Initial boundary value problem for a class of p-Laplacian equations with logarithmic nonlinearity

Abstract: In this paper, we discuss global existence, boundness, blow-up and extinction properties of solutions for the Dirichlet boundary value problem of the p-Laplacian equations with logarithmic nonlinearity ut − div(|∇u|p−2∇u) + β|u|q−2u = λ|u|r−2u ln |u|, where 1 < p < 2, 1 < q ≤ 2, r > 1, β, λ > 0. Under some appropriate conditions, we obtain the global existence of solutions by means of the Galerkin approximations, then we prove that weak solution is globally bounded and blows up at positive infinity by virtue of potential well theory and the Nehari manifold. Moreover, we obtain the decay estimate and the extinction of solutions.


Introduction
In this paper, we consider the following p-Laplacian equations with logarithmic nonlinearity.
in Ω, u = 0, on ∂Ω × (0, T ), where 1 < p < 2, 1 < q ≤ 2, r > 1, β, λ > 0, T ∈ (0, +∞], Ω ∈ R N is a bounded domain with smooth boundary and u 0 (x) ∈ L ∞ (Ω) ∩ W 1,p 0 (Ω) is a nonzero non-negative function. Problem (1.1) is a class of parabolic equation with logarithmic nonlinearity, it is worth pointing out that the interest in studying problem (1.1) relies not only on mathematical purposes, but also on their significance in real models. Among the fields of mathematical physics, biosciences and engineering, problem (1.1) is one of the most important nonlinear evolution equations. For example, in the combustion theory, we can use the function u(x, t) to represent temperature, the −div(|∇u| p−2 ∇u) term to represent thermal diffusion, β|u| q−2 u to represent absorption, and λ|u| r−2 u ln |u| to be the source. In the diffusion theory, we can use u(x, t) to represent the density of a type of population at position x at time t, the −div(|∇u| p−2 ∇u) term represents the diffusion of density, λ|u| r−2 u ln |u| and β|u| q−2 u represents the absorption and the sources, respectively. We refer the reader to [1,2] and the references therein for further details on more practical applications of problem (1.1).
The research with logarithmic nonlinearity terms is the current research hotspot. The literature on the evolution equations with logarithmic nonlinearity term is very interesting, we refer the readers to [3][4][5][6] and the references therein. At the same time, the study of p-Laplacian equations has also achieved many important results. The study of p-Laplacian equations can be divided into two cases, namely 1 < p < 2 and p > 2. For the case of p > 2, most researchers discuss the global existence and blow-up of solutions of the equations (see [7][8][9][10]). For the case of 1 < p < 2, the extinction and attenuation estimation of solutions are mainly discussed, we refer the readers to [11][12][13].
In particular, there are also some papers concerning properties such as global existence or extinction for the problem (1.1) for special cases.
In [14], Liu studied a more general form is a bounded domain with smooth boundary and u 0 (x) ∈ L ∞ (Ω) ∩ W 1,p 0 (Ω) is a nonzero non-negative function. The author gave the extinction properties and attenuation estimates of the solutions by comparison principle and differential inequality.
In [15], Cao and Liu considered the following initial-boundary value problem for a nonlinear evolution equation with logarithmic source where 1 < p < 2, u 0 ∈ H 1 0 (Ω), T ∈ (0, +∞), k ≥ 0, Ω ⊂ R n (n ≥ 1) is a bounded domain with smooth boundary ∂Ω. They proved the global existence of weak solutions and studied the asymptotic behavior of solutions and gave some decay estimates and growth estimates by constructing a family of potential wells and using the logarithmic Sobolev inequality.
In [16], Pan et al. considered the following pseudo-parabolic equation with p-Laplacian and logarithmic nonlinearity terms in a bounded domain Ω ⊂ R n (n ≥ 1) with smooth boundary, where u 0 ∈ W 1,p 0 (Ω)\{0} and the parameters p, q satisfy 2 < p < q < p(1 + 2 n ). They gave the upper and lower bound estimates of blow-up time and blow-up rate, and established a weak solution with high initial energy.
In [17], Xiang and Yang studied the following initial-boundary value problem for the fractional p-Laplacian Kirchhoff type equations where 0 < s < 1 < p < 2, 1 < q ≤ 2, r > 1, µ, λ > 0, Ω ⊂ R N is a bounded domain with Lipschitz boundary, M : [0, ∞) → (0, ∞) is a continuous function. By flexible application of differential inequalities, they gave the extinction and the decay estimates of solutions. Inspired by the above work, we study problem (1.1). Compared with problem (1.2) and (1.5), the focus of our work is partial differential equations with logarithmic nonlinearity. If the nonlinear term λ|u| r−2 u ln |u| in problem (1.1) is transformed into λ|u| r−2 u, then the problem (1.1) can be transformed into problem (1.2). For more research results of the logarithmic nonlinear p-Laplacian equations, we refer the readers to [18][19][20][21] and the references therein. But as far as we know, no work has dealt with the global existence and extinction properties of solutions for problem (1.1) with both the absorption and source effects as well as the term of logarithmic nonlinearity. To state our main results, we need the following two definitions.
where (·, ·) means the inner product of L 2 (Ω). Definition 1.2 (Extinction of solutions). Let u(t) be a weak solution of problem (1.1). We call u(t) is an extinction of solutions if there exists T > 0 such that u(x, t) > 0 for all t ∈ (t, T ) and u(x, t) ≡ 0 for all t ∈ [T, +∞). Further, we respectively define the energy functional J(u) and the Nehari functional I(u) of problem (1.1) as and By the subsequent Lemma 2.1 and a simple calculation, we can obtain for 0 < σ < 2 * − r. So J(u) and I(u) are well-defined for u ∈ W 1,p 0 (Ω). Next, the potential well W and its corresponding set V are defined by and define the Nehari manifold Next, we state our main results. (1) If λ < R 0 , then the weak solution of (1.1) satisfies (2) If 2N/(N + 2s) < p < 2 and λ < R 0 or 1 < p ≤ 2N/(N + 2s) and λ < R 1 , then the nonnegative solutions of (1.1) vanish in finite time, and and C is the embedding constant, Γ(p, Ω) will be given in section 3. Theorem 1.4 Let r = p and p > q. If 0 < J(u 0 ) < R 2 and I(u 0 ) < 0, then the solution u(x, t) of (1.1) is non-extinct in finite time, where Theorem 1.5 Assume I(u 0 ) > 0, r > p and q = 2, then the nonnegative weak solution of problem (1.1) vanishes in finite time and and C p * is the embedding constant, l 1 , v 2 , s 2 will be given in section 3. The paper is organized as follows. In section 2, we give some necessary Lemmas such as some properties for Nehari functional and known results for ODEs. In section 3, we present the proof of the main theorems.

