Critical non-linearity for some evolution equations with Fujita-type critical exponent

Abstract

We consider the Cauchy problem for a class of non-linear evolution equations in the form \[L(\partial_t,\partial_x) u=F(\partial_t^\ell u), \quad (t,x)\in [0,\infty)\times \mathbb{R}^n;\] here, $L(\partial_t,\partial_x)$ is a linear partial differential operator with constant coefficients, of order $m\geq 1$ with respect to the time variable $t$, and $\ell$ is a natural number satisfying $0\leq \ell\leq m-1$. For several different choices of $L$, many authors have investigated the existence of global (in time) solutions to this problem when $F(s)=|s|^p$ is a power non-linearity, looking for a \textit{critical exponent} $p_c>1$ such that global small data solutions exist in the supercritical case $p>p_c$, whereas no global weak solutions exist, under suitable sign assumptions on the data, in the subcritical case $1<p<p_c$. In the present paper we consider a more general non-linear term in the form $F(s)=|s|^pμ(|s|)$; for a large class of models, we provide an integral condition on $μ$ which allows to distinguish more precisely the region of existence of a global (in time) small data solution from that in which the problem admits no global (in time) weak solutions, refining the existing results about the critical exponents for power type non-linearities.