Abstract
We investigate a slow diffusion equation with nonlocal source and inner absorption subject to homogeneous Dirichlet boundary condition or homogeneous Neumann boundary condition. Based on an auxiliary function method and a differential inequality technique, lower bounds for the blow-up time are given if the blow-up occurs in finite time.
Highlights
We investigate a slow diffusion equation with nonlocal source and inner absorption subject to homogeneous Dirichlet boundary condition or homogeneous Neumann boundary condition
Our main interest lies in the following slow diffusion equation with nonlocal source term and inner absorption term: ut = Δum + up ∫ uqdx − kus, Ω
0, (x, t) ∈ ∂Ω × (0, t∗), (3b) where Ω ⊂ R3 is a bounded domain with smooth boundary
Summary
The bounds for the blow-up time of the blow-up solutions to nonlinear diffusion equations have been widely studied in recent years. Payne and Schaefer in [7, 8] used a differential inequality technique and an auxiliary function method to obtain a lower bound on blow-up time for solution of the heat equation with local source term under boundary condition (3a) or (3b). Song [9] considered the lower bounds for the blow-up time of the blow-up solution to the nonlocal problem (1)-(2) when m = 1 and p = 0, subject to homogeneous boundary condition (3a) or (3b); for the case k = 0, we refer to [10]. Motivated by the works above, we investigate the lower bounds for the blow-up time of the blow-up solutions to Mathematical Problems in Engineering the nonlocal problem (1)-(2) with homogeneous boundary condition (3a) or (3b).
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