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Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem. (English) Zbl 0785.35011

Summary: We investigate the behavior of the solution u(x,t) of

u t=Δu+u p in n ×(0,T),u(x,0)=φ(x)in n ,

where Δ= i=1 n 2 / x i 2 is the Laplace operator, p>1 is a constant, T>0, and φ is a nonnegative bounded continuous function in n . The main results are for the case when the initial value φ has polynomial decay near x=. Assuming φλ(1+|x|) -a with λ, a>0, various questions of global (in time) existence and nonexistence, large time behavior or life span of the solution u(x,t) are answered in terms of simple conditions on λ, a, p and the space dimension n.

35B40Asymptotic behavior of solutions of PDE
35B30Dependence of solutions of PDE on initial and boundary data, parameters
35K55Nonlinear parabolic equations
35K15Second order parabolic equations, initial value problems