Self-similar extinction for a fast diffusion equation with weighted absorption

Abstract

Finite time extinction of any bounded solution to the fast diffusion equation with spatially inhomogeneous absorption $$ \partial_tu=Δu^m-|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ with $N\geq1$ and exponents $$ p>1, \quad m_c=\frac{(N-2)_+}{N}<m<1, \quad σ>σ_*:=\frac{2(p-1)}{1-m}, $$ is established. Moreover, the existence of self-similar solutions of the form $$ U(x,t)=(T-t)^αf(|x|(T-t)^β), \quad α=\frac{σ+2}{(1-m)(σ-σ_*)}, \ β=\frac{p-m}{(1-m)(σ-σ_*)}, $$ with $f(0)>0$, $f'(0)=0$ and $$ \lim\limits_{ξ\to\infty}ξ^{(σ+2)/(p-m)}f(ξ)=L\in(0,\infty). $$ is proved, together with some unbounded self-similar solutions as well. The property of finite time extinction is in striking contrast to the standard fast diffusion equation with absorption (that is, $σ=0$), where the strict positivity of solutions for any $t\in(0,\infty)$ is well-known.

Figures (1)

• Figure 1: Trajectories $l_C$ going out of $Q_1$. Numerical experiments for $m=0.5$, $p=2$, $N=3$ and $\sigma=4.5$.

Theorems & Definitions (22)

• Definition 1.1
• Theorem 1.2: Self-similar solutions with extinction
• Theorem 1.3: Finite time extinction and lower rates
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof