Properties of Weak Solutions for a Pseudoparabolic Equation with Logarithmic Nonlinearity of Variable Exponents

Abstract

In this paper, a new pseudoparabolic equation with logarithmic nonlinearity of variable exponents is investigated. By using the energy functional and the classical potential well, we obtain the global existence and blow-up results of weak solutions with variable exponents. These results extend and improve some recent results in which the blow-up results were showed involving the logarithmic nonlinearity with variable exponents.

1. Introduction

Consider the following a pseudoparabolic equation problem with variable exponents and logarithmic nonlinear term where is a bounded domain with smooth boundary , . The exponent functions and are satisfying the following conditions:

(H1) are measurable functions defined on such that where

(H2) , , where and .

Pseudoparabolic-type equations appear in many areas of applications, such as unidirectional propagation of nonlinear diffusive long waves, aggregation of populations, and nonstationary processes in semiconductor groups with input sources, which have been considered by different authors; see [1–8] and the references therein. They studied the global existence, uniqueness, asymptotic behavior, and blow-up of solutions with pseudoparabolic equations.

Let us consider the following pseudoparabolic equations of the form

In the case of with , they are usually called the pseudoequations with variable exponents; see [9–13]; they investigated the existence, nonexistence, and blow-up of weak solutions and also considered a long-time behavior of the weak solutions.

In the case of logarithmic nonlinearity , the Dirichlet boundary value problem for equation (4), which is refereed to [14], has been studied by many authors. Moreover, such problems have been extensively investigated by using the method of potential wells and the logarithmic Sobolev inequality. Many results concerning global existence and nonexistence have been proved; see [15–23].

In the recent monograph [24], Xiangyu Zhu, Bin Guo, and Menglan Liao studied the following problem

They obtained the global existence and blow-up results of weak solutions with arbitrarily high initial energy.

Motivated by these works, we consider the problem (1) with the logarithmic nonlinearity . The aim of this paper is to study the existence and blow-up of weak solutions for a pseudoparabolic equation with the logarithmic nonlinearity of variable exponents. To the best of our knowledge, there are only a few works about the logarithmic nonlinearity with variable exponents; see [25, 26]. So, the study of problem (1) is a new and interesting topic. The main difficulties for treating the problem are caused by the existence of the gap between the norm and the module. In addition, owing to the term leads to some difficulties in deploying the potential well method.

The outline of this work is the following: In Section 2, we shall introduce some definitions and the function spaces of type; in Section 3, we obtain the global existence and blow-up results of weak solutions by analyzing the controlled relationship between the energy functional and the sources and using the method of potential wells.

2. Preliminaries

In this section, we introduce some notations and basic properties of the generalized Lebesgue space and Lebesgue-Sobolev space .

Let satisfy the condition (2); we give the definition of variable exponent Lebesgue space that consists of all measurable functions such that equipped with the

The variable exponent Lebesgue space is a special case of Orlicz-Musielak spaces studied by Tsutsumi in [27].

It is obvious that is a Banach space in [28]. It follows directly from the definition that

By Corollary 3.3.4 [28], we have , and we also give some lemmas needed in our main results as the following.

Lemma 1. Let be a bounded domain of and satisfies and , and then for all , the following result hold

Proof. Set , and then we clearly have In view of (8), we easily obtain Substituting (10) into (11), we get immediately (9).

Lemma 2 (see [28, 29]). Let be a bounded domain of and satisfies and , then where is the optimal embedding positive constant which depends on and . In particular, the space has an equivalent norm given by .

Lemma 3 (see [28, 29]). If , is a measurable function and Then the embedding is continuous and compact.

3. Main Results

In this section, the existence and blow-up of weak solutions will be studied. For this purpose, we give the definition of weak solutions as follow.

Definition 4. A function is called a weak solution of problem (1) if with , and for all , the following equality holds

Take into account of (2), we get , and then, for any , define the functional and Nehari functional as follows: we also define energy functional

Let where . Define

In the following, in order to consider the weak solutions of the problem (1), we give some sets and functionals.

Furthermore, for any , we define , and . Obviously, is nonincreasing (nondecreasing) with respect to .

