Papers by Jesús Ildefonso Díaz
Extracto del discurso leido por el Profesor Diaz en el acto de su recepcion en la Real Academia d... more Extracto del discurso leido por el Profesor Diaz en el acto de su recepcion en la Real Academia de Ciencias Exactas, Fisicas y Naturales. El discurso completo ha sido publicado por la Academia.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1981
SynopsisWe obtain existence and uniqueness of solutions with compact support for some nonlinear e... more SynopsisWe obtain existence and uniqueness of solutions with compact support for some nonlinear elliptic and parabolic problems including the equations of one-dimensional motion of a non-newtonian fluid. Precise estimates for the support of these solutions are obtained, and the optimality of our hypotheses is discussed.
Advances in Nonlinear Analysis, 2021
We consider a class of semilinear equations with an absorption nonlinear zero order term of power... more We consider a class of semilinear equations with an absorption nonlinear zero order term of power type, where elliptic condition is given in terms of Gauss measure. In the case of the superlinear equation we introduce a suitable definition of solutions in order to prove the existence and uniqueness of a solution in ℝN without growth restrictions at infinity. A comparison result in terms of the half-space Gaussian symmetrized problem is also proved. As an application, we give some estimates in measure of the growth of the solution near the boundary of its support for sublinear equations. Finally we generalize our results to problems with a nonlinear zero order term not necessary of power type.
Advances in Nonlinear Analysis, 2019
We consider the mathematical treatment of a system of nonlinear partial differential equations ba... more We consider the mathematical treatment of a system of nonlinear partial differential equations based on a model, proposed in 1972 by J. Newman, in which the coupling between the Lithium concentration, the phase potentials and temperature in the electrodes and the electrolyte of a Lithium battery cell is considered. After introducing some functional spaces well-adapted to our framework, we obtain some rigorous results showing the well-posedness of the system, first for some short time and then, by considering some hypothesis on the nonlinearities, globally in time. As far as we know, this is the first result in the literature proving existence in time of the full Newman model, which follows previous results by the third author in 2016 regarding a simplified case.
Advanced Nonlinear Studies, 2020
We consider the high-dimensional equation ∂ t u - Δ u m + u - β χ { u > 0 } = 0 {\partia... more We consider the high-dimensional equation ∂ t u - Δ u m + u - β χ { u > 0 } = 0 {\partial_{t}u-\Delta u^{m}+u^{-\beta}{\chi_{\{u>0\}}}=0} , extending the mathematical treatment made in 1992 by B. Kawohl and R. Kersner for the one-dimensional case. Besides the existence of a very weak solution u ∈ 𝒞 ( [ 0 , T ] ; L δ 1 ( Ω ) ) {u\in\mathcal{C}([0,T];L_{\delta}^{1}(\Omega))} , with u - β χ { u > 0 } ∈ L 1 ( ( 0 , T ) × Ω ) {u^{-\beta}\chi_{\{u>0\}}\in L^{1}((0,T)\times\Omega)} , δ ( x ) = d ( x , ∂ Ω ) {\delta(x)=d(x,\partial\Omega)} , we prove some pointwise gradient estimates for a certain range of the dimension N, m ≥ 1 {m\geq 1} and β ∈ ( 0 , m ) {\beta\in(0,m)} , mainly when the absorption dominates over the diffusion ( 1 ≤ m < 2 + β {1\leq m<2+\beta} ). In particular, a new kind of universal gradient estimate is proved when m + β ≤ 2 {m+\beta\leq 2} . Several qualitative properties (such as the finite time quenching phenomena and the finite ...
Dynamical Systems and Differential Equations, AIMS Proceedings 2015 Proceedings of the 10th AIMS International Conference (Madrid, Spain)
We extend some previous results in the literature on the Steiner rearrangement of linear second o... more We extend some previous results in the literature on the Steiner rearrangement of linear second order elliptic equations to the semilinear concave parabolic problems and the obstacle problem.
