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.
Green's Function and Infinite Time Bubbling in the Semilinear Heat Equation at the Critical Sobolev Exponent
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.
I will discuss the singularity formations for the harmonic map flows from a general two-dimensional domain to $S^2$. We construct Type II blow up solutions in general domains with single bubbles, multiple bubble tree, and reverse bubbling. Joint work with Juan Davila and Manuel del Pino.
By Weiming Ni, East China Normal University
In this talk, I shall survey some of the recent new developments in directed movements in population dynamics for two competing species in general environments. A new type of limiting system will be derived and preliminary studies will be described.