Electron. J. Diff. Eqns., Vol. 2008(2008), No. 10, pp. 1-18.

Blowup and life span bounds for a reaction-diffusion equation with a time-dependent generator

Ekaterina T. Kolkovska, Jose Alfredo Lopez-Mimbela, Aroldo Perez

Abstract:
We consider the nonlinear equation
$$ \frac{\partial}{\partial t} u (t) = k (t) \Delta _{\alpha }u (t) + u^{1+\beta } (t),\quad u(0,x)=\lambda \varphi (x),\; x\in \mathbb{R} ^{d}, $$
where $\Delta _{\alpha }:=-(-\Delta)^{\alpha /2}$ denotes the fractional power of the Laplacian; $0<\alpha \leq 2$, $\lambda$, $\beta >0$ are constants; $ \varphi$ is bounded, continuous, nonnegative function that does not vanish identically; and $k$ is a locally integrable function. We prove that any combination of positive parameters $d,\alpha,\rho,\beta$, obeying $0<d\rho\beta /\alpha<1$, yields finite time blow up of any nontrivial positive solution. Also we obtain upper and lower bounds for the life span of the solution, and show that the life span satisfies $T_{\lambda\varphi}\sim \lambda^{-\alpha \beta /(\alpha -d\rho \beta )}$ near $\lambda=0$.

Submitted August 24, 2007. Published January 21, 2008.
Math Subject Classifications: 60H30, 35K55, 35K57, 35B35.
Key Words: Semilinear evolution equations; Feynman-Kac representation; critical exponent; finite time blowup; nonglobal solution; life span.

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Ekaterina T. Kolkovska
Centro de Investigación en Matemáticas
Apartado Postal 402, 36000 Guanajuato, Mexico
email: todorova@cimat.mx
José Alfredo López-Mimbela
Centro de Investigación en Matemáticas
Apartado Postal 402, 36000 Guanajuato, Mexico
email: jalfredo@cimat.mx
Aroldo Pérez Pérez
Universidad Juárez Autónoma de Tabasco
División Académica de Ciencias Básicas
Km. 1 Carretera Cunduacán-Jalpa de Méndez
C.P. 86690 A.P. 24, Cunduacán, Tabasco, Mexico
email: aroldo.perez@dacb.ujat.mx

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