Universal self-similarity of porous media equation with absorption: the critical exponent case. (Q1428647)

The authors consider the \(N\)-dimensional Cauchy problem \[ \begin{cases} u_t= \Delta_x u^m- u^q\quad &\text{in }\mathbb{R}^N\times (0,\infty),\\ u(x,0)= u_0(x)\geq 0\quad & \text{in }\mathbb{R}^N,\;u_0\in L^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N),\end{cases} \] where \(m> 1\) and \(q= q_{\text{crit}}\equiv m+{2\over N}\). They study the global dynamics of general solution with \(L^1\)-initial data having mild decay as \(|x|\to\infty\). They derive the exact spatio-temporal profile of a solution when \(t\to\infty\).