Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 66, pp. 1-11.
Title: Life span of nonnegative solutions to certain quasilinear
parabolic Cauchy problems
Author: Hendrik J. Kuiper (Arizona State Univ., Tempe, USA)
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 $T>0$ such that there is a solution for $0 \leq t < T$.
This is illustrated for the case $\rho \equiv 1$ and $h \equiv 1$ with
initial condition $u(x,0)=\sigma u_0(x)$, $\sigma > 0$,
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.