Fujita exponent for an inhomogeneous pseudoparabolic equation

Abstract

We consider the Cauchy problem of the inhomogeneous pseudoparabolic equation $u_t-k\mathrm{\Delta }{u}_t=\mathrm{\Delta }u+|u{|}^p+\omega \left(x\right)$ in ${ℝ}^n$ with nonnegative initial value $u_0\left(x\right)$, where $k>0$ and $p>1$. When the equation is homogeneous, i.e., $\omega \equiv 0$, it was shown by Cao, Yin and Wang (2009) that the critical Fujita exponent $p_c$ is given by $p_c=1+2∕n$, which means for $p\in \left(1,p_c\right)$, the distribution of the initial data has no effect on the blow-up phenomena; for $p\in \left(p_c,\infty \right)$, the distribution of the initial data does have effect on the blow-up phenomena. We study the effect of the inhomogeneous term $\omega \left(x\right)$ on the critical Fujita exponent $p_c$, and we show $p_c=\infty$ if $n=1,2$, and $p_c=1+2∕\left(n-2\right)$ if $n\ge 3$.