On large time behavior of solutions of higher order evolution inequalities with fast diffusion

Abstract

We obtain stabilization conditions and large time estimates for weak solutions of the inequality $$ \sum_{|α| = m} \partial^α a_α(x, t, u) - u_t \ge f (x, t) g (u) \quad \mbox{in } Ω\times (0, \infty), $$ where $\Omega$ is a non-empty open subset of ${\mathbb{R}}^n$, $m,n\ge 1$, and $a_{\alpha }$ are Caratheodory functions such that $$|a_{\alpha }(x,t,\zeta )|\le A{\zeta }^p,|\alpha |=m,$$ with some constants $A>0$ and $0<p<1$ for almost all $(x,t)\in \Omega \times (0,\infty )$ and for all $\zeta \in [0,\infty )$. For solutions of homogeneous differential inequalities, we give an exact universal upper bound.