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Abstract We investigate the following nonlinear parabolic equations with nonlocal source and nonlinear boundary conditions: { ( g ( u ) ) t = ∑ i , j = 1 N ( a i j ( x ) u x i ) x j + γ 1 u m ( ∫ D u l d x ) p − γ 2 u r in D × ( 0 , t ∗ ) , ∑ i , j = 1 N a i j ( x ) u x i ν j = h ( u ) on ∂ D × ( 0 , t ∗ ) , u ( x , 0 ) = u 0 ( x ) ≥ 0 in D ‾ , $$ \textstyle\begin{cases} (g(u) )_{t} =\sum_{i,j=1}^{N} (a^{ij}(x)u_{x_{i}} ) _{x_{j}}+\gamma _{1}u^{m} (\int _{D} u^{l}{\,\mathrm{d}}x ) ^{p}-\gamma _{2}u^{r}& \mbox{in } D\times (0,t^{*}), \\ \sum_{i,j=1}^{N}a^{ij}(x)u_{x_{i}}\nu _{j}=h(u) & \mbox{on } \partial D\times (0,t^{*}), \\ u(x,0)=u_{0}(x)\geq 0 &\mbox{in } \overline{D}, \end{cases} $$ where p and γ 1 $\gamma _{1}$ are some nonnegative constants, m, l, γ 2 $\gamma _{2}$ , and r are some positive constants, D ⊂ R N $D\subset \mathbb{R}^{N}$ ( N ≥ 2 $N\geq 2$ ) is a bounded convex region with smooth boundary ∂D. By making use of differential inequality technique and the embedding theorems in Sobolev spaces and constructing some auxiliary functions, we obtain a criterion to guarantee the global existence of the solution and a criterion to ensure that the solution blows up in finite time. Furthermore, an upper bound and a lower bound for the blow-up time are obtained. Finally, some examples are given to illustrate the results in this paper.

Abstract In this paper, we deal with the blow-up and global solutions of the following p-Laplacian parabolic problems with Neumann boundary conditions: { ( g ( u ) ) t = ∇ ⋅ ( | ∇ u | p − 2 ∇ u ) + k ( t ) f ( u ) in Ω × ( 0 , T ) , ∂ u ∂ n = 0 on ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) ≥ 0 in Ω ‾ , $$\textstyle\begin{cases} (g(u) )_{t} =\nabla\cdot ( {|\nabla u|^{p-2}}\nabla u )+k(t)f(u) & \mbox{in } \Omega\times(0,T), \\ \frac{\partial{u}}{\partial n}=0 &\mbox{on } \partial\Omega\times (0,T), \\ u(x,0)=u_{0}(x)\geq0 & \mbox{in } \overline{\Omega}, \end{cases} $$ where p > 2 $p>2$ and Ω is a bounded domain in R n $\mathbb{R}^{n}$ ( n ≥ 2 $n\geq 2$ ) with smooth boundary ∂Ω. By introducing some appropriate auxiliary functions and technically using maximum principles, we establish conditions to guarantee that the solution blows up in some finite time or remains global. In addition, the upper estimates of blow-up rate and global solution are specified. We also obtain an upper bound of blow-up time.

This paper deals with the global existence and blow-up of the solution for a class of nonlinear reaction diffusion problems. The purpose of this paper is to establish conditions on the data to guarantee the blow-up of the solution at some finite time, and conditions to ensure that the solution remains global. In addition, an upper bound for the 'blow-up time', an upper estimate of the 'blow-up rate', and an upper estimate of the global solution are also specified. Finally, as applications of the obtained results, some examples are presented. [ABSTRACT FROM AUTHOR]