Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values
The following initial value problem is considered \[ \partial_t u=\Delta u^m+u^p,\quad x\in \mathbb{R}^N,\;t>0 \qquad u(x,0)=u_0(x),\quad x\in\mathbb{R}^N, \] where \(1<m<p\), \(N\geq 1\), and \(u_0(x)\) is a nonnegative bounded and continuous function. The problem describes a combustion process in a stationary medium, where \(u\) represents the temperature, and it is assumed that thermal conductivity and volume heat source depend on some powers of \(u\). It is well known that this problem has a unique, nonnegative and bounded solution in some weak sense at least locally in time. The paper establishes some sufficient conditions implying that the considered solution exists only on a finite time interval (blow up) or that it has an infinite life span. It concentrates on the case of initial values \(u_0\) having slow decay \(u_0\sim\lambda|x|^a\), \(\lambda>0\), \(a\geq 0\), near \(x=\infty\). The problems of global existence and nonexistence, large time behavior or life span are investigated in terms of \(\lambda\) and \(a\).
scientific article; zbMATH DE number 1383357