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

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 $u_t=\sum_{i=1}^nD_i\bigl(a_i(x,t)|D_iu|^{p_i(x)-2}D_iu\bigr)+\sum_{i=1}^Kb_i(x,t)|u|^{\sigma_i(x,t)-2}u$. Two different cases are studied. In the first case $a_i\equiv a_i(x)$, $p_i\equiv2$, $\sigma_i\equiv\sigma_i(x,t)$, and $b_i(x,t)\geq0$. 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 $\min\sigma_j(x,t)>2$ and either $b_j>0$, or $b_j(x,t)\geq0$ and $\int_\Omega b_j^{-\rho(t)}(x,t)\,dx<\infty$ with some $\rho(t)>0$ depending on $\sigma_j$. In the case of the quasilinear equation with the exponents $p_i$ and $\sigma_i$ depending only on $x$, we show that the solutions may blow up if $\min\sigma_i\geq\max p_i$, $b_i\geq0$, and there exists at least one $j$ for which $\min\sigma_j>\max p_j$ and $b_j>0$. We extend these results to a semilinear equation with nonlocal forcing terms and quasilinear equations which combine the absorption ($b_i\leq0$) and reaction terms.