Life span of nonnegative solutions to certain quasilinear parabolic Cauchy problems

Kuiper, Hendrik J.

Electronic Journal of Differential Equations, Tome 2003 (2003).

Résumé

Summary: 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 $T \le C_0 \sigma^{-\vartheta}$.

Détail

Classification : 35K55, 35B33, 35B30
Keywords: nonlinear parabolic equation, blow-up, lifespan, critical exponent