Seminário de EDP's

Às 10h teremos: 

Pêdra Andrade - Instituto Superior T´ecnico - Universidade de Lisboa

Título: C1,α regularity for fully nonlinear integro-differential equations with degeneracies.

Abstract: In this talk, we will discuss regularity estimates for a class of de-generate and singular fully nonlinear integro-differential equations.
In the degenerate case, we show that there exists at least one viscos-ity solution of class C1,α loc for some constant α ∈ (0, 1). Additionally, under suitable conditions on σ, we establish regularity estimates in H¨older spaces for any viscosity solution. We also explore the sin-gular setting and demonstrate H¨older regularity estimates for the gradient of the solutions.

às 10h50 haverá bolo e café

Às 11h 10min teremos:

J. López-Gómez - Complutense University of Madrid

Título: New trends in Lotka-Volterra diffusive competition

Abstract: This talk discusses several recent findings on the dynamics of the spatially-heterogeneous diffusive Lotka-Volterra competing species model. First, it delivers a general (optimal) singular perturbation result generalizing, very substantially, the pioneering theorem of Hutson, López-Gómez, Mischaikow and Vickers (1994) for their mutant model, later analyzed, very sharply, by W. M. Ni and his collaborators. Then, it establishes that, as soon as any steady-state solution of the non-spatial model is linearly unstable somewhere in the inhabiting territory, Ω, any steady state of the spatial counterpart perturbing from it therein as the diffusion rates separate away from zero must be linearly unstable. From this feature one can derive a number of rather astonishing consequences, as the multiplicity of the coexistence steady states when the non-spatial model exhibits founder control competition somewhere in Ω, say Ω𝑏𝑏𝑏𝑏, even if Ω𝑏𝑏𝑏𝑏 is negligible empirically. Actually, this is the first existing multiplicity result for small diffusion rates.
Finally, based on the Picone identity, we can establish a new, rather striking, uniqueness result valid for general spatially heterogeneous models. This result generalizes, very substantially, those of W. M. Ni and collaborators for the autonomous model.