Abstract
Abstract This paper is devoted to discussing the blow-up problem of the positive solution of the following degenerate parabolic equations: ( r ( u ) ) t = div ( ∣ ∇ u ∣ p ∇ u ) + f ( x , t , u , ∣ ∇ u ∣ 2 ) , ( x , t ) ∈ D × ( 0 , T ∗ ) , ∂ u ∂ ν + σ u = 0 , ( x , t ) ∈ ∂ D × ( 0 , T ∗ ) , u ( x , 0 ) = u 0 ( x ) , x ∈ D ¯ . \left\{\begin{array}{ll}{(r\left(u))}_{t}={\rm{div}}(| \nabla u{| }^{p}\nabla u)+f\left(x,t,u,| \nabla u\hspace{-0.25em}{| }^{2}),& \left(x,t)\in D\times \left(0,{T}^{\ast }),\\ \frac{\partial u}{\partial \nu }+\sigma u=0,& \left(x,t)\in \partial D\times \left(0,{T}^{\ast }),\\ u\left(x,0)={u}_{0}\left(x),& x\in \overline{D}.\end{array}\right. Here p > 0 p\gt 0 , the spatial region D ⊂ R n ( n ≥ 2 ) D\subset {{\mathbb{R}}}^{n}\hspace{0.33em}\left(n\ge 2) is bounded, and its boundary ∂ D \partial D is smooth. We give the conditions that cause the positive solution of this degenerate parabolic problem to blow up. At the same time, for the positive blow-up solution of this problem, we also obtain an upper bound of the blow-up time and an upper estimate of the blow-up rate. We mainly carry out our research by means of maximum principles and first-order differential inequality technique.
Highlights
IntroductionThe blow-up problem of the degenerate parabolic equations has attracted the attention and research of many scholars (see, for example [1,2,3,4,5,6,7,8,9])
Over the past decade, the blow-up problem of the degenerate parabolic equations has attracted the attention and research of many scholars
For the positive blow-up solution of this problem, we obtain an upper bound of the blow-up time and an upper estimate of the blow-up rate
Summary
The blow-up problem of the degenerate parabolic equations has attracted the attention and research of many scholars (see, for example [1,2,3,4,5,6,7,8,9]). The purpose of this paper is to study the blow-up positive solutions of the following degenerate parabolic problems:. In problem (1), p > 0, the spatial region D ⊂ n (n ≥ 2) is bounded, and its boundary ∂D is smooth, T∗ represents the blow-up time of the solution, ∂ represents the external normal derivative, the function. Ding studied the blow-up problem of the following nondegenerate parabolic equations in the paper [13]:. With the aid of maximum principles and first-order differential inequality technique, he gave the conditions for the blowup of the positive solution of problem (2). In problem (3), p > 0, the spatial region D ⊂ n (n ≥ 2) is bounded, and its boundary ∂D is smooth They used first-order differential inequality technique to give the conditions that make the positive solution of problem (3) blow up.
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