The Second Critical Exponent for a Fast Diffusion Equation with Potential

Abstract:

      This paper considers a fast diffusion equation with potential $u_t=\Delta u^m-V(x)u^m+u^p$ in $\mathbb{R}^n\times(0,T)$, where $1-\frac{2}{\alpha m+n}1$, $n\geq 2$, $V(x)\sim \frac{\omega}{\mid x\mid^2}$ with $\omega\geq 0$ as $|x|\rightarrow \infty$, and $\alpha$ is the positive root of $\alpha m(\alpha m+n-2)-\omega=0$. The critical Fujita exponent was determined as $p_c=m+\frac{2}{\alpha m+n}$ in a previous paper of the authors. In the present paper, we establish the second critical exponent to identify the global and non-global solutions in their co-existence parameter region $p>p_c$ via the critical decay rates of the initial data. With $u_0(x)\sim |x|^{-a}$ as $|x|\rightarrow \infty$, it is shown that the second critical exponent $a^*=\frac{2}{p-m}$, independent of the potential parameter $\omega$, is quite different from the situation for the critical exponent $p_c$.