Blow-up solutions for a class of nonlinear parabolic equations with Dirichlet boundary conditions

In the present paper, the blow up of smooth local solutions for a. class of nonlinear parabolic equations u,(t) =del(a(u)delu) + f(x, u, q, t) (q = \delu\(2)) with Dirichlet boundary conditions are studied. By constructing an auxiliary function and using Hopf's maximum principles on it, the sufficient conditions for blow-up solutions are obtained and the upper bound of "blow-up time" is given under some suitable assumptions on a, f and initial date. The obtained results are applied to some examples in which a and f are exponential functions.. (C) 2003 Elsevier Science Ltd. All rights reserved.