Global and Blow-Up Solutions for a Class of Nonlinear Parabolic Problems under Robin Boundary Condition

Abstract

We discuss the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions: $(b(u){)}_t=\nabla ·(h(t)k(x)a(u)\nabla u)+f(x,u,|\nabla u{|}^2,t)$, in $D\times (0,T)$, $(\partial u/\partial n)+\gamma u=0$, on $\partial D\times (0,T)$, $u(x,0)=u_0(x)>0$, in $\overline{D}$, where $D\subset {\mathbb{R}}^N(N\ge 2)$ is a bounded domain with smooth boundary $\partial D$. Under some appropriate assumption on the functions $f$, $h$, $k$, $b$, and $a$ and initial value $u_0$, we obtain the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for “blow-up time,” and an upper estimate of “blow-up rate.” Our approach depends heavily on the maximum principles.