Blow-up phenomena in porous medium equation systems with nonlinear boundary conditions

This paper deals with the blow-up phenomena for the following porous medium equation systems with nonlinear boundary conditions {u(t) = Delta u(m) + k(1)(t)f(1)(v), v(t) = Delta v(n) + k(2)(t)f(2)(u) in Omega x (0, t*), partial derivative u/partial derivative nu = g(1)(u), partial derivative v/partial derivative nu = g(2)(v) on partial derivative Omega x (0, t*), u(x, 0) = u(0)(x) >= 0, v(0)(x, 0) = v(0)(x) >= 0 in (Omega) over bar, where m, n > 1, Omega subset of R-N(N >= 2) is bounded convex domain with smooth boundary. Using a differential inequality technique and a Sobolev inequality, we prove that under certain conditions on data, the solution blows up in finite time. We also derive an upper and a lower bound for blow-up time. In addition, as applications of the abstract results obtained in this paper, an example is given. (C) 2019 Elsevier Ltd. All rights reserved.