*(English)*Zbl 0936.35034

The following initial value problem is considered

where $1<m<p$, $N\ge 1$, and ${u}_{0}\left(x\right)$ 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 ${u}_{0}\sim \lambda {\left|x\right|}^{a}$, $\lambda >0$, $a\ge 0$, near $x=\infty $. The problems of global existence and nonexistence, large time behavior or life span are investigated in terms of $\lambda $ and $a$.