Abstract : This minisymposium will be focused on different models concerning mathematical biology and mathematical physics, both from an analytical and applied point of view.
In the first part, we will give an overview on different problems in population dynamics, such as invasion, spreading of populations and living in regions or in graphs.
The second part will deal with problems involving the Schrödinger operator, coming from mathematical Physics. Precisely, it will be discussed results such as existence of ground state solutions and stability of solutions.
Organizer(s) : Pablo Álvarez-Caudevilla, Cristina Brändle, Eduardo Colorado, Tatsuya Watanabe
[04143] The effect of diffusion on principal eigenvalues for Hamilton-Jacobi equations
Format : Talk at Waseda University
Author(s) :
Cristina Brändle (U. Carlos III Madrid)
Abstract : We deal with the ergodic problem of viscous Hamilton–Jacobi equations with superlinear Hamiltonian, inward-pointing
drift and a positive potential function that vanishes at infinity. We characterize the generalized principal eigenvalue with respect to diffusion and also specify the necessary and sufficient condition so that the spectral
function contains a plateau.
[03901] Population models with an interface region inside the domain
Format : Talk at Waseda University
Author(s) :
Pablo Alvarez-Caudevilla (Universidad Carlos III de Madrid)
Pablo Alvarez Caudevilla (Universidad Carlos III de Madrid)
Abstract : We will discuss several models that might be regarded as migration models of populations moving from one part of a domain to the other and becoming part of the population living on the other side. Different situations assuming symmetry of movement between both sides of the domain, following a logistic model in their own environment and assuming spatial heterogeneities, are going to be discussed. Through such a common boundary both populations are coupled, acting as a permeable membrane on which their flow moves in and out.
We will describe the precise interplay between the stationary solutions with respect to the parameters involved in the problem, in particular the growth rate of the populations and the coupling parameter involved on the boundary where the interchange of flux is taking place.
[04005] A bifurcation result for a fractional semilinear Neumann problem
Format : Talk at Waseda University
Author(s) :
Luca Vilasi (University of Messina)
Abstract : We will examine a parameterized elliptic problem governed by the Neumann fractional Laplacian on a bounded domain of $\mathbb R^N$, $N\geq 1$, with a general nonlinearity. This problem arises, in particular, when looking for steady state solutions to Keller-Segel systems in which the diffusion of the chemical is non-local. By variational arguments we will show the existence of non-trivial solutions, local minima of the corresponding energy functional, that branch off the null one for small values of the parameter. We will also derive some regularity results, as well as other qualitative properties of the solutions.
[04254] Systems of coupled nonlinear Schrödinger equations
Format : Talk at Waseda University
Author(s) :
Eduardo Colorado (Universidad Carlos III de Madrid)
Abstract : Along the talk we will see some results about existence of solutions to general coupled systems of nonlinear Schrödinger (NLS) equations (and systems of coupled NLS-nonlinear Korteweg-de Vries equations).
To do so, we will use variational techniques once one pass to the elliptic system obtained by looking for standing solitary wave solutions (or standing-travelling wave solutions) for the NLS coupled system (or the NLS-NKdV coupled system).
00563 (2/2) : 3E @G709 [Chair: Pablo Álvarez-Caudevilla]
[03153] Stable standing waves for a Schrödinger system with nonlinear chi^3 response
Format : Talk at Waseda University
Author(s) :
Tatsuya Watanabe (Kyoto Sangyo University)
Abstract : In this talk, we consider standing wave solutions for a certain nonlinear Schrodinger system which appears in nonlinear optics. This two-component system contains a cubic nonlinear term which is called $\chi^3$-interaction, and has a strong coupling on one side only.
Oliveira-Pastor (2021) showed the existence of ground state solutions for the corresponding stationary problems, and investigated their stability and instability. In our study, by considering the solvability of a constraint minimization problem, we show the existence of stable standing wave solutions. We also investigate the correspondence between minimizers and ground state solutions.
This work is based on joint research with Mathieu Colin (University of Bordeaux).
[03664] Critical nonlocal problems driven by the fractional Laplacian
Format : Online Talk on Zoom
Author(s) :
Raffaella Servadei (Università degli Studi di Urbino Carlo Bo)
Abstract : Critical problems are particularly relevant for their relations with many applications where a lack of compactness occurs.
The fractional Laplacian operator appears in concrete applications in many different fields. This is one of the reason why,
recently, nonlocal fractional problems are widely studied in the literature.
Aim of this talk is to discuss some recent results about existence and multiplicityof solutions for fractional nonlocal
equations with critical growth assumptions on the nonlinear term.
[04405] Variational and topological methods on non-compact Randers spaces
Format : Online Talk on Zoom
Author(s) :
Giovanni Molica Bisci (University of Urbino Carlo Bo)
Abstract : Motivated by a wide interest in the literature, the leading purpose of this talk is to present some recent abstract results on non-compact Randers spaces and their applications to quasilinear elliptic equations. The main approach is based on novel abstract Sobolev embedding results as well as on some variational and topological methods.
