Jesús Ildefonso Díaz Díaz

The countable branches of nodal solutions bifurcating from the infinity for a sublinear semilinear equation are described with two different approaches. In the one-dimensional case we use plane phase methods of ordinary differential equations. The general N-dimensional problem can be studied by using topological methods and we sketch here some previous results by the second author in collaboration with João-Paulo Dias. One of the main motivations of the present paper was the ambiguity of the mathematical treatment of the Schrödinger equation for the infinite well potential. We study the classes of flat solutions (i.e. with zero normal derivative at the boundary) and solutions with compact support of the semilinear problem which allow to offer a kind of "alternative approach" to the infinite well potential for the Schrödinger equation.

Sublinear function

Infinity

10.4171/pm/1962

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Citations (7)

Local strong solutions of a parabolic system related to the Boussinesq approximation for buoyancy-driven flow with viscous heating

Advances in Differential Equations (2008)

Jesús Ildefonso Díaz Díaz Jean‐Michel Rakotoson Paul G. Schmidt

We propose a modification of the classical Navier-Stokes-Boussinesq system of equations, which governs buoyancy-driven flows of viscous, incompressible fluids. This modification is motivated by unresolved issues regarding the global solvability of the classical system in situations where viscous heating cannot be neglected. A simple model problem leads to a coupled system of two parabolic equations with a source term involving the square of the gradient of one of the unknowns. In the present paper, we establish the local-in-time existence and uniqueness of strong solutions for the model problem. The full system of equations and the global-in-time existence of weak solutions will be addressed in forthcoming work.

Boussinesq approximation (buoyancy)

10.57262/ade/1355867327

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Citations (6)

Finite Speed of Propagation and Waiting Time for Local Solutions of Degenerate Equations in Viscoelastic Media or Heat Flows with Memory

Journal of Elliptic and Parabolic Equations (2016)

S. N. Antont︠s︡ev Jesús Ildefonso Díaz Díaz

10.1007/bf03377402

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New unexpected limit operators for homogenizing optimal control parabolic problems with dynamic reaction flow on the boundary of critically scaled particles

arXiv (Cornell University) (2024)

Jesús Ildefonso Díaz Díaz Alexander Podolskiy Т. А. Шапошникова

We pass to the limit in the homogenization of an optimal control problem associated with a parabolic equation with a dynamic boundary condition. New unexpected terms appear due to the critical scale.

10.48550/arxiv.2406.18712

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On the convergence of controls and cost functionals in some optimal control heterogeneous problems when the homogenization process gives rise to some strange terms

Journal of Mathematical Analysis and Applications (2021)

Jesús Ildefonso Díaz Díaz Alexander Podolskiy Т. А. Шапошникова

Homogenization

Weak convergence

10.1016/j.jmaa.2021.125559

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Citations (8)

A nonlocal memory strange term arising in the critical scale homogenization of diffusion equations with dynamic boundary conditions