Electron. J. Diff. Eqns., Vol. 2003(2003), No. 66, pp. 1-11.

Life span of nonnegative solutions to certain quasilinear parabolic Cauchy problems

Hendrik J. Kuiper

Abstract:
We consider the problem
$$ \rho(x)u_t-\Delta u^m=h(x,t)u^{1+p}, \quad x \in \mathbb{R}^N, \; t>0, $$
with nonnegative, nontrivial, continuous initial condition,
$$ u(x,0)=u_0(x) \not\equiv 0, \quad u_0(x)\ge 0, \; x \in \mathbb{R}^N. $$
An integral inequality is obtained that can be used to find an exponent $p_c$ such that this problem has no nontrivial global solution when $p \leq p_c$. This integral inequality may also be used to estimate the maximal $T greater than 0$ such that there is a solution for $0 \leq t less than T$. This is illustrated for the case $\rho \equiv 1$ and $h \equiv 1$ with initial condition $u(x,0)=\sigma u_0(x)$, $\sigma greater than 0$, by obtaining a bound of the form $T \le C_0 \sigma^{-\vartheta}$.

Submitted May 15, 2003. Published June 13, 2003.
Math Subject Classifications: 35K55, 35B33, 35B30.
Key Words: Nonlinear parabolic equation, blow-up, lifespan, critical exponent.

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Hendrik J. Kuiper
Department of Mathematics
Arizona State University
Tempe, AZ 85287-1804 USA
email: kuiper@asu.edu

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