On the blow-up of solutions to anisotropic parabolic equations with variable nonlinearity

Abstract

The aim of this paper is to establish sufficient conditions of the finite time blow-up in solutions of the homogeneous Dirichlet problem for the anisotropic parabolic equations with variable nonlinearity \( u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} } \) . Two different cases are studied. In the first case ai ≡ ai(x), pi ≡ 2, σi ≡ σi(x, t), and bi(x, t) ≥ 0. We show that in this case every solution corresponding to a “large” initial function blows up in finite time if there exists at least one j for which min σj(x, t) > 2 and either bj > 0, or bj(x, t) ≥ 0 and Σπbj−ρ(t)(x, t) dx 0 depending on σj. In the case of the quasilinear equation with the exponents pi and σi depending only on x, we show that the solutions may blow up if min σi ≥ max pi, bi ≥ 0, and there exists at least one j for which min σj > max pj and bj > 0. We extend these results to a semilinear equation with nonlocal forcing terms and quasilinear equations which combine the absorption (bi ≤ 0) and reaction terms.