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

We study existence and uniqueness of weak solutions to (F) $\partial\_t u+ (-Δ)^\alphau+h(t, u)=0 $ in $(0,\infty)\times\R^N$,with initial condition $u(0,\cdot)=ν$ in $\R^N$, where $N\ge2$, the operator $(-Δ)^α$is the fractional Laplacian with $α\in(0,1)$, $ν$ isa bounded Radon measure and $h:(0,\infty)\times\R\to\R$ is a continuous function satisfying a subcritical integrability condition.In particular, if $h(t,u)=t^βu^p$ 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 $ν=kδ\_0$, where $δ\_0$ 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 $1 \textless{} p \textless{} p^*\_β$, 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 $(F)\_\infty$ $\partial\_t u+ (-Δ)^αu+t^βu^p=0$ 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 $1\textless{} p \textless{} p^{**}\_β$, then $u\_\infty\equiv U\_p$, where $U\_p$ is the maximal solution of the differential equation $y'+t^βy^p=0$ on $\R\_+$.

No reviews yet.

No comments yet.