Instantaneous blow-up for evolution inequalities of Sobolev type with nonlinear convolution terms

Abstract

We consider evolution inequalities of Sobolev type involving nonlinearities of the form $ |x|^{\sigma-N}*|u|^p $ and $ |x|^{\sigma-N}*|\nabla u|^p $, where $ * $ is the convolution product in $ \mathbb{R}^N $, $ p>1 $ and $ 0<\sigma<N $. For each case, we prove the existence of a critical exponent $ p_{cr}(\sigma, N)\in (1, \infty] $ depending on the parameter $ \sigma $ and the dimension $ N $, in the following sense: if $ 1<p\leq p_{cr}(\sigma, N) $, then there is no local weak solutions; if $ p>p_{cr}(\sigma, N) $, then local weak solutions exist for some initial data.