Thus, we have

Lemma 5 (see [10]). If is a weak solution to problem (1), the energy functional is nonincreasing with respect to as long as belongs to the existence interval of weak solutions and

Lemma 6. The potential depth is positive.

Proof. For fixed , from Lemma 1-Lemma 3, and by simple calculations, for each , we have where and is the optimal embedding constant Therefore, from (21), we have It follows from the definitions of , , and (23) that As a result, from (23), (24), and the definition of , we obtain Next, we give the properties of and .

Lemma 7. For any , and satisfy .

Proof. Consider the following problem of elliptic equation ’s inequality shows ; here is the first eigenvalue of problem (26). Then we get On the other hand, for , the continuous embedding implies that Combining (27) with (28), we have For any , recalling (19) and (24), we have (29) and (30) yield which implies .
For any , in view of the Sobolev embedding , as well as the definition of and , we have where , is the optimal embedding constant.
Case 1. For , on the basis of the discussion above, we can easily obtain which implies that Case 2. For , according to discussion above, we can easily get which implies that Set From (34) to (36), we get The Gagliardo-Nirenberg inequality and the continuous embedding imply that there exists a positive constant such that where as a result of (2). Furthermore, noticing that , then we can obtain Moreover, we have In addition, we combine (39) with (41) to obtain where , .
Obviously, the right-hand side of the above inequality remains bounded away from no matter what the sign of and are due to (30) and (38). Therefore, .

In fact, we can get a lower bound of that is independent of .

Proposition 8. If , then for any , , here is given in (48).

Proof. In the following, we will divide the proof of this part into two cases.
Case 1. For , the Sobolev embedding (see [30]) and (9) yield which implies where is the optimal embedding constant for .
Case 2. For , the Gagliardo-Nirenberg inequality and the embedding inequality indicate that there exists a positive constant such that where , , and . A simple computation shows that . By the same argument of Lemma 7, we easily obtain By (45) and (47), we get , here

Theorem 9. Assume that with , we have (1)if and , the weak solution of problem (1) exists globally and as (2)if and , the weak solution of problem (1) blows up in finite time

Proof. Denote by the maximal existence time of weak solutions for problem (1). If there exists a global solution, i.e., , we denote by the -limit of . (1)If and , we claim that for all . By contradiction, there exists a such that for and . Choosing in (14), we obtainTherefore, it follows from (49) that for . Further, (20) yield , which implies . Therefore, . By the definition of , we obtain Noticing that for , according to (49), we have , which contradicts (50). Therefore, for all . Further, we get for all , . Moreover, it follows from (24) that which indicates that the boundedness of is independent of , .
For any , then , by (49) and (20), respectively. The second inequality implies . Noticing that the definition of and the first inequality, we derive ; further, . Thus, , which indicates . Therefore, the weak solution of problem (1) exists globally and as . (2)If and , we claim that for all . By contradiction, there exists a such that for and . Similar to case (1), we get , which implies . Therefore, . By the definition of , we obtainNoticing that for , it follows from (49) that , which contradicts (52). Therefore, for all . Further, we get for all , . Suppose , then for any , then , by (49) and (20), respectively. The second inequality implies . Noticing that the definition of and the first inequality, we derive ; further, . Thus, , which indicates . This result contradicts which is that is obtained by the same argument of (23). Therefore, , ; that is, the weak solution of problem (1) blows up in finite time.

For , in particular, we have the following Proposition 10 that is easily proved based on Theorem 9(2), here we omit the proof. Meanwhile, we give the following Theorem 11 to illustrate there exists such that is arbitrarily large, and the corresponding solution to problem (1) with as initial datum blows up in finite time as well.

Proposition 10. If satisfies , then and the weak solution of problem (1) blows up in finite time.

Theorem 11. For any , there exists initial value such that , and the weak solution of problem (1) blows up in finite time.

Proof. Assume that and , are two arbitrary disjoint open subdomains of . Furthermore, we assume that is an arbitrary nonzero function. Recalling (2), then we take large enough such that We fix such a number and choose a function satisfying . Extend and to be in and . Set , then , and it follows that By Proposition 10, and the weak solution of problem (1) blows up in finite time.

Data Availability

No real data were used to support this study. The data used in this study are hypothetical.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (11526035) and the Department of Education for Jilin Province (JJKH20220040KJ).