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Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values. (English) Zbl 0936.35034

The following initial value problem is considered

t u=Δu m +u p ,x N ,t>0u(x,0)=u 0 (x),x N ,

where 1<m<p, N1, 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 λ|x| a , λ>0, a0, near x=. The problems of global existence and nonexistence, large time behavior or life span are investigated in terms of λ and a.

MSC:
35B40Asymptotic behavior of solutions of PDE
35K65Parabolic equations of degenerate type
35K15Second order parabolic equations, initial value problems
35K55Nonlinear parabolic equations
80A25Combustion, interior ballistics