On the two critical exponents for a dispersive equation with weak dissipation

Abstract

In this paper, we study a nonlinear Cauchy problem written in normal form \begin{equation*} \begin{cases} u_{tt} + A u+ Bu_t =D|u|^α, & t\geq 0, x \in \mathbb{R}^n\\ u(0,x)=u_0(x),\,u_t(0,x)=u_1(x). \end{cases} \end{equation*} Here~ $Af={\mathcal{F}}^{-1}(a(\xi {)}^2\mathcal{F}f)$, with $a(\xi )$ homogeneous of order $k>0$ under the assumption that the Hessian matrix of $a(\xi )$ is nonsingular, the operator $Bf={\mathcal{F}}^{-1}(b(\xi )\mathcal{F}f)$ represents a weak dissipation, in the sense that $b(\xi )\sim 1$ as $|\xi |\to \infty$ and %, in several models, $D=B$ but we also consider the case $Df=\nu \ast (-\Delta {)}^{\delta }(I-\Delta {)}^{-\delta }$, with $\nu$ a bounded measure and $\delta \ge k/2$. We derive $L^p-L^q$ estimates of the linear solution operator to the considered model and we prove the transition from a Fujita-type exponent to a Strauss-type exponent for the global-in-time existence of small data solution for the corresponding nonlinear problems.