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 u t = Δ u + a ∫ Ω u p d x , x , t ∈ Q T , n · ∇ u + g u u = 0 , x , t ∈ S T , u x , 0 = u 0 x , x ∈ Ω with nonlocal sources under nonlinear heat-loss boundary conditions, where a , p > 0 is constant, Q T = Ω × 0 , T , S T = ∂ Ω × 0 , T , and Ω is a bounded region in R N , 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.