A doubly critical semilinear heat equation in the $L^1$ space

Abstract

We study the existence and nonexistence of a Cauchy problem of the semilinear heat equation ${\partial }_{t}u=\Delta u+|u{|}^{p-1}u$ in ${\mathbb{R}}^N\times (0,T)$, $u(x,0)=\varphi (x)$ in ${\mathbb{R}}^N$, in $L^1({\mathbb{R}}^N)$. Here, $N\ge 1$, $p=1+2/N$ and $\varphi \in L^1({\mathbb{R}}^N)$ is a possibly sign-changing initial function. Since $N(p-1)/2=1$, the $L^1$ space is scale critical and this problem is known as a doubly critical case. It is known that a solution does not necessarily exist for every $\varphi \in L^1({\mathbb{R}}^N)$. Let $X_q:=\{\varphi \in L_{\mathrm{loc}}^1({\mathbb{R}}^N)|{\int }_{{\mathbb{R}}^N}|\varphi |{\left[\log (e+|\varphi |)\right]}^{q}dx<\infty \}(\subset L^1({\mathbb{R}}^N))$. In this paper we construct a local-in-time mild solution in $L^1({\mathbb{R}}^N)$ for $\varphi \in X_q$ if $q\ge N/2$. We show that, for each $0\le q<N/2$, there is a nonnegative initial function ${\varphi }_0\in X_q$ such that the problem has no nonnegative solution, using a necessary condition given by Baras-Pierre [Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 185--212]. Since $X_q\subset X_{N/2}$ ($q\ge N/2$), $X_{N/2}$ becomes a sharp integrability condition. We also prove a uniqueness in a certain set of functions which guarantees the uniqueness of the solution constructed by our method.

12 pages

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