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+ (-Δ)^\alphau+h(t, u)=0 $ in $(0,\infty )\times {\mathbb{R}}^N$,with initial condition $u(0,\cdot )=\nu$ in ${\mathbb{R}}^N$, where $N\ge 2$, the operator $(-\Delta {)}^{\alpha }$is the fractional Laplacian with $\alpha \in (0,1)$, $\nu$ isa bounded Radon measure and $h:(0,\infty )\times \mathbb{R}\to \mathbb{R}$ is a continuous function satisfying a subcritical integrability condition.In particular, if $h(t,u)=t^{\beta }{u}^p$ with $β\textgreater{}-1$ and $0 \textless{} p \textless{} p^*\_β:=1+\frac{2α(1+β)}{N}$, 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 $(0,\infty )\times {\mathbb{R}}^N$ as $k\to \infty$ for $p\in (0,1]$ and the limit of $u\_k$ exists as $k\to \infty$ when $1 \textless{} p \textless{} p^*\_β$, we denote it by $u\_\infty$.When $1+\frac{2α(1+β)}{N+2α}:=p^{**}\_β\textless{} p \textless{} p^*\_β$,$u\_\infty$ is the minimal self-similar solution of $(F)\_\infty$ $\partial \_tu+(-\Delta {)}^{\alpha }u+t^{\beta }{u}^p=0$ in $(0,\infty )\times {\mathbb{R}}^N$ with the initial condition $u(0,\cdot )=0$ in ${\mathbb{R}}^N\setminus \{0\}$ and it satisfies $u\_\infty (0,x)=0$ for $x\ne 0$.While if $1\textless{} p \textless{} p^{**}\_β$, then $u\_\infty \equiv U\_p$, where $U\_p$ is the maximal solution of the differential equation $y^{\prime }+t^{\beta }{y}^p=0$ on $\mathbb{R}\_+$.

arXiv commentNonlinear Analysis, Theory, Methods and Applications, to appear