Global existence and blow-up results for p-Laplacian parabolic problems under nonlinear boundary conditions

This paper is devoted to studying the global existence and blow-up results for the following p-Laplacian parabolic problems: {(h(u))t=∇⋅(|∇u|p−2∇u)+f(u)in D×(0,t∗),∂u∂n=g(u)on ∂D×(0,t∗),u(x,0)=u0(x)≥0in D‾.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textstyle\begin{cases} (h(u) )_{t} =\nabla\cdot (|\nabla u|^{p-2}\nabla u )+f(u) &\mbox{in } D\times(0,t^{*}), \\ \frac{\partial u}{\partial n}=g(u) &\mbox{on } \partial D\times (0,t^{*}), \\ u(x,0)=u_{0}(x)\geq0 & \mbox{in } \overline{D}. \end{cases} $$\end{document} Here p>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p>2$\end{document}, the spatial region D in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{N}$\end{document} (N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\geq2$\end{document}) is bounded, and ∂D is smooth. We set up conditions to ensure that the solution must be a global solution or blows up in some finite time. Moreover, we dedicate upper estimates of the global solution and the blow-up rate. An upper bound for the blow-up time is also specified. Our research relies mainly on constructing some auxiliary functions and using the parabolic maximum principles and the differential inequality technique.

(1. 2) In (1.2), p ≥ 2, the spatial region D in R N (N ≥ 2) is bounded and convex, and ∂D is smooth. By constructing some auxiliary functions and using the differential inequality technique, they established the conditions on functions f , g, h, and u 0 to ensure that the solution u blows up at some time. In addition, an upper bound and a lower bound of the blow-up time were obtained. The method in [1] is not suitable for the study of problem (1.1) because of the different boundary conditions of problem (1.1) and problem (1.2). Zhang et al. [16] and Zhang [22] dealt with the following problem: In (1.3), the spatial region D in R N (N ≥ 2) is bounded, and ∂D is smooth. By constructing some auxiliary functions and using parabolic maximum principles, they set up the conditions on functions a, f , g, h, and u 0 to guarantee that the solution either blows up in a finite time or exists globally. Moreover, an upper estimate of the blow-up rate and an upper bound of the blow-up time are given. They also obtained an upper estimate of the global solution. We intend to use the methods in [16] and [22] to study problem (1.1).
Since the principal parts of the two equations are different in problems (1.1) and (1.3), the auxiliary functions in papers [16] and [22] are not suitable for problem (1.1). Therefore, the key to our research is to construct some new auxiliary functions. By using these new auxiliary functions, parabolic maximum principles, and differential inequality techniques, we complete the study of problem (1.1). We proceed as follows. In Sect. 2, we set up some conditions to ensure that the solution blows up in a finite time. An upper estimate of the blow-up solution and an upper bound of the blow-up time are also given. Section 3 is devoted to finding some conditions to guarantee that the solution exists globally. At the same time, we also obtain an upper estimate of the global solution. In Sect. 4, as applications of the abstract results, two examples are presented.
In this paper, for convenience, we use a comma to denote partial differentiation, for example, u ,i = ∂u ∂x i , u ,ij = ∂ 2 u ∂x i ∂x j . We also adopt summation convection, for example,

Blow-up solution
In order to study the blow-up solution of (1.1), we define We also construct the following two auxiliary functions: Since g(s) is a positive C 2 (R + ) function, we have which means that the function has an inverse function -1 . With the aid of the above two auxiliary functions, we can get the following Theorem 2.1 that is the main result on the blow-up solution.
Theorem 2.1 Let u be a nonnegative classical solution of (1.1). Assume that the following three assumptions are true: Then the solution u must blow up in a finite time t * and Proof For the auxiliary function P(x, t) defined in (2.3), by calculating, we have and With (2.9), we get Using the first equation of (1.1), we obtain It follows from (2.9)-(2.11) that and (2.14) Inserting (2.13) and (2.14) into (2.12), we arrive at It follows from the first equation of (1.1) that The regularity assumptions on functions f , g, and h in Sect. 1, parabolic maximum principles [27], and (2.20) imply that under the following three possible cases, P may take its nonnegative maximum value: (a) for t = 0, (b) at a point where |∇u| = 0, (c) on the boundary ∂D × (0, t * ). First, we consider case (a). With (2.2), we deduce Then, we consider case (b). Assume that (x,t) ∈ D × (0, t * ) is a point where |∇u(x,t)| = 0. Now we have Hence, we obtain It follows from (2.22), (2.1), and (2.5) that Finally, we consider case (c). Making use of the boundary condition of (1.1), we get Parabolic maximum principles, (2.21), and (2.23)-(2.24) guarantee that the maximum value of P in D × [0, t * ) is nonpositive. Hence, we have from which we obtain the following differential inequality: At the pointx ∈ D, where u 0 (x) = M 0 , integrating (2.25) from 0 to t, we derive (2.26) which implies that u must blow up in a finite time t * . In fact, suppose that u is a global solution, then for any t > 0, we deduce Taking the limit as t → +∞ in (2.27), we arrive at which contradicts (2.6). This contradiction suggests that u must blow up in a finite time t * .
Letting t → t * in (2.26), we have For each fixed x ∈ D, integrating (2.25) from t tot (0 < t <t < t * ), we get (2.28) Passing to the limit ast → t * in (2.28), we obtain from which we deduce The proof is complete.

Global solution
In order to complete the study of the global solution to (1.1), we define We also construct the following two auxiliary functions: Here g(s) is a positive C 2 (R + ) function to ensure This implies that the inverse function -1 of the function exists. The following Theorem 3.1 is the main result of the global solution to problem (1.1).
Theorem 3.1 Let u be a nonnegative classical solution of (1.1). Assume that the following three assumptions are satisfied: (3.7) Then u must be a global solution and Proof By using the reasoning process (2.8)-(2.19) for the auxiliary function Q defined in (3.3), we have It follows from (3.7) and (3.8) that The parabolic maximum principle guarantees that in the following three possible cases, Q may take its nonpositive minimum value: (a) for t = 0, (b) at a point where |∇u| = 0, (c) on the boundary ∂D × (0, t * ). First, case (a) is considered. By (3.2), we deduce Then, case (b) is considered. Repeating the reasoning process of (2.23) and using (2.22), (3.1), and (3.5), we have where (x,t) ∈ D × (0, t * ) is a point where |∇u(x,t)| = 0. Finally, case (c) is considered. With the aid of the reasoning process in (2.24), it is easy to get ∂Q ∂n = ηe u ∂u ∂n = ηe u g > 0 in ∂D × 0, t * . (3.11) Combining (3.9)-(3.11) and parabolic maximum principles, we can obtain that the minimum value of Q in D × [0, t * ) is nonnegative. In other words, we have which implies that the following differential inequality holds: For each fixed x ∈ D, integrating (3.12) from 0 to t, we deduce which guarantees that u must be a global solution. In fact, if we assume that u blows up at a finite time t * , then the following conclusion holds: Letting t → t *in (3.13), we have m 0 e -τ g(τ ) dτ + t * < +∞, which contradicts (3.6). This shows that u must be a global solution. It follows from (3.13) that The proof is complete.

Applications
In this section, we give two examples to illustrate the results of Theorems 2.1 and 3.1.
Example 4.1 Let u be a nonnegative classical solution of the following problem: where the spatial region