On blow-up results for solutions of inhomogeneous evolution equations and inequalities. II

Abstract

The main purpose of this work is to further develop ideas and methods from a recent paper by the authors. In particular, we obtain new blow-up results for solutions of the inequality $$ |u|_t \geq \Delta [|u|^\sigma u] + |u|^{q} + \omega (x) $$ on the half-space ${\mathbb R}^1_+ \times {\mathbb R}^n$, where $n\geq 1$, $\sigma\geq 0$, $q>1+\sigma$, and $\omega: {\mathbb R}^n \to {\mathbb R}^1$ is a globally integrable function such that $\int_{{\mathbb R}^n} \omega (x) dx >0$, and establish that for $n>2$ the critical blow-up exponent $q^*=n(1+\sigma)/(n-2)$ is of the blow-up type.