Global existence and blow-up analysis for parabolic equations with nonlocal source and nonlinear boundary conditions

Abstract

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.

Keywords

• Nonlinear parabolic equations
• Blow-up
• Nonlocal source
• Lower bound