .

The rest is proved similar to (i).

(iii) Consider the basis u m ( x , t ) as (i); we claim that u m ∈ N + .

Assuming that the assertion is not tenable, there exists a t 0 > 0 such that u m ∈ N + for t ∈ ( 0 , t 0 ) and u m ( t 0 ) ∈ N . Letting φ = u m in (2.3) and

∫ 0 t ∫ Ω ⨍ Ω ∣ u m ∣ m ( x ) − 1 u m d x u m d x d τ = ∫ 0 t ⨍ Ω ∣ u m ∣ m ( x ) − 1 u m d x ∫ Ω u m d x d τ = 0 ,

we find

1 2 ∥ u m ∥ H 1 2 − 1 2 ∥ u m ( 0 ) ∥ H 1 2 + ∫ 0 t ∫ Ω ∣ ∇ u m ∣ r ( x ) d x − ∫ Ω ∣ u m ∣ m ( x ) + 1 d x d s = 0 .

Then,

(4.12) 1 2 d d t ∥ u m ∥ H 1 2 = − I ( u m ) ,

which yields boundedness of ∥ u m ∥ H 1 2 and ∫ 0 t ∥ u m τ ∥ H 1 2 d τ ≠ 0 for Ω × ( 0 , t 0 ) . Furthermore, by (4.4), we have u m ( t 0 ) ∈ J J ( u m ( 0 ) ) . Hence, u m ( t 0 ) ∈ N J ( u m ( 0 ) ) . Due to the definition of λ J ( u 0 ) , we obtain

(4.13) ∥ u m ( t 0 ) ∥ H 1 2 ≥ λ J ( u 0 ) .

With the help of (4.12), by the fact I ( u m ) > 0 , we have

∥ u m ( t 0 ) ∥ H 1 2 < ∥ u m ( x , 0 ) ∥ H 1 2 ≤ λ J ( u 0 ) ,

which contradicts (4.13). Then, for all t 0 > 0 , u m ∈ N + .

Furthermore, we can obtain u m ( t ) ∈ J J ( u m ( 0 ) ) ∩ N + . In addition, (4.4) implies

J ( u m ( 0 ) ) ≥ J ( u m ( t ) ) ≥ 1 r + − 1 m − + 1 ∫ Ω ∣ ∇ u m ∣ r ( x ) d x + 1 m − + 1 I ( u m )

and

J ( u m ( 0 ) ) ≥ ∫ 0 t ‖ u m τ ‖ H 1 2 d τ ,

which imply the boundedness of ∫ Ω ∣ ∇ u m ∣ r ( x ) d x and ∫ 0 t ∥ u m τ ∥ H 1 2 d τ . As in the proof of (i), then Problem (1.1) has a unique global weak solution u . Theorem 2.2 is proved.□

5 Asymptotic behavior

In this section, we shall prove the asymptotic behavior of solutions to Problem (1.1) for the sub-sharp-critical initial energy, sharp-critical initial energy, and supercritical initial energy.

Proof of Theorem 2.3

From Proposition 2.1, we have u ∈ W δ for 1 ≤ δ < δ ¯ or δ 1 < δ < δ 2 with δ 1 < 1 < δ 2 , and particularly I ( u ) > 0 . Then, the norm ∥ u ∥ H 1 is equivalent to the norm ∥ ∇ u ∥ 2 on H 1 ( Ω ) ; from (4.12), we obtain
(5.1) 1 2 d d t ∥ u ∥ H 1 2 = − I ( u ) = − ∫ Ω ∣ ∇ u ∣ r ( x ) d x + ∫ Ω ∣ u ∣ m ( x ) + 1 d x ≤ − ∫ Ω ∣ ∇ u ∣ r ( x ) d x + B 1 m + + 1 max ∫ Ω ∣ ∇ u ∣ r ( x ) d x m + + 1 r − , ∫ Ω ∣ ∇ u ∣ r ( x ) d x m − + 1 r + = B 1 m + + 1 max ∫ Ω ∣ ∇ u ∣ r ( x ) d x m + + 1 − r − r − , ∫ Ω ∣ ∇ u ∣ r ( x ) d x m − + 1 − r + r + − 1 ∫ Ω ∣ ∇ u ∣ r ( x ) d x ≤ C B 1 m + + 1 max ∫ Ω ∣ ∇ u ∣ r ( x ) d x m + + 1 − r − r − , ∫ Ω ∣ ∇ u ∣ r ( x ) d x m − + 1 − r + r + − 1 ∥ ∇ u ∥ 2 2 ≤ C B 1 m + + 1 max ∫ Ω ∣ ∇ u ∣ r ( x ) d x m + + 1 − r − r − , ∫ Ω ∣ ∇ u ∣ r ( x ) d x m − + 1 − r + r + − 1 ∥ u ∥ H 1 2 .

