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 ut=Δu+a∫Ωupdx,x,t∈QT,n·∇u+guu=0,x,t∈ST,ux,0=u0x,x∈Ω with nonlocal sources under nonlinear heat-loss boundary conditions, where a,p>0 is constant, QT=Ω×0,T, ST=∂Ω×0,T, and Ω is a bounded region in RN,N≥1 with a smooth boundary ∂Ω. 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.