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

Abstract

AbstractWe investigate the following nonlinear parabolic equations with nonlocal source and nonlinear boundary conditions: $$ \textstyle\begin{cases} (g(u) )_{t} =\sum_{i,j=1}^{N} (a^{ij}(x)u_{x_{i}} ) _{x_{j}}+\gamma _{1}u^{m} (\int _{D} u^{l}{\,\mathrm{d}}x ) ^{p}-\gamma _{2}u^{r}& \mbox{in } D\times (0,t^{*}), \\ \sum_{i,j=1}^{N}a^{ij}(x)u_{x_{i}}\nu _{j}=h(u) & \mbox{on } \partial D\times (0,t^{*}), \\ u(x,0)=u_{0}(x)\geq 0 &\mbox{in } \overline{D}, \end{cases} $${(g(u))t=∑i,j=1N(aij(x)uxi)xj+γ1um(∫Duldx)p−γ2urin D×(0,t∗),∑i,j=1Naij(x)uxiνj=h(u)on ∂D×(0,t∗),u(x,0)=u0(x)≥0in D‾, where p and $\gamma _{1}$γ1 are some nonnegative constants, m, l, $\gamma _{2}$γ2, and r are some positive constants, $D\subset \mathbb{R}^{N}$D⊂RN ($N\geq 2$N≥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.