This article is concerned with the existence, uniqueness and numerical approximation of boundary blow up solutions for elliptic PDE's as ∆u = f (u) where f satisfies the so-called Keller-Osserman condition. We characterize existence of such solutions for non-monotone f . As an example, we construct an infinite family of boundary blow up solutions for the equation ∆u = u 2 (1 + cos u) on a ball. We prove uniqueness (on balls) when f is increasing and convex in a neighborhood of infinity and we discuss and perform some numerical computations to approximate such boundary blow-up solutions.2000 AMS Mathematics Subject Classification. 35J60.
The existence of the global attractor of a weakly damped, forced Korteweg-de Vries equation in the phase space L 2 ðRÞ is proved. An optimal asymptotic smoothing effect of the equation is also shown, namely, that for forces in L 2 ðRÞ; the global attractor in the phase space L 2 ðRÞ is actually a compact set in H 3 ðRÞ: The energy equation method is used in conjunction with a suitable splitting of the solutions; the dispersive regularization properties of the equation in the context of Bourgain spaces are extensively exploited. # 2002 Elsevier Science (USA)
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