Blow‐up analysis for a class of nonlinear reaction diffusion equations with Robin boundary conditions

E-ISSN： 1099-1476|41|4|1683-1696

ISSN： 0170-4214

Source： MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Vol.41, Iss.4, 2018-03, pp. : 1683-1696

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Abstract

This paper is devoted to the study of the blow‐up phenomena of following nonlinear reaction diffusion equations with Robin boundary conditions: $\{\begin{matrix} {(g (u))}_{t} = \nabla \cdot (ρ (| \nabla u |^{2}) \nabla u) + k (t) f (u) & in Ω \times (0, t^{*}), \\ \frac{\partial u}{\partial ν} + γ u = 0 & on \partial Ω \times (0, t^{*}), \\ u (x, 0) = u_{0} (x) & in \bar{Ω} . \end{matrix}$ Here, $Ω \subset R^{n} (n ⩾ 2)$ is a bounded convex domain with smooth boundary. With the aid of a differential inequality technique and maximum principles, we establish a blow‐up or non–blow‐up criterion under some appropriate assumptions on the functions f,g,ρ,k, and u₀. Moreover, we dedicate an upper bound and a lower bound for the blow‐up time when blowup occurs.