Blow-up and global existence for nonlinear reaction-diffusion equations under Neumann boundary conditions

Abstract

In this paper, we study the blow-up and global solutions of the following nonlinear reaction-diffusion equations under Neumann boundary conditions: $$\left \{ \textstyle\begin{array}{l@{\quad}l} (g(u) )_{t} =\nabla\cdot(a(u)b(x)\nabla u)+f(x,u) &\mbox{in } D\times(0,T), \frac{\partial u}{\partial n}=0 &\mbox{on } \partial D\times(0,T), u(x,0)=u_{0}(x)>0 & \mbox{in } \overline{D}, \end{array}\displaystyle \right . $$ where $D\subset\mathbb{R}^{N}$ ( $N\geq2$ ) is a bounded domain with smooth boundary ∂D. By constructing auxiliary functions and using maximum principles and a first-order differential inequality technique, sufficient conditions for the existence of the blow-up solution, an upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, sufficient conditions for the existence of global solution, and an upper estimate of the global solution are specified under some appropriate assumptions on the functions a, b, f, g, and initial value $u_{0}$ .