Now, we shall show that
(5.2) B 1 m + + 1 max ∫ Ω ∣ ∇ u ∣ r ( x ) d x m + + 1 − r − r − , ∫ Ω ∣ ∇ u ∣ r ( x ) d x m − + 1 − r + r + − 1 < 0 .
Considering (2.1) and I ( u ) > 0 , we obtain
J ( u ) > 1 r + − 1 m − + 1 ∫ Ω ∣ ∇ u ∣ r ( x ) d x ,
i.e.,
(5.3) ∫ Ω ∣ ∇ u ∣ r ( x ) d x < ( m − + 1 ) r + m − + 1 − r + J ( u ) ≤ ( m − + 1 ) r + m − + 1 − r + J ( u 0 ) .
By (5.3), we see that
(5.4) B 1 m + + 1 ∫ Ω ∣ ∇ u ∣ r ( x ) d x m − + 1 − r + r + < B 1 ( m + + 1 ) r + m − + 1 − r + r + ( m − + 1 ) ( m − + 1 − r + ) J ( u 0 ) m − + 1 − r + r + < 1 ,
which implies ∫ Ω ∣ ∇ u ∣ r ( x ) d x < 1 , i.e.,
(5.5) max ∫ Ω ∣ ∇ u ∣ r ( x ) d x m + + 1 − r − r − , ∫ Ω ∣ ∇ u ∣ r ( x ) d x m − + 1 − r + r + = ∫ Ω ∣ ∇ u ∣ r ( x ) d x m − + 1 − r + r + .
Hence, from (5.4) and (5.5), we obtain (5.2).

Considering (5.4), there exists ψ > 0 such that
(5.6) − ψ ≔ C B 1 ( m + + 1 ) r + m − + 1 − r + r + ( m − + 1 ) ( m − + 1 − r + ) J ( u 0 ) m − + 1 − r + r + − 1 .
From (5.1)–(5.6), it follows that
1 2 d d t ∥ u ∥ H 1 2 < − ψ ∥ u ∥ H 1 2 .
By Gronwall’s inequality, for ψ > 0 , we deduce
∥ u ∥ H 1 2 < ∥ u 0 ∥ H 1 2 e − 2 ψ t , 0 ≤ t < ∞ .
The existence of the global solution u ( t ) of (1.1) is proved; then, we claim that u ∈ W for t > 0 .

Arguing by contradiction, assuming t 0 > 0 is the first time such that I ( u ( t 0 ) ) = 0 . By the definition of M , we have J ( u ( t 0 ) ) ≥ M . But
(5.7) 0 < J ( u ( t 0 ) ) = M − ∫ 0 t 0 ∥ u τ ∥ H 1 2 d τ = d 1 ≤ M ,
for any t 0 > 0 . Then, we obtain
(5.8) J ( u ( t 0 ) ) = M .
Considering (5.7) and (5.8), we obtain
∫ 0 t 0 ∥ u τ ∥ H 1 2 d τ = 0 ,
i.e., u t ≡ 0 for 0 ≤ t ≤ t 0 , which contradicts I ( u 0 ) > 0 (from (4.12)). Then, we deduce that u ∈ W for 0 < t < ∞ .

