Blow-up of solutions to parabolic equations with nonstandard growth conditions

We study the phenomenon of finite time blow-up in solutions of the homogeneous Dirichlet problem for the parabolic equation u(t) = div (a(x, t)vertical bar del u vertical bar(p(x)-2)del u) + b(x, t)vertical bar u vertical bar(sigma(x, t)-2)u with variable exponents of nonlinearity p(x), sigma(x, t) is an element of (1, infinity). Two different cases are studied. In the case of semilinear equation with p(x) congruent to 2, a(x, t) equivalent to 1, b(x. t) >= b(-) > 0 we show that the finite time blow-up happens if the initial function is sufficiently large and either min(Omega) sigma(x, t) = sigma(-)(t) > 2 for all t > 0, or sigma(-)(t) >= 2, sigma(-)(t) SE arrow 2 as t -> infinity and integral(infinity)(1) e(s(2-sigma-(s))) ds < infinity. In the case of the evolution p(x)-Laplace equation with the exponents p(x), sigma(x) independent of t, we prove that every solution corresponding to a sufficiently large initial function exhibits a finite time blow-up if b(x, t) >= b(-) > 0, a(t) (x, t) <= 0, b(t)(x, t) >= 0, min sigma(x) > 2 and max p(x) <= min sigma(x). (C) 2010 Elsevier B.V. All rights reserved.