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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.


MSC:
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