Blow-Up Solutions and Global Existence for a Kind of Quasilinear Reaction-Diffusion Equations

Abstract

In this paper, we study the blow-up solutions and global existence for a quasilinear reaction-diffusion equation including a gradient term and nonlinear boundary condition: $$ \left{ \begin{alignedat}{2} (g(u)){t}&=\nabla\cdot(a(u)\nabla u)+f(x,u,|\nabla u|^{2},t)&\quad &\text{in} \ D\times(0,T)\ %\[0.5em] %\displaystyle \tfrac{\partial u}{\partial n}&=r(u)& &\rm{on} \ \partial D\times(0,T)\ %%\[0.5em] u(x,0)&=u{0}(x)>0& &\rm{in} \ \overline{D}, \end{alignedat} \right. $$ where $D\subset R^{N}$ is a bounded domain with smooth boundary $\partial D$. The sufficient conditions are obtained for the existence of a global solution and a blow-up solution. 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 specified under some appropriate assumptions for the nonlinear system functions $f, g, r,a$, and initial value $u\_{0}$ by constructing suitable auxiliary functions and using maximum principles.