A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces

We consider the scalar semilinear heat equation $u_t-\mathrm{\Delta }u=f(u)$, where $f\text{\hspace{0.17em}⁣}:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is continuous and non-decreasing but need not be convex. We completely characterise those functions $f$ for which the equation has a local solution bounded in $L^q(\mathrm{\Omega })$ for all non-negative initial data $u_0\in L^q(\mathrm{\Omega })$, when $\mathrm{\Omega }\subset {\mathbb{R}}^d$ is a bounded domain with Dirichlet boundary conditions. For $q\in (1,\mathrm{\infty })$ this holds if and only if ${\mathrm{lim\hspace{0.17em}sup}}_{s\to \mathrm{\infty }}{s}^{-(1+2q\mathrm{/}d)}f(s)<\mathrm{\infty }$; and for $q=1$ if and only if ${\int }_1^{\mathrm{\infty }}{s}^{-(1+2\mathrm{/}d)}F(s)\mathrm{d}s<\mathrm{\infty }$, where $F(s)={\sup }_{1\le t\le s}f(t)\mathrm{/}t$. This shows for the first time that the model nonlinearity $f(u)=u^{1+2q\mathrm{/}d}$ is truly the `boundary case’ when $q\in (1,\mathrm{\infty })$, but that this is not true for $q=1$. The same characterisation results hold for the equation posed on the whole space ${\mathbb{R}}^d$ provided that in addition ${\mathrm{lim\hspace{0.17em}sup}}_{s\to 0}f(s)\mathrm{/}s<\mathrm{\infty }$.