Group of Nonlinear Elliptic and Parabolic Equations

Department of Mathematical Analysis and Applied Mathematics
Institute of Interdisciplinary Mathematics (IMI)

At present, the Research Group consists of 9 members working in Madrid: J. López-Gómez, the Principal Researcher of the Group, together with 8 of his most recent PhD students: A. Tellini, who defended his Thesis in 2013, I. Antón, who defended her Thesis on July 2015, D. González de la Aleja Gallego, who defended his Thesis on December 2015, L. Maire, who defended his Thesis on July 2017, S. Fernández-Rincón and J. C. Sampedro, who at present are completing their PhD Thesis under the supervision of J. López-Gómez, E. Muñoz Hernández, who is completing his Thesis under the joint supervision of F. Zanolin, at the University of Udine, and of J. López-Gómez, and M. Fencl, who is completing his PhD under the joint supervision of M. Kucera at Praha and J. López-Gómez. Outside Madrid, the Research Group is integrated by some of the most recognized experts in Partial Differential Equations and Ordinary Differential Equations: P. H. Rabinowitz (University of Wisconsin, Madison, USA), former President of the Section of Mathematics of the Academy of Sciences of the USA, who has received important international recognitions and honors, L. Véron (Tours University, France), P. Omari (University of Trieste, Italy), F. Zanolin (University of Udine, Italy), who is supervising, together with J. López-Gómez, the PhD Thesis of E. Muñoz Martínez, and M. Kucera (University of Praha, Czek Republic), who is supervising, together with J. López-Gómez, the PhD Thesis of M. Fencl.

Bibliographical founds of the Faculties of Sciences of the Complutense University of Madrid, as well as the bibliographical founds of the Politechnic University and the King Juan Carlos University, were have postdoctorsal possitions several members of the Research Team.

The general interest of this Group is analyzing the effects of the spatial and temporal heterogeneities on the dynamics of a variety of parabolic systems of interest in Physics, Reaction-Diffusion and Population Dynamics. To reach this aim a huge variety of techniques of PDEs, ODEs, Operator Theory, Nonlinear Analysis and Numerical Analysis are brought together. At present, the main programmes runned by the Group are the next ones: (i) analyzing the structure of the set of nodal solutions of a class of 1-D semilinear problems (M. Fencl, J. López-Gómez, P. H. Rabinowitz and A. Tellini), (ii) analyzing the structure of the set of bounded variation positive solutions of a class of quasilinear 1-D problems in capilarity theory (J. López-Gómez and P. Omari), (iii) ascertaining the, very complex, structure of the set of suharmonics of the periodic predator-prey model of Lotka-Volterra (J. López-Gómez, E. Muñoz and F. Zanolin), (iv) analyzing the effects of the spatial heterogeneities on the dynamics of a class of diffusive competing species models of Lotka-Volterra type (S. Fernández-Rincón and J. López-Gómez), (v) characterizing the existence of positive periodic solutions in a class of periodic-parabolic logistic equations of interest in Population Dynamics (D. Aleja, I. Antón, J. López-Gómez and A. Tellini), (vi) finding optimal uniqueness results of large positive solutions in a general class of semilinear elliptic equations (L. Maire, J. López-Gómez and L. Véron), (vii) applying the degree of Fitzpatrick, Pejsachowicz and Rabier for Fredholm operators of index zero, to get existence results in a general class of quasilinear elliptic BVPs of interest in Physics (J. López-Gómez and J. C. Sampedro).