Global Existence and Blow-Up for the Pseudo-parabolic p(x)-Laplacian Equation with Logarithmic Nonlinearity

In this paper, we study the initial boundary value problem of the pseudo-parabolic p(x)-Laplacian equation with logarithmic nonlinearity. The existence of the global solution is obtained by using the potential well method and the logarithmic inequality. In addition, the sufficient conditions of the blow-up are obtained by concavity method.


Introduction
The main purpose of this paper is to study the existence and blow-up of solutions for the following pseudo-parabolic p(x)-Laplacian equation with logarithmic nonlinearity: where Δ p(x) u = div(|∇u| p(x)−2 ∇u) is the p(x)-Laplacian, Ω ⊂ ℝ N (N ≥ 1) is a bounded domain with smooth boundary Ω, u 0 ∶ Ω → ℝ is the initial function, and p, q ∶ Ω → ℝ + are continuous functions which satisfy the following conditions: (1) with Pseudo-parabolic equations are characterized by the occurrence of a time derivative appearing in the highest order term, which can be wildly used in some physical and biological scenarios, such as the seepage of homogeneous fluids through a fissured rock, the heat conduction involving two temperature systems, the unidirectional propagation of nonlinear, dispersive, long waves, fluid flow in fissured porous media, two phase flow in porous media with dynamical capillary pressure and the aggregation of populations (see [1,2]). For problem (1), we can also give an example about the non-stationary process in semiconductors in the presence of sources, where Δu represents the linear dissipation of free charge current, Δu t − u t represents the free electron density rate and the nonlinear term including the p(x)-Laplacian and logarithmic nonlinearity stands for the source of free electron current (see [3]).
In particular, the pseudo-parabolic equations involving the p(x)-Laplacian can be used to study electrorheological fluids which are characterized by their ability to change the mechanical properties under the influence of the external electromagnetic, more introductions on physical motivations can be found in [4,5] and the references contained therein.
In the past years, many authors make efforts to the investigation of the existence and blow-up of solutions for such kinds of equations. In Chen et al. [6], investigated the following semilinear heat equation with logarithmic nonlinearity they obtained the existence and blow-up at +∞ of solutions of problem (2), they further proved that the global weak solution decayed exponentially in the case of p = 2 , and decayed algebraically in the case of p > 2 . Subsequently, Peng and Zhou [7] considered the following initial boundary value problem of semilinear heat equation with logarithmic nonlinearity where p satisfies they obtained the existence of global solution, finite time blow-up and the upper bound of blow-up time of problem (3). For problem (2) u t − Δu = u log(|u|), where p (z) = |z| p−2 z . In case of 2 < p < 2 * , the existence or nonexistence of global weak solutions as well as finite time blow-up phenomenon were obtained in [8]. In He et al. [9], considered the following nonlinear equation they obtained result of decay estimation and finite blow-up of solutions where 2 < p < q < p(1 + 2 n ) . Recently, Boudjeriou [10] considered the following nonlinear equation which involved the p(x)-Laplacian operator with nonlinearities of variable exponent type. They proved that the local solutions of problem (6) blew up in finite time under suitable conditions. More results about partial differential equations involving the p(x)-Laplacian operator, we refer to [11][12][13][14] and the references therein.
A powerful technique for solving the existence of the above problem is the so-called potential well method, which was established by Tsutsumi [15], Levine [16], Payne and Sattinger in [17]. Liu et al. [18,19] generalized and improved the method by introducing a family of potential wells which include the known potential well as a special case. On the other hand, the physical interpretation of blow-up phenomena is generally thought of as a dramatic increase in temperature which leads to ignition of a chemical reaction. Many researchers applied different methods to derive the sufficient conditions for finite time and infinite time blowup result. In particular, Levine [16] studied the abstract equation where p and A are positive linear operators defined on a dense subdomain D of a real or complex Hilbert Space, in which they obtained the blow-up solutions, under abstract conditions This work has been recognized as a creative and elegant tool for giving criteria for the blow-up, which is called the concavity method. Nowadays, it is one of the most useful method for blow-up of solutions for evolution equations.
Inspired by the above works, by using the potential well method and concavity method, we consider the existence and blow-up of solutions for problem (1) which combine with pseudo-parabolic, variable exponent and logarithmic nonlinearity term. To our best knowledge, it is the first attempt to study the properties of the solutions for such kind of equations.
Here we give some important definitions as follows: for u 0 ∈ W 1,p(x) 0 (Ω) , we define the energy functional E and Pohozaev functional J as (Ω)�{0} ∶ J(u) = 0 be the Pohozaev manifold [10]. Related to the above manifold we have the real number To introduce our main results, we first give the definition of weak solutions and blow-up for problem (1).

