Global Existence and Blow-Up of Solutions to a Parabolic Nonlocal Equation Arising in a Theory of Thermal Explosion

Abstract

Focusing on the physical context of the thermal explosion model, this paper investigates a semilinear parabolic equation $\left\{\begin{array}{cc}{u}_t=\Delta u+a{\int }_{\Omega }{u}^{p}dx,\hfill & \left(x,t\right)\in Q_T,\hfill \\ n·\nabla u+g\left(u\right)u=0,\hfill & \left(x,t\right)\in S_T,\hfill \\ u\left(x,0\right)=u_0\left(x\right),\hfill & x\in \Omega \hfill \end{array}\right.$ with nonlocal sources under nonlinear heat-loss boundary conditions, where $a,p>0$ is constant, $Q_T=\Omega \times \left(0,T\right]$ , $S_T=\partial \Omega \times \left(0,T\right]$ , and $\Omega$ is a bounded region in $R^N,N\ge 1$ with a smooth boundary $\partial \Omega$ . First, we prove a comparison principle for some kinds of semilinear parabolic equations under nonlinear boundary conditions; using it, we show a new theorem of subsupersolutions. Secondly, based on the new method of subsupersolutions, the existence of global solutions and blow-up solutions is presented for different values of $p$ . Finally, the blow-up rate for solutions is estimated also.