Assuming that t 1 > 0 is the initial time, for t > t 1 , we have u ( x , t ) ∈ W . Then, from (5.1)–(5.4), we can see that there exists a constant ς > 0 as
− ς ≔ C B 1 ( m + + 1 ) r + m − + 1 − r + r + ( m − + 1 ) ( m − + 1 − r + ) J ( u 1 ) m − + 1 − r + r + − 1 ,
such that
∥ u ∥ H 1 2 < ∥ u ( t 1 ) ∥ H 1 2 e − 2 ς ( t − t 1 ) , t ∈ ( t 1 , + ∞ ) .
We denote by ω ( u 0 ) = ∩ t ≥ 0 { u ( s ) : s ≥ t } ¯ the ω -limit of u . From (4.4) and (4.12), we obtain
∥ ω ∥ H 1 2 < ∥ u 0 ∥ H 1 2 ≤ λ J ( u 0 ) , J ( ω ) ≤ J ( u 0 ) ,
for all ω ∈ ω ( u 0 ) , which implies that ω ∉ N J ( u 0 ) and ω ∈ J J ( u 0 ) , i.e., ω ∉ N . Therefore, ω ( u 0 ) ∩ N = ∅ , which means ω ( u 0 ) = { 0 } . Then, u ( t ) → 0 as t → + ∞ .

Theorem 2.3 is proved.□

6 Blowup

In this section, we shall discuss the blowup of weak solutions in finite-time with the sub-sharp-critical initial energy and sharp-critical initial energy.

Lemma 6.1

Let u ( t ) be a weak solution of (1.1) with u 0 ∈ V and J ( u 0 ) < d , then

∫ Ω ∣ ∇ u ∣ r ( x ) d x > r + ( m − + 1 ) m − + 1 − r + M .

Proof

Let u ( t ) be the weak solution of (1.1) with J ( u 0 ) < d and I ( u 0 ) < 0 and T be the maximal existence time. By Proposition 2.1, we can see that u ( x , t ) ∈ V δ , i.e., I ( u ) < 0 for 0 < t < T . Recalling Lemma 2.4 (ii), take δ = 1 , and we obtain

∫ Ω ∣ ∇ u ∣ r ( x ) d x > 1 B 1 m + + 1 r + m − + 1 − r + .

From Lemma 2.6, we have

□ M < m − + 1 − r + r + ( m − + 1 ) ∫ Ω ∣ ∇ u ∣ r ( x ) d x .

Proof of Theorem 2.4

We shall prove u ( t ) blowing up in finite-time with u 0 ∈ V .

