Fractional heat equations with subcritical absorption having a measure as initial data

Abstract

We study existence and uniqueness of weak solutions to (F) $\partial\_t u+ (-\Delta)^\alphau+h(t, u)=0 $ in $(0,\infty)\times\R^N$,with initial condition $u(0,\cdot)=\nu$ in $\R^N$, where $N\ge2$, the operator $(-\Delta)^\alpha$is the fractional Laplacian with $\alpha\in(0,1)$, $\nu$ isa bounded Radon measure and $h:(0,\infty)\times\R\to\R$ is a continuous function satisfying a subcritical integrability condition.In particular, if $h(t,u)=t^\beta u^p$ with $\beta\textgreater{}-1$ and $0 \textless{} p \textless{} p^*\_\beta:=1+\frac{2\alpha(1+\beta)}{N}$, we prove that there exists a unique weak solution $u\_k$ to (F) with $\nu=k\delta\_0$, where $\delta\_0$ is the Dirac mass at the origin. We obtain that $u\_k\to\infty$ in $(0,\infty)\times\R^N$ as $k\to\infty$ for $p\in(0,1]$ and the limit of $u\_k$ exists as $k\to\infty$ when $1 \textless{} p \textless{} p^*\_\beta$, we denote it by $u\_\infty$.When $1+\frac{2\alpha(1+\beta)}{N+2\alpha}:=p^{**}\_\beta\textless{} p \textless{} p^*\_\beta$,$u\_\infty$ is the minimal self-similar solution of $(F)\_\infty$ $\partial\_t u+ (-\Delta)^\alpha u+t^\beta u^p=0 $ in $(0,\infty)\times\R^N$ with the initial condition $u(0,\cdot)=0$ in $\R^N\setminus\{0\}$ and it satisfies $u\_\infty(0,x)=0$ for $x\neq 0$.While if $1\textless{} p \textless{} p^{**}\_\beta$, then $u\_\infty\equiv U\_p$, where $U\_p$ is the maximal solution of the differential equation $y'+t^\beta y^p=0$ on $\R\_+$.