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The behavior of the life span for solutions to ${u}_{t}={\Delta }u+a\left(x\right){u}^{p}$ in ${ℝ}^{d}$. (English) Zbl 0914.35056
The author of this interesting paper studies the life span ${T}^{*}\left(\lambda ,{\Phi }\right)$ of the positive, bounded solution $u\left(x,t\right)$ to the Cauchy problem for the nonlinear reaction diffusion equation ${u}_{t}={\Delta }u+a\left(x\right){u}^{p}$ $\left(x\in {ℝ}^{d}$, $t\in \left(0,T\right)$, $p>1\right)$ under initial condition $u\left(x,0\right)=\lambda {\Phi }\left(x\right)$, where $\lambda >0$, $0\le a\in {C}^{\alpha }\left({ℝ}^{d}\right)$, $0\le {\Phi }\in {C}_{b}\left({ℝ}^{d}\right)$. The initial function has the property ${\Phi }\left(x\right)\le \delta exp\left[-\gamma |x{|}^{2}\right]$ $\left(\delta ,\gamma >0\right)$ or it is bounded. The asymptotic behavior of ${T}^{*}\left(\lambda ,{\Phi }\right)$ as $\lambda \to 0$ in the case that ${T}^{*}\left(\lambda ,{\Phi }\right)<\infty$, for all $\lambda >0$ and as $\lambda \to \infty$ in all cases is studied accurately. The asymptotic order depends on $a,{\Phi },p$ and $d$ in case that $\lambda \to 0$, while on the other hand in the case that $\lambda \to \infty$, it depends only on whether there is a point ${x}_{0}$ such that $a\left({x}_{0}\right)$, ${\Phi }\ne 0$, or whether the supports of $a$ and ${\Phi }$ are separated by a positive distance.
##### MSC:
 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions of PDE 35K15 Second order parabolic equations, initial value problems
##### Keywords:
blow-up; asymptotic order