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Zbl 0914.35056
Pinsky, Ross G.
The behavior of the life span for solutions to $u_t=\Delta u+a(x) u^p$ in $\bbfR^d$. (English)
[J] J. Differ. Equations 147, No.1, 30-57 (1998). ISSN 0022-0396

The author of this interesting paper studies the life span $T^*(\lambda,\Phi)$ of the positive, bounded solution $u(x,t)$ to the Cauchy problem for the nonlinear reaction diffusion equation $u_t=\Delta u+a(x)u^p$ $(x\in\bbfR^d$, $t\in(0,T)$, $p>1)$ under initial condition $u(x,0)= \lambda \Phi(x)$, where $\lambda>0$, $0\le a\in C^\alpha (\bbfR^d)$, $0\le\Phi\in C_b(\bbfR^d)$. The initial function has the property $\Phi(x)\le\delta \exp[-\gamma| x|^2]$ $(\delta,\gamma>0)$ or it is bounded. The asymptotic behavior of $T^*(\lambda,\Phi)$ as $\lambda\to 0$ in the case that $T^*(\lambda, \Phi)<\infty$, 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 $a(x_0)$, $\Phi\ne 0$, or whether the supports of $a$ and $\Phi$ are separated by a positive distance.
[Dimitar Kolev (Sofia)]

MSC 2000:: *35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions of PDE
35K15 Second order parabolic equations, initial value problems

Keywords: blow-up; asymptotic order

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Zentralblatt MATH Berlin [Germany]