Initial boundary value problem for fractional p-Laplacian Kirchhoff type equations with logarithmic nonlinearity

Abstract: In this paper, we study the initial boundary value problem for a class of fractional pLaplacian Kirchhoff type diffusion equations with logarithmic nonlinearity. Under suitable assumptions, we obtain the extinction property and accurate decay estimates of solutions by virtue of the logarithmic Sobolev inequality. Moreover, we discuss the blow-up property and global boundedness of solutions.


Introduction
In this paper, we study the extinction and the blow-up for the following fractional p-Laplacian Kirchhoff type equations with logarithmic nonlinearity. |u(x) − u(y)| p−2 (u(x) − u(y)) |x − y| N+sp dy, where u(x) ∈ C ∞ and u(x) has compact support in Ω, B γ (x) ⊂ R N is the ball with center x and radius γ. u 0 (x) ∈ L ∞ (Ω) ∩ W s,p 0 (Ω) is a nonzero non-negative function, where L ∞ (Ω) and W s,p 0 (Ω) are Lebesgue space and fractional Sobolev space respectively, which will be given in section 2. M(·) is a Kirchhoff function with the following assumptions (M 1 ) 0 < s < 1, M(τ) := a + bθτ θ−1 for τ ∈ R + 0 := [0, +∞) ( a > 0, b ≥ 0 are two constants), θ ≥ 1; (M 2 ) M : R + 0 → R + 0 \{0} is continuous and there exits m 0 ≥ 0 such M(τ) ≥ m 0 for all τ ≥ 0. It is worth pointing out that the interest in studying problems like (1.1) relies not only on mathematical purposes, but also on their significance in real models. For example, in the study of biological populations, we can use u(x, t) to represent the density of the population at x at time t, the term (−∆) s p u represents the diffusion of density, µ|u| q−2 u represents the internal source and λ|u| r−2 u ln |u| denotes external influencing factors. For more practical applications of problems like (1.1), please refer to the studies [1][2][3].
Compared with integer-order equations, it is very difficult to study the problem (1.1), which contains both non-local terms (including fractional p-Laplacian operators and Kirchhoff functions) and logarithmic nonlinearity. For the fractional order theory, we refer the readers to the studies [4][5][6]. In [7,8], the authors use Sobolev space and Nehari manifold to study the existence of solutions for fractional equations. In [9,10], the solutions for fractional equations are discussed by virtue of Nehari manifold and fibrillation diagram. By using different methods from above, the properties of the solutions for such partial differential equations are considered by the method of variational principle and topological theory in the the literature [11][12][13]. Moreover, the authors prefer to use potential well theory, Galerkin approximation and Nehari manifold method to prove the existence of solutions, decay estimation and blow-up, we refer the reader to the literature [14][15][16].
Existence, extinction and blow-up of solutions are three important topics which regard parabolic problems; in particular, the study of extinction properties has made great progress in recent years. In [17], Liu considered the following initial boundary value problem for the fractional p-Laplacian equation u t − div(|∇u| p−2 ∇u) + βu q = λu r , x ∈ Ω, t > 0, (1.2) where 1 < p < 2, q ≤ 1 and r, λ, β > 0. By employing the differential inequality and comparison principle, they obtained the extinction and the non-extinction of weak solutions. In [18], Sarra Toualbia et al. considered the following initial boundary value problem of a nonlocal heat equations with logarithmic nonlinearity where p ∈ (2, +∞). By using the logarithmic Sobolev inequality and potential well method, they obtained decay, blow-up and non-extinction of solutions. In [19], Xiang and Yang studied the first initial boundary value problem of the following fractional p-Kirchhoff type Under suitable assumptions, they proved the extinction and non-extinction of solutions and perfected the Gagliardo-Nirenberg inequality. For more information on the extinction properties of the solution, please refer to the studies [20][21][22][23].
Inspired by the above works, we overcome the research difficulties of logarithmic nonlinearity, p-Laplace operator and Kirchhoff coefficients in problem (1.1) based on the potential well theory, Nehari manifold and differential inequality methods, we give the extinction and the blow-up properties of solutions. In addition, we give the global boundedness of the solution by appropriate assumptions. To the best of our knowledge, it is the first result in the literature to investigate the extinction and blow-up of solutions for fractional p-Laplacian Kirchhoff type with logarithmic nonlinearity.
In order to introduce our main results, we first give some related definitions and sets. Definition 1.1(Weak solution). A function u(x, t) is said to be a weak solution of problem (1.1), if (x, t) ∈ Ω × [0, T ), u ∈ L p 0, T ; W s,p 0 (Ω)) ∩ C(0, T ; L 2 (Ω) , u t ∈ L 2 0, T ; L 2 (Ω) , u(x, 0) = u 0 (x) ∈ W s,p 0 (Ω), for all v ∈ W s,p 0 (Ω), t ∈ (0, T ), the following equation holds Define the following two functionals on W s,p 0 (Ω) where 0 < σ < p * s − r, then we can claim that E(u) and I(u) are well-defined in W s,p 0 (Ω). Further, by arguing essentially as in [24], one can prove the that u → Ω |u| r ln |u|dx is continuous from W s,p 0 (Ω) to R. It follows that E(u) and I(u) are continuous. Define some sets as follows Let λ 1 be the first eigenvalue of the problem and φ(x) > 0 a.e. in Ω be the eigenfunction corresponding to the eigenvalue First, we give some results satisfying I(u 0 ) > 0 and q = 2. Theorem 1.1 Assume that I(u 0 ) > 0, r = p and q = 2. Let m 0 be as in assumption (M 2 ), and let , where L(p, Ω) and R are given in Lemma 2.1 and Lemma 2.5. Then, there exist positive constants C 1 , C 2 , T 1 and T 2 such that (i) If λ < λ 1 P 1 , then the weak solution of (1.1) vanishes in the sense of · 2 as t → +∞.
(i) If E(u 0 ) < 0 , r = p > q and θ = 1, then the weak solution u(x, t) blows up at +∞; (ii) If 0 < E(u 0 ) ≤ h and I(u 0 ) ≥ 0, then the weak solution u(x, t) is globally bounded. The rest of the paper is organized as follows. In Section 2, we give some related spaces and lemmas. In Section 3, we give the proof process for the main results of problem (1.1).

Preliminaries
In order to facilitate the proof of the main results, we start this section by introducing some symbols and Lemmas that will be used throughout the paper.
In this section, we assume that 0 < s < 1 < p < 2 and Ω ∈ R N (N > 2s) is a bounded domain with Lipschitz boundary. We denote by u i (i ≥ 1) the norm of Lebesgue space L i (Ω). Let W s,p (Ω) be the linear space of Lebesgue measurable functions u from R N to R such that the restriction to Ω of any function u in W s,p (Ω) belongs to L p (Ω) and We further give a closed linear subspace W s,p 0 (Ω) = u ∈ W s,p (Ω)|u(x) = 0 a.e. in R N \Ω .
As shown in [19], it can be concluded that is an equivalent norm of W s,p 0 (Ω). Next we give the necessary Lemmas.
Then by the definition of I(u), we obtain where C r+σ is the embedding constant for W s,p 0 (Ω) → L r+σ (Ω). We can get If 0 < u ≤ R, then it follows from the definition of R that a − λ 1 σ C r+σ r+σ u r+σ−p ≥ 0, thus (i) holds.
By a simple calculation, we get We claim that u(x, t) ∈ W for any t > 0. If it is false, there exists a t 0 ∈ R + 0 \{0} such that u(t 0 ) ∈ ∂W, which implies I(u(x, t 0 )) = 0 or E(u(x, t 0 )) = h.
Remark 2 Compared with problem (1.4), we not only discuss the extinction of weak solutions of problem (1.1) with logarithmic nonlinearity, but also prove that the weak solutions are globally bounded and blow up at infinity.