Differential Equations and Nonlinear Analysis Workshop

Peter Bates (Michigan State University) Spectral convergence and Turing instability in systems with long range diffusion Abstract: Many physical and biological processes occur with long range interaction, giving rise to equations with nonlocal in space operators in place of the usual Laplacian. These operators are diffusive-like but are bounded rather than unbounded as is the case of the diffusion operator. We study systems which include such nonlocal operators and through a spectral convergence result when a certain scaling parameter becomes small, show that Turing instabilities also occur, producing patterned stable states. This is joint work with Guangyu Zhao. Gabriel Bengochea (Universidad Autnoma de la Ciudad de Mexico) Operational solution of fractional differential equations Abstract: In this talk we construct an operational calculus which is based on a modified shift operator. This operator acts on an abstract space of formal Laurent series. By means of this calculus we solve fractional differential equations. We adopt Weyl’s definition of derivatives of fractional order. Alfonso Castro (Harvey Mudd College) Existence of singular solutions for a semilinear elliptic boundary value problem Abstract: Existence of countably many continua of singular solutions for elliptic boundary value problems of the form ∆u + f(u) = 0 for ‖x‖ < 1, and u(x) = 0 for ‖x‖ = 1 will be discussed. The role of the Pohozaev identity in finding shooting initial conditions will be emphasized. This is part of joint work with Professors Victor Ardila and Jose Caicedo of the Universidad Nacional de Colombia. David G. Costa (University of Nevada) On compactness properties of critical Sobolev embeddings and applications Abstract: We consider the Schrodinger operator L = −∆ + V (x) in R and compactness properties concerning the embedding of its associated functional space. The situations in dimensions N > 2 and N = 2 are very different in many ways but we’ll present an approach in which those differences can be seen as the realization of a common phenomenon. We also give applications to critical equations involving the operator L with a singular potential V (x).