Arguing by contradiction, assume that the solution is global in time. By Proposition 2.1 and u 0 ∈ V , we obtain u ∈ V δ for t ∈ [ 0 , + ∞ ) . Define
G ( t ) = ∫ 0 t ∥ u ∥ H 1 2 d τ + ( T 0 − t ) ∥ u 0 ∥ H 1 2 , t ∈ [ 0 , T 0 ] ,
where 0 < T 0 < + ∞ . Obviously, we have G ( t ) > 0 for any t ∈ [ 0 , T 0 ] . From the continuity of G ( t ) with respect to t , we deduce that there exists a constant θ > 0 , such that G ( t ) ≥ θ for t ∈ [ 0 , T 0 ] . Then,
(6.1) G ′ ( t ) = ∥ u ∥ H 1 2 − ∥ u 0 ∥ H 1 2 = ∥ u ∥ 2 2 + ∥ ∇ u ∥ 2 2 − ∥ u 0 ∥ 2 2 − ∥ ∇ u 0 ∥ 2 2 = 2 ∫ 0 t ( u , u τ ) d τ + 2 ∫ 0 t ( ∇ u , ∇ u τ ) d τ ,
and (4.12) gives
(6.2) G ″ ( t ) = 2 ( u , u t ) + 2 ( ∇ u , ∇ u t ) = − 2 I ( u ) .
By (6.1) and applying the Cauchy-Schwarz inequality, we have
(6.3) ( G ′ ( t ) ) 2 = 4 ∫ 0 t ( u , u τ ) d τ + ∫ 0 t ( ∇ u , ∇ u τ ) d τ 2 = 4 ∫ 0 t ( u , u τ ) d τ 2 + ∫ 0 t ( ∇ u , ∇ u τ ) d τ 2 + 2 ∫ 0 t ( u , u τ ) d τ ∫ 0 t ( ∇ u , ∇ u τ ) d τ ≤ 4 ∫ 0 t ∥ u ∥ 2 2 d τ ∫ 0 t ∥ u τ ∥ 2 2 d τ + ∫ 0 t ∥ ∇ u ∥ 2 2 d τ ∫ 0 t ∥ ∇ u τ ∥ 2 2 d τ + 2 ∫ 0 t ∥ u ∥ 2 2 d τ 1 2 ∫ 0 t ∥ u τ ∥ 2 2 d τ 1 2 ∫ 0 t ∥ ∇ u ∥ 2 2 d τ 1 2 ∫ 0 t ∥ ∇ u τ ∥ 2 2 d τ 1 2 ≤ 4 ∫ 0 t ∥ u ∥ 2 2 d τ ∫ 0 t ∥ u τ ∥ 2 2 d τ + ∫ 0 t ∥ ∇ u ∥ 2 2 d τ ∫ 0 t ∥ ∇ u τ ∥ 2 2 d τ + ∫ 0 t ∥ u ∥ 2 2 d τ ∫ 0 t ∥ ∇ u τ ∥ 2 2 d τ + ∫ 0 t ∥ ∇ u ∥ 2 2 d τ ∫ 0 t ∥ u τ ∥ 2 2 d τ ≤ 4 ∫ 0 t ∥ u ∥ 2 2 d τ + ∫ 0 t ∥ ∇ u ∥ 2 2 d τ ∫ 0 t ∥ u τ ∥ 2 2 d τ + ∫ 0 t ∥ ∇ u τ ∥ 2 2 d τ ≤ 4 ∫ 0 t ∥ u ∥ H 1 2 d τ ∫ 0 t ∥ u τ ∥ H 1 2 d τ ≤ 4 G ( t ) ∫ 0 t ∥ u τ ∥ H 1 2 d τ .
Considering (6.2) and (6.3), it is easy to see that
G ″ ( t ) G ( t ) − m − + 3 4 ( G ′ ( t ) ) 2 ≥ G ( t ) G ″ ( t ) − ( m − + 3 ) ∫ 0 t ∥ u τ ∥ H 1 2 d τ = G ( t ) − 2 I ( u ) − ( m − + 3 ) ∫ 0 t ∥ u τ ∥ H 1 2 d τ .
Let
ξ ( t ) = − 2 I ( u ) − ( m − + 3 ) ∫ 0 t ∥ u τ ∥ H 1 2 d τ .
With the help of (2.1), we know that
J ( u ) ≥ 1 r + − 1 m − + 1 ∫ Ω ∣ ∇ u ∣ r ( x ) d x + 1 m − + 1 I ( u ) .
Hence,
ξ ( t ) ≥ 2 m − + 1 r + − 1 ∫ Ω ∣ ∇ u ∣ r ( x ) d x − 2 ( m − + 1 ) J ( u 0 ) + ( m − − 1 ) ∫ 0 t ∥ u τ ∥ H 1 2 d τ .
Now, we shall discuss the following two cases.

Case 1: 0 < J ( u 0 ) < M . By Lemma 6.1, we find
(6.4) ξ ( t ) ≥ 2 ( m − + 1 ) M − 2 ( m − + 1 ) J ( u 0 ) + ( m − − 1 ) ∫ 0 t ∥ u τ ∥ H 1 2 d τ = ρ > 0 .
Then,
G ″ ( t ) G ( t ) − m − + 3 4 ( G ′ ( t ) ) 2 ≥ θ ρ > 0 , t ∈ [ 0 , T 0 ] ,
which yields
( G − ϑ ( t ) ) ″ = − ϑ G ϑ + 2 ( t ) ( G ″ ( t ) G ( t ) − ( ϑ + 1 ) ( G ′ ( t ) ) 2 ) < 0 , ϑ = m − − 1 4 .
Therefore, similar to the proof of [40, Theorem 4.3], there exists a T > 0 such that
lim t → T G − ϑ ( t ) = 0
and
lim t → T G ( t ) = + ∞ ,
which is a contradiction with T = + ∞ .

