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.