Blow-up phenomena for a parabolic system with gradient nonlinearity under nonlinear boundary conditions

In this paper, we consider the following reaction diffusion systems with gradient nonlinearity under nonlinear boundary condition {u(t) = Delta u + u(p)v(q) - vertical bar del u vertical bar(alpha), (x, t) is an element of Omega x (0, t*); v(t) = Delta v + u(r)v(s) - vertical bar del v vertical bar(alpha), (x, t) is an element of Omega x (0, t*); partial derivative u/partial derivative v = g(u), partial derivative u/partial derivative v = h(v), (x, t) is an element of partial derivative Omega x (0, t*); u(x, 0) = u(0) (x), v(x, 0) = v(0) (x), x is an element of Omega where Omega subset of R-N (N >= 1) is a bounded region with smooth boundary partial derivative Omega, p, q, r, s >= 0, alpha > 1, t* is a possible blow-up time when blow-up occurs. By constructing an appropriate auxiliary functions, and by means of Payne-Weinberger or Scott's method, a lower bound on blow-up time when blow-up occurs is derived. (C) 2017 Elsevier Ltd. All rights reserved.