Initial behavior of the free boundary for a porous media equation with strong absorption (Q2747830)

The author considers the Cauchy problem for a diffusion equation with strong absorption: NEWLINE\[NEWLINE u_t=\Delta u^m\;-c u^p, \quad x\in \mathbb{R}^N,\;t>0, NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(x,0)=u_0(x) \geq 0, \qquad x\in \mathbb{R}^N, NEWLINE\]NEWLINE \noindent where \(c>0\), \(m>0\), \(0<p<\min(1,p)\). The aim of the paper is to prove asymptotic estimates on the support of the solution under the following condition: \(\|x\|^a\;u_0(x) \to A\) uniformly as \(\|x\|\to \infty\), for some \(a, A >0\). Namely, it is shown that NEWLINE\[NEWLINEu(x,t) \equiv 0 \text{ for } 0<t<T_0, \quad \|x\|>c\;t^{-\frac 1 {a(1-p)}};\tag{a}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(x,t)\geq C >0 \text{\quad for } t<\frac {T_0}2, \quad c_1 t^{-\frac 1 {a(1-p)}}<\|x\|<c_2 t^{-\frac 1 {a(1-p)}}.\tag{b}NEWLINE\]NEWLINE{}.