Lower bound for the blow-up time for a general nonlinear nonlocal porous medium equation under nonlinear boundary condition

In this paper, we study the blow-up phenomenon for a general nonlinear nonlocal porous medium equation in a bounded convex domain (Ω∈Rn,n≥3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\varOmega\in \mathbb{R}^{n}, n\geq 3)$\end{document} with smooth boundary. Using the technique of a differential inequality and a Sobolev inequality, we derive the lower bound for the blow-up time under the nonlinear boundary condition if blow-up does really occur.


Introduction
Liu in paper [1] studied the blow-up phenomena for the solution of the following problems: under the Robin boundary condition He obtained a lower bound for the blow-up time of the system when the solution blows up.
In paper [2], the authors also studied equations (1.1) and (1.2) subject to either homogeneous Dirichlet boundary condition or homogeneous Neumann boundary condition. The lower bounds for the blow-up time under the above two boundary conditions were obtained. Equation (1.1) is used in the study of population dynamics (see [3]). For other systems in porous medium, one could see [4]. There have been a lot of papers in the literature on studying the question of blow-up for the solution of parabolic problems under a homogeneous Dirichlet boundary condition and Neumann boundary condition(one can see [5][6][7][8][9][10][11][12]). Some authors have started to consider the blow-up of these problems under Robin boundary conditions (see [13][14][15][16][17]). In papers [18][19][20][21], the authors studied the blow-up phenomena for the heat equation under nonlinear boundary conditions. Some new results about the nonlinear evolution equations may be founded in [22][23][24]. These papers have mainly focused on the bounded convex domain in R 3 . Recently, there have been some papers starting to study the blow-up problems in R n (n ≥ 3) (see [25][26][27][28][29]). We continue the work of [2] for a more general equation. Until now, the authors have not found any paper dealing with lower bound for the blow-up time of a nonlinear nonlocal porous medium equation under nonlinear boundary condition in R n (n ≥ 3). In this sense, the result obtained in this paper is new and interesting. In this paper, we consider the blow-up phenomena of the solution for the following equation: with the following boundary initial conditions: where Ω is a bounded convex domain in R n , n ≥ 3, with sufficiently smooth boundary, is the Laplace operator, ∂Ω is the boundary of Ω, and t * is the possible blow-up time, ∂u ∂ν is the outward normal derivative of u. We assume where s > max{ 2n 2n-1 , p + q + 1 -m}.

Some useful inequalities
We will use the following useful inequalities later in the proof.

Lemma 2.1
We suppose that u is a nonnegative function and σ , m are positive constants, then we have the result as follows: where ρ 0 := min ∂Ω |x · ν|, ν is the outward normal vector of ∂Ω and d := max ∂Ω |x|.
Applying the divergence theorem, we obtain Because Ω is a convex domain, we have ρ 0 := min ∂Ω |x · ν| > 0. Then we derive Lemma 2.2 Supposing that u ∈ W 1,2 (Ω) and n ≥ 3, we have where C = C(n, Ω) is a Sobolev embedding constant depending on n and Ω.

Lower bound for the blow-up time
In this section it is useful in the sequel to define an auxiliary function of the following form: We will derive a differential inequality for φ(t). From the inequality, we can establish the following theorem.
from which it follows that the blow-up time t * is bounded below. We have where Θ -1 is the inverse function of Θ, and a(t), b(t) are defined in (3.21), (3.22) respectively.
First we compute Integrating by parts, we have Using the result of Lemma 2.1, we obtain M . Now we estimate the third term of the right-hand side of (3.4). Using Hölder's inequality, we have Then we obtain σ +m-1 ) 2 , ε 1 is a positive constant which will be defined later. From the above deductions, we get Combining (3.4) and (3.5), we obtain where Using Hölder's and Young's inequalities, we have where r 7 = n-2-x 1 n n-2 ( n-2
From the definition of Θ(t), we have dΘ(t) dt = b(t)e 4H(t) > 0. We get Θ(t) is a strictly increasing function. So we can get t * ≥ Θ -1 1 4φ 4 (0) , from which we complete the proof of Theorem 3.1.