Spatial Heterogeneities in Nonlinear Parabolic Problems

Brief description:

The most realistic parabolic models in applied, or social, sciences and engineering are those with spatially heterogeneous coefficients, as in nature spatial (and temporal) heterogeneities determine not only the evolution of Earth ecosystems, but the evolution of the universe as a whole… since the Big-Bang. In this project we are proposing to analyze the effects of spatial heterogeneities on the dynamics of some paradigmatic models in Ecology and Environmental Sciences. The first one is a generalized class of logistic equations, where we expect to characterize their dynamics within the metasolutions regime by establishing the uniqueness of the underlying large solutions of the model. The second one is the diffusive Lotka-Volterra competition model, where we expect to characterize its dynamics for small diffusivities. The third one is a one-dimensional boundary value problem of degenerate type, with a vanishing weight function in front of the nonlinearity, where we expect to ascertain the internal fine structure of the set of nodal solutions in terms of a certain spectral parameter.

Researchers:

Julián López-Gómez (Full Proffessor, Faculty of Mathematics, UCM)
Sergio Fernández-Rincón (Contract FPU, University Personal under Formation, UCM)
Andrea Tellini (Juan de la Cierva Postdoctoral Contract, UCM)
Luis Maire Marín (Without contract, University Personal under Formation, UCM)

Expected results:

As far as concerns to the first aim, we expect to get uniqueness of the large solution of –𝚫𝚫u=a(x)f(u) in any bounded C² domain, Ω, provided –𝚫𝚫u=f(u) has a unique large solution, for every continuous function a(x) such that a(x)>0 for all xεΩ. As far as concerns the second aim, we expect to characterize, completely, all the admissible global dynamics for small diffusivities. In particular, we expect to classify all possible limiting behaviors of the coexistence states of the model under low level of aggressions between the species, as well as establishing the uniqueness of the coexistence state when the associated non-spatial model, the one obtained by switching off to zero the diffusivities, exhibits either permanence, or dominance. Moreover, we also expect getting multiple coexistence steady states when the non-spatial model exhibits founder control competition somewhere. Concerning the nodal solutions, we expect to construct examples of weight functions for which the set of nodal solutions consists of a single component, and others for which the set of nodal solutions consist of an arbitrarily large number of components. As all those findings go far beyond established paradigms and certainly open new research horizons, they should deserve a huge attention and hence, the expected impact is huge.