| Autor: |
Laister, Robert, Robinson, James C., Sierzega, Mikolaj, Vidal-López, Alejandro |
| Rok vydání: |
2014 |
| Předmět: |
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| Zdroj: |
Annales de l'Institut Henri Poincar\'e - Analyse Non-Lin\'eaire, 33 (6). pp. 1519-1538 (2016) |
| Druh dokumentu: |
Working Paper |
| DOI: |
10.1016/j.anihpc.2015.06.005 |
| Popis: |
We consider the scalar semilinear heat equation $u_t-\Delta u=f(u)$, where $f\colon[0,\infty)\to[0,\infty)$ 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 $L^q(\Omega)$ for all non-negative initial data $u_0\in L^q(\Omega)$, when $\Omega\subset{\mathbb R}^d$ is a bounded domain with Dirichlet boundary conditions. For $q\in(1,\infty)$ this holds if and only if $\limsup_{s\to\infty}s^{-(1+2q/d)}f(s)<\infty$; and for $q=1$ if and only if $\int_1^\infty s^{-(1+2/d)}F(s)\,{\rm d}s<\infty$, where $F(s)=\sup_{1\le t\le s}f(t)/t$. This shows for the first time that the model nonlinearity $f(u)=u^{1+2q/d}$ is truly the `boundary case' when $q\in(1,\infty)$, 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 $\limsup_{s\to0}f(s)/s<\infty$. |
| Databáze: |
arXiv |
| Externí odkaz: |
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