The Fujita Exponent for Semilinear Heat Equations with Quadratically Decaying Potential or in an Exterior Domain

Details The Fujita Exponent for Semilinear Heat Equations with Quadratically Decaying Potential or in an Exterior Domain The Fujita Exponent for Semilinear Heat Equations with Quadratically Decaying Potential or in an Exterior Domain

2008-05-09

arxiv.org/abs/0805.1313v2

Mathematics

Analysis of PDEs

24 pages, replaces and improves previous version

Scientific paper

Consider the equation u_t=\Delta u-Vu +au^p \text{in} R^n\times (0,T); u(x,0)=\phi(x)\gneq0, \text{in} R^n, where $p>1$, $n\ge2$, $T\in(0,\infty]$, $V(x)\sim\frac\omega{|x|^2}$ as $|x|\to\infty$, for some $\omega\neq0$, and $a(x)$ is on the order $|x|^m$ as $|x|\to\infty$, for some $m\in (-\infty,\infty)$. A solution to the above equation is called global if $T=\infty$. Under some additional technical conditions, we calculate a critical exponent $p^*$ such that global solutions exist for $p>p^*$, while for $1

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