Apr
26
3:00 PM15:00

Nonlinear Fractional Elliptic Equations with Measure Data

By Laurent Véron, University of Tours

We study the existence of weak solutions to

(-∆)^αu+g (u) =∨      (E)

in a bounded regular domain  Ω in R^N (N>1)  which vanish in R^N \ Ω, where (-∆)^α denotes the fractional Laplacian with 0<α<1, is a Radon measure and g is a non-decreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of a weak solution for problem (E) for any measure. In the case where is a Dirac measure, we characterize the asymptotic behavior of the solution. When g(r)=|r|^(k-1) r with k supercritical, we show that the absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved.

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Apr
26
1:30 PM13:30

Green's Function and Infinite Time Bubbling in the Semilinear Heat Equation at the Critical Sobolev Exponent

By Mónica Musso, Pontifical Catholic University of ChileCenter for Analysis of Partial Differential Equations

We discuss some new results on globally defined in time positive solutions of the semilinear heat equation with critical power nonlinearity and Dirichlet boundary conditions in a bounded domain. For any given number k we can find a solution that, as time grows, blows up exactly at k points of the domain with a bubbling profile that can be precisely computed. This is joint work with Carmen Cortazar and Manuel del Pino.

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