ar X iv : 0 80 9 . 18 05 v 1 [ m at h . A P ] 1 0 Se p 20 08 Solutions of some nonlinear parabolic equations with initial blow - up
نویسندگان
چکیده
Abstract We study the existence and uniqueness of solutions of ∂tu − ∆u + u q = 0 (q > 1) in Ω×(0,∞) where Ω ⊂ R is a domain with a compact boundary, subject to the conditions u = f ≥ 0 on ∂Ω× (0,∞) and the initial condition limt→0 u(x, t) = ∞. By means of Brezis’ theory of maximal monotone operators in Hilbert spaces, we construct a minimal solution when f = 0, whatever is the regularity of the boundary of the domain. When ∂Ω satisfies the parabolic Wiener criterion and f is continuous, we construct a maximal solution and prove that it is the unique solution which blows-up at t = 0. 1991 Mathematics Subject Classification. 35K60.
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