Case 2: J ( u 0 ) ≤ 0 . It is easy to obtain (6.4) directly. The rest is similar to the proof of Case 1.
By the continuity of J ( u ) and I ( u ) with respect to t , there exists a sufficiently small t 0 > 0 such that J ( u ( t 0 ) ) > 0 and I ( u ( t 0 ) ) < 0 for J ( u 0 ) = M > 0 and I ( u 0 ) < 0 . From (6.2), for 0 < t ≤ t 0 , we can see that u t ≠ 0 for 0 < t ≤ t 0 . Hence, we obtain
J ( u ( t 0 ) ) = M − ∫ 0 t 0 ∥ u τ ∥ H 1 2 d τ = d 0 < M .
Taking t = t 0 as the initial time and recalling Proposition 2.1, we obtain u ( x , t ) ∈ V for t > t 0 . The rest is proved similar to (i).

The proof of Theorem 2.4 is complete.□

7 Conclusions

In this study, we have studied the existence, uniqueness, and asymptotic behavior of global solutions and the blowup phenomena of solutions for a mixed pseudo-parabolic r ( x ) -Laplacian type equation. Compared with the studies in [48,9], we replace the terms Δ u and div ( ∣ ∇ u ∣ p − 2 ∇ u ) in the equations with the r ( x ) -Laplacian term div ( ∣ ∇ u ∣ r ( x ) − 2 ∇ u ) according to actual physical background, where the imbedding theorems play an important role. By combining the Galerkin method and the theory of potential wells, we overcome the difficulties in a priori estimation of the r ( x ) -Laplacian term and nonlinear term ⨍ Ω ∣ u ∣ m ( x ) − 1 u d x in the equation.

From r ( x ) and m ( x ) satisfying the conditions given by (1.1) and m ( x ) ≤ n + 2 n − 2 , we see that (i) m ( x ) < n r − n − r − − 1 ≤ n + 2 n − 2 for 1 < r − ≤ 2 and (ii) m ( x ) ≤ n + 2 n − 2 < n r − n − r − − 1 for r − > 2 . Similarly, from r ( x ) and m ( x ) satisfying the conditions given by (1.1) and m ( x ) ≤ n + r ( x ) n − r ( x ) , we have (i) m ( x ) < n r − n − r − − 1 ≤ n + r ( x ) n − r ( x ) for 1 < r − ≤ 2 ; and (ii) m ( x ) ≤ n + r ( x ) n − r ( x ) and m ( x ) < n r − n − r − − 1 for r − > 2 . Obviously, (i) if r ( x ) = 2 and m ( x ) is a constant, our results are consistent with those in [48] and (ii) if r ( x ) and m ( x ) are constants, our results reduce to those in [9].

Another major innovation of this article is to find λ ∗ in Lemma 2.5 by the mean value theorem of integrals, and the proof method in Lemma 2.5 can be directly applied to find λ ∗ in the proof of Lemma 4.1(ii) in [37] and in [23].

For all that, there exist some problems to be further studied such as the asymptotic behavior for J ( u 0 ) ∈ ( M , d ] and the blowup for J ( u 0 ) > M , since (5.4) in the proofs is related to M . In addition, further studies can be conducted to strengthen the advantages of variable exponents by reflecting on the corresponding references [8,12,19,28,39,49,51].

Acknowledgements

The authors would like to thank the editors and the referees for their very helpful comments and suggestions.

Funding information: This research was supported by the National Natural Science Foundation of China (No. 12071491).
Author contributions: The authors contributed equally to the writing of this manuscript. All authors read and approved the final manuscript.
Conflict of interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this manuscript.

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Received: 2023-03-27

Revised: 2023-11-02

Accepted: 2024-01-03

Published Online: 2024-02-24

This work is licensed under the Creative Commons Attribution 4.0 International License.

Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation

Abstract

1 Introduction

2 Preliminaries and main results

Lemma 2.1

Lemma 2.2

Lemma 2.3

Definition 2.1

Definition 2.2

Lemma 2.4

Proof

Lemma 2.5

Proof

Lemma 2.6

Proof

Remark 2.1

Lemma 2.7

Proof

Lemma 2.8

Proposition 2.1

Lemma 2.9

Proof

Theorem 2.1

Theorem 2.2

Theorem 2.3

Theorem 2.4

Remark 2.2

3 Local solutions

Proof of Theorem 2.1

4 Global existence and uniqueness

Proof of Theorem 2.2

5 Asymptotic behavior

Proof of Theorem 2.3

6 Blowup

Lemma 6.1

Proof

Proof of Theorem 2.4

7 Conclusions

Acknowledgements

References

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