The blow-up properties of solutions to semilinear heat equations with nonlinear boundary conditions

Abstract

This paper deals with the blow-up properties of solutions to semilinear heat equation \(u_t-u_{xx}=u^p \text{ in }(0,1)\times(0,T)\) with the nonlinear boundary conditions \(u_x(0,t)=0,u_x(1,t)=u^q\text{ on }[0,T)\). The necessary and sufficient conditions for the solution to have a finite time blow-up and the exact blow-up rates are established. It is also proved that the blow-up will occur only at the boundary x = 1.