Thermal blow-up in a subdiffusive medium due to a nonlinear boundary flux

Abstract

Abstract A fractional heat equation is used to model thermal diffusion in a one-dimensional bar that exhibits subdiffusive behavior. The left end of the bar is subjected to a nonlinear influx of heat. For the boundary constraint at the right end of the bar, two cases are considered, namely a homogeneous Neumann condition and a homogeneous Dirichlet condition. By reducing both cases to a nonlinear Volterra equation, it is shown that a blow-up always occurs. The asymptotic behavior near the blow-up is determined for both cases. It is also shown that the solution for the Neumann case dominates that of the Dirichlet case.