The following initial value problem is considered
where , , and is a nonnegative bounded and continuous function. The problem describes a combustion process in a stationary medium, where represents the temperature, and it is assumed that thermal conductivity and volume heat source depend on some powers of . 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 having slow decay , , , near . The problems of global existence and nonexistence, large time behavior or life span are investigated in terms of and .