Fractional heat equations involving initial measure data and subcritical absorption
Résumé
We study the existence and uniqueness of weak solution to (F) $\partial_t u+ (-\Delta)^\alpha u+h(t, u)=0 $ in $(0,\infty)\times\R^N$, is given by the initial condition $u(0,\cdot)=\nu$ in $\R^N$, where $N\ge1$, the nonlocal operator $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(0,1)$, $\nu$ is a bounded Radon measure and $h:(0,\infty)\times\R\mapsto\R$ is a continuous function satisfying a subcritical integrability condition. In particular of $h(t,u)=t^\beta u^p$ with $\beta>-1$, there exists a unique 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 $p\in(1,1+\frac{2\alpha(1+\beta)}{N})$, in this case denoting it by $u_\infty$. When $p\in(1+\frac{2\alpha(1+\beta)}{N+2\alpha},1+\frac{2\alpha(1+\beta)}N)$, $u_\infty$ is the unique 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 $\R^N\setminus\{0\}$. When $p\in(1,1+\frac{2\alpha(1+\beta)}{N+2\alpha})$, $u_\infty$ is a trivial self-similar solution to $(F)_\infty$, precisely, $$u_\infty(t,x)=(\frac{1+\beta}{p-1})^{\frac1{p-1}} t^{-\frac{1+\beta}{p-1}},\quad (t,x)\in(0,\infty)\times\R^N.$$
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