Preliminaries and Lemmas
In this section, we will recall some important results of variable exponentials on Lebesgue or Sobolev spaces. For more details on variable exponential Sobolev spaces, please refer to [20][21][22][23].
Let Ω ⊂ ℝ N be a smooth bounded domain and p ∈ L ∞ (Ω) be a measurable function with p + = ess sup x∈Ω p(x) and p − = ess inf x∈Ω p(x).
The variable exponent Lebesgue space The variable exponent Sobolev space W 1,p(x) (Ω) is defined by with the norm So with these norms, the space L p(x) (Ω) and W 1,p(x) (Ω) are reflexive and separable Banach spaces. The closure of Related to the properties of logarithms and Lebesgue space L p(x) (Ω) , we have the following logarithm inequality and generalized Hölder-type inequality.

Main Results
In this section, we will give the three theorems which are, respectively, related to global existence, local existence and finite time blow-up of solutions involving problem (1). The existence or nonexistence of the global solution is obtained by using the potential well method and the logarithmic inequality. The finite time blow-up phenomenon is obtained by concavity method.
Proof First we define the potential well associated with the problem (1) as (Ω) , we take a basis w j ∞ j=1 and define the finite dimensional space (Ω) as m → +∞ . We look for the approximate solutions of the following form where the coefficients mj (t) = u m , w j 2 satisfy the system of ODES The existence of local solution of system (13,14) is guaranteed by Peano's theorem. Multiplying the equality of (13) by � m j (t) , summing for j from 1 to m and integrating with respect to time from 0 to t, it yields From (12,13,15) and the continuity of E, we get E(u m (0)) → E(u 0 ) . According to the assumption that E u 0 < Γ , we have E u 0m < Γ for sufficiently large m. Therefore, we obtain for sufficiently large m. We will show that t 0,m = +∞ and for sufficiently large m. Suppose that (17) does not hold and let t * ∈ 0, t 0m be the smallest time for which u m (t * ) ∉ W, then by the continuity of the u m (t) , we get u m (t * ) ∈ W . Hence, it follows that or If (18) is true, it contradicts with (16). While if (19) is true, then u m (t * ) ∈ P , E(u m (t * )) ≥ inf u∈P E(u) = Γ , which also contradicts with (16). Consequently, we have u m (t) ∈ W, ∀t ≥ 0.
On the other hand, since u m (t) ∈ W and (7), it follows that Combining the above inequality with (16), we derive from where it follows that t 0,m = +∞. By (20), there exists functions u, and a subsequence of u m ∞ m=1 which we still denote by u m ∞ m=1 such that By (21,22) and Aubin-Lions-Simon Lemma ( [25], Corollary 4), we get   On the other hand, by a direct calculation, we have Hence, by Lion's Lemma (see [25], Lemma 1.3), it yields weakly star in L ∞ (0, +∞); L q � (x) (Ω) .
Multiplying the equality of (13) by m j (t) , summing over j form 1 to m and integrating over (0, T), it yields From (24)  (Ω)) and a.e. t ∈ [0, ∞). In view of Definition 1 and the above discussions, we get the global existence of the solution of problem (1). The proof of Theorem 1 is complete. ◻ , ∀x ∈ Ω , then the problem (1) admit a local weak solution. Moreover, u satisfy the energy inequality Proof Similar to the proof of Theorem 1, by means of Galerkin method, we consider the approximate solution of problem (1) as u m (x, t) = ∑ m j=1 mj (t)w j (x) , which satisfies the following equations Journal of Nonlinear Mathematical Physics (2022) 29:41-57 Due to u 0 ∈ W 1,p(x) 0 (Ω) , then it exists b mj , j = 1, 2, ⋯ , m such that The existence of local solution of system (40) (41) is guaranteed by Peano's theorem.
Multiplying the equality of (40) by mj (t) and summing over j from 1 to m, we obtain By virtue of Lemma 1, we have where ∈ (0, 1) and 1 Combing the above inequality with (44), we obtain We discuss the proof into two cases as follows. Case 1 Suppose ‖u m ‖ ≥ 1 , then Integrating inequality (51) with respect to time from 0 to t, we get Multiplying the equality of (40) by � mj (t) and summing over j from 1 to m, and integrating over (0,t) yields

Conclusion
In this paper, by virtue of the potential well method and logarithmic inequalities, we obtain the global existence of solutions of problem (1). What's more, the finite time blow-up phenomenon is obtained by concavity method. The first attempt to study the properties of the solutions for such kind of equations will enrich the research of mathematical physics equations.
Author Contributions All authors completed the paper together. All authors read and approved the final manuscript.
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Conflict of interest
The authors declare that they have no competing interests.
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