Preliminaries and Lemmas
Lemma 2.1 ( [22]) Let α be positive number, then where Γ(p, Ω) := |Ω| ep + 1 e(p * −p) C p * p * , C p * is the best constant of embedding from W 1,p 0 (Ω) to L p * (Ω). Proof. For convenience, we provide complete proof. As we know, for 1 < p < 2, Using the properties of logarithmic, we have Taking σ = p * − p in Lemma 2.1, and by Sobobev embedding inequality, we obtain By direct calculation and Eq (2.1), we have The proof is completed.
The remainder of the proof is the same as that in [15]. Proof of Theorem 1. 2 We first consider the case 0 < J(u 0 ) < d and I(u 0 ) > 0. Choosing ν = u in Definition 1.1, we obtain Next, we proof that u(x, t) ∈ W for any t > 0. If there exists a t 0 > 0, such that u(x, t 0 ) ∈ ∂W, namely I(u(x, t 0 )) = 0 or J(u(x, t 0 )) = d.
Since u 0 (x) ∈ L ∞ (Ω) ∩ W 1,p 0 (Ω) is a nonzero non-negative function, ρ m ∈ (0, 1) and I(u 0 ) ≥ 0, then we have and From the results above, we can derive that the weak solution of Eq (3.19) is globally bounded. Then, we discuss that weak solution blows up at infinity. Let M(t) = 1 2 Ω |u(x, t)| 2 dx, then we have By (3.26) and the following equation which implies Making ν = u t in Definition 1.1, we get By a simple calculation, we get Therefore, the following inequality holds The proof is completed.
Proof of Theorem 1. 5 We consider first the case p < r < 2 and 2N/(N + 2) < p < 2. Choosing ν = u in Definition 1. where Since l 0 < p * , by the Sobolev embedding theorem and the Hölder's inequality, we get