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

: 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 ﬁ nite-time blowup of solutions with sub-sharp-critical initial energy and sharp-critical initial energy, respectively.


Introduction
Consider the initial boundary value problem of semilinear pseudo-parabolic equation with ( ) r x -Laplacian: 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: 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 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 u u u , , , Δ Δ 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: Considering the physical significance of practical problems, such as atoms and ions, researchers often replace the term u Δ in the pseudo-parabolic (1.2) with the p-Laplacian term . 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: 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 is the natural generalization of the usual term 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: They constructed a suitable control function, improved the regularity of the approximate solution, obtained a new energy inequality, and proved that if 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) 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: (i) In Section 2, we give some preliminaries and notations.Meanwhile, we state our main results of this article.(ii) In Sections 3, we prove the local existence of solution using the standard Galerkin method.(iii) In Section 4, we establish the existence and uniqueness of global solutions with subcritical initial energy, critical initial energy, and supercritical initial energy, respectively.(iv) In Section 5, we prove the decay estimate of global solutions with sub-sharp-critical initial energy, sharpcritical initial energy, and supercritical initial energy, respectively.(v) In Section 6, we discuss the global nonexistence of solutions with sub-sharp-critical initial energy and sharp-critical initial energy, respectively.(vi) In Section 7, we make some conclusions and discussions.

Preliminaries and main results
Throughout this article, let C be a general positive constant that may change from line to line.For ≤ ≤ ∞ p 1 , denote the ( ) where D u α is the α-order weak derivative of u, and ( ) when = p 2. Furthermore, we define the following set: Define the Banach space of the Orlicz-Sobolev type as follows: equipped with the following Luxemburg norm:

Ω
Global existence and finite-time blowup  3 and the conjugate space is , where ( ) , for all ∈ x Ω. Correspondingly, the space ( ) can be defined as: and it can be equipped with the following norm: , which means the conservation law for Problem (1.1).Hence, we define Then, there is a continuous and compact embedding ( ) ( ) be the imbedding constant for ( ) Next, the energy functional ( ) J u and the Nehari functional ( ) I u are defined as: and the Nehari manifold is as follows: Based on the definitions of ( ) J u and ( ) I u , we introduce sets where d is the depth of the potential well, which can be defined as: u Define a family of potential wells where > δ 0. In the following, let us give some sets and functionals in order to consider the weak solution with high energy level: 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 ( ) and satisfies for any ( ) ( ) . Moreover, the following equality , where ( ) h δ is a function of δ and satisfies Global existence and finite-time blowup  5 ( ) Proof. and Proof.From the definition of ( ) J u , we have . Furthermore, from the mean value theorem of integrals, we have that In addition, by (2.4), we obtain , and we have the following lemma.Proof.If ∈ u , then ( ) = I u 0 and ≠ u 0. According to the definitions of ( ) J u and ( ) I u , we obtain By Lemma 2.4 (iii), we have In this case, we call d the critical initial energy and M the sharp-critical initial energy, respectively.Lemma 2.7.For ( ) 2), we obtain Proof.Now, we need to prove , respectively, and if successful, then the lemma is proved.
Proposition 2.1.Assume that u is the weak solution of (1.1), . Furthermore, if ( ) > I u 0 0 , then for any and ≠ u u , .Due to the property of Ga ˆteaux derivative, we find ; and the weak solution is unique for ( ) ; and the weak solution is unique for ( ) ; and the weak solution is unique for Theorem 2.3.[Asymptotic behavior] Let ( ) u x t , be the global bounded weak solution in Theorem 2.2.
, then there exists a constant > δ 0 such that , then there exist constants > t 0 1 and > ς 0 such that , by Lemmas 2.1 and 2.2, we have .

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 for every > T 0, and define the norm on as follows: Next, for every > T 0 and ∈ u , we shall prove that there is a unique ∈ v satisfying where of the Laplace operator in are the characteristic values.By [47,30,48], we have that ( . Construct the following approximation functions: where ( ) g t jm satisfies the initial value problem of the following equations: From (3.2) and (3.3), we have where for all j, by standard existence theory for ordinary differential equations, the initial value Problem (3.4) admits a local solution.
Multiplying the jth equation of (3.2) by ( ) ′ g t jm , summing up with respect to j, we obtain Consider the third term on the left-hand side of the aforementioned equation, we have Then, combining Gronwall's inequality and (3.7), we have Multiplying the jth equation of (3.2) by ( ) ′ g t jm , summing on j, and integrating with respect to the time variable from 0 to t, we obtain Hence, from (3.8) and (3.11), there exist a v and a subsequence 2), we obtain the existence of a weak solution v of (3.1) with the aforementioned regularity.In addition, we have 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 v 1 2 , we obtain where [ ] ∈ θ 0, 1 1 . From (3.12), it is easy to know that ≡ v v 2 , for any > T 0, consider the set: T

According to
Step I, for any ∈ u X T and the unique solution ∈ v of Problem (3.1), we can define T is a contractive map.(i) For given ∈ u X T , similar to (3.5), we have Similar to (3.6), we obtain Combining (3.13) and (3.14), from Gronwall's inequality, we have Choose a sufficiently small T such that 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 ( ) and testing the both sides by  v , we have where . From Lemma 2.9, we obtain where Considering (3.15) and (3.16), we obtain which gives i.e., Global existence and finite-time blowup  15 for some < δ 1 T as long as T is sufficiently small.Then, the map for ≤ ≤ s m 1 , in which Then, by Lemma 2.2, there exists a subsequence of { } u m , denoted by the same symbol satisfying

∫∫ ∫∫
; similar to the proof of Lemma 2.9, we have for each = s 2, 3,… .From Propo- sition 2.1, we know that ( ) ∈ u x t W , s .Similar to the proof process of (i), we obtain The rest is proved similar to (i).

Global existence and finite-time blowup  7 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.

2 ,
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] then there exist > T 0 and a unique solution of (1.1) over [ ] T 0, .Theorem 2.2.[Global existence and uniqueness] Assume that (

.
Due to (i), Problem (1.1) with the initial condition (4.11) has a unique global solution ( ) 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.

.
As in the proof of (i), then Problem (1.1) has a unique global weak solution u.Theorem 2.2 is proved.□Inthis 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.