System Modelling and Optimization, 1994
Progress in Nonlinear Differential Equations and Their Applications
Energy Methods for Free Boundary Problems, 2002
ABSTRACT
Energy Methods for Free Boundary Problems, 2002
In this chapter the energy method is applied to study the space and time localization of weak sol... more In this chapter the energy method is applied to study the space and time localization of weak solutions for nonlinear degenerate parabolic equations and systems of such equations.
James Serrin. Selected Papers, 2014
Pure and Applied Geophysics, 2008
Pure and Applied Geophysics, 2008
Knowledge of the Earth's structure and dynamics calls for a multidisciplinary study that makes us... more Knowledge of the Earth's structure and dynamics calls for a multidisciplinary study that makes use of the most advanced methods of Physics, Chemistry, Mathematics and Information Technology, in the framework, or in a close collaboration with, the different branches of Earth Sciences such as Geology, Geophysics and Geodesy. The research to be developed includes subjects ranging from data acquisition, both with traditional techniques and with the most advanced resources of our time; data treatment and processing; to the development of new modelling methodologies for the simulation and reproduction and prediction of the terrestrial processes on a local, regional and, by far the most ambitious, global scale. The large amount of (geological, geophysical and geodetic) high precision observation data about the Earth available acquired both from the planet itself and from space, grows increasingly. Currently one can obtain huge amounts of high precision data that cover the widest areas desired. Often the time or space distribution of these data are almost continuous, and considerable of data has been obtained by unconventional techniques. In view of the privileged situation at present, we must reconsider the connection and utilization of that abundant data with the more theoretical studies that offer new and more refined mathematical models, capable of making the most of the technological breakthroughs in observation. In the words of Jacques-Louis Lions (1928-2001): ''If we accept that any mathematical models isolated of any experimental data have no predictive value, similarly, we must recognize that the most abundant data banks without good mathematical models produce nothing more than confusion''. What is required is to develop new mathematical, analytical and numerical, models and methods for data processing and interpretation, considering their ever-increasing quality, variety in origin (terrestrial and space), type (it is becoming more and more possible to measure larger numbers of parameters simultaneously that can be related to
Mathematical Methods in the Applied Sciences, 2002
In this paper, we study the number of steady solutions of a non‐linear model arising in Climatolo... more In this paper, we study the number of steady solutions of a non‐linear model arising in Climatology. By applying a shooting method we show the existence of infinitely many steady solutions for some values of a parameter (the solar constant). This method allows us to determine how many times a solution attains the critical temperature (−10°C) at which the coalbedo is assumed to be discontinuous. Copyright © 2002 John Wiley & Sons, Ltd.
Communications in Partial Differential Equations, 1992
Differential and Integral Equations
It is well-known that the pressure of a lubricating fluid filling the gap between two solid surfa... more It is well-known that the pressure of a lubricating fluid filling the gap between two solid surfaces satisfies the Reynolds equation involving the distance function, h, between both planes, as a crucial coefficient. Nevertheless, in most of the applications the function h is not known a priori. Here we consider the simple case in which the surfaces are two parallel planes and assume prescribed the total force applied upon one of the surfaces. We give some sufficient conditions on the total force in order to solve this inverse problem. We show that in the incompressible case, such a condition is also necessary.
Advances in Differential Equations
We propose a modi…cation of the classical Navier-Stokes-Boussinesq system of equations, which gov... more We propose a modi…cation of the classical Navier-Stokes-Boussinesq system of equations, which governs buoyancy-driven ‡ows of viscous, incompressible ‡uids. This modi…cation 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.
Doklady Mathematics
The paper studies the asymptotic behavior of the optimal control for the Poisson type boundary va... more The paper studies the asymptotic behavior of the optimal control for the Poisson type boundary value problem in a domain perforated by holes of an arbitrary shape with Robintype boundary conditions on the internal boundaries. The cost functional is assumed to be dependent on the gradient of the state and on the usual norm of the control. We consider the so-called "critical" relation between the problem parameters and the period of the structure ε → 0. Two "strange" terms arise in the limit. The paper extends, by first time in the literature, previous papers devoted to the homogenization of the control problem which always assumed the symmetry of the periodic holes.
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Papers by Jesús Ildefonso Díaz