Stabilization of a hyperbolic/elliptic system modelling the viscoelastic-gravitational deformation in a multilayered Earth
Alicia Arjona Jesús Ildefonso Díaz
In the last 30 years several mathematical studies have been devoted to the viscoelastic-gravitational coupling in stationary and transient regimes either for static case or for hyperbolic case. However, to the best of our knowledge there is a lack of mathematical study of the stabilization as $t$ goes to infinity of a viscoelastic-gravitational models crustal deformations of multilayered Earth. Here we prove that, under some additional conditions on the data, the difference of the viscoelastic and elastic solutions converges to zero, as $t$ goes to infinity, in a suitable functional space. The proof of that uses a reformulation of the hyperbolic/elliptic system in terms of a nonlocal hyperbolic system.
keywords: Stabilization LaSalle's Invariance principle weak solution viscoelastic-gravitational hyperbolic/elliptic equations Lyapunov function.
Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions
Sabri Bensid Jesús Ildefonso Díaz

We study stability of the nonnegative solutions of a discontinuous elliptic eigenvalue problem relevant in several applications as for instance in climate modeling. After giving the explicit expresion of the S-shaped bifurcation diagram $\left( \lambda ,{{\left\| {{\mu }_{\lambda }} \right\|}_{\infty }} \right)$ we show the instability of the decreasing part of the bifurcation curve and the stability of the increasing part. This extends to the case of non-smooth nonlinear terms the well known 1971 result by M.G. Crandall and P.H. Rabinowitz concerning differentiable nonlinear terms. We point out that, in general, there is a lacking of uniquenees of solutions for the associated parabolic problem. Nevertheless, for nondegenerate solutions (crossing the discontinuity value of u in a transversal way) the comparison principle and the uniqueness of solutions hold. The instability is obtained trough a linearization process leading to an eigenvalue problem in which a Dirac delta distribution appears as a coefficient of the differential operator. The stability proof uses a suitable change of variables, the continuuity of the bifurcation branch and the comparison principle for nondegenerate solutions of the parabolic problem.

keywords: Nonlinear eigenvalue problem discontinuous nonlinearity S-shaped bifurcation curve stability free boundary energy balance climate models
On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions
Gregorio Díaz Jesús Ildefonso Díaz
This paper deals with several qualitative properties of solutions of some stationary equations associated to the Monge--Ampère operator on the set of convex functions which are not necessarily understood in a strict sense. Mainly, we focus our attention on the occurrence of a free boundary (separating the region where the solution $u$ is locally a hyperplane, thus, the Hessian $D^{2}u$ is vanishing, from the rest of the domain). In particular, our results apply to suitable formulations of the Gauss curvature flow and of the worn stones problems intensively studied in the literature.
keywords: Monge--Ampère equation Gauss curvatures surfaces free boundary problem.
On the retention of the interfaces in some elliptic and parabolic nonlinear problems
Luis Alvarez Jesús Ildefonso Díaz
We study some retention phenomena on the free boundaries associated to some elliptic and parabolic problems of reaction-diffusion type. This is the case, for instance, of the waiting time phenomenon for solutions of suitable parabolic equations. We find sufficient conditions in order to have a discrete version of the waiting time property (the so called nondiffusion of the support) for solutions of the associated family of elliptic equations and prove how to pass to the limit in order to get this property for the solutions of the parabolic equation.
keywords: free boundary waiting time interfaces. nonlinear problems
On a climate model with a dynamic nonlinear diffusive boundary condition
Jesús Ildefonso Díaz L. Tello
This work studies the sensitivity of a global climate model with deep ocean effect to the variations of a Solar parameter $Q$. The model incorporates a dynamic and diffusive boundary condition. We study the number of stationary solutions according to the positive parameter $Q$.
keywords: diffusive boundary condition comparison principle. p-laplacian Climate atmosphere - ocean model upper and lower solutions
On the free boundary for quenching type parabolic problems via local energy methods
Jesús Ildefonso Díaz
We extend some previous local energy method to the study the free boundary generated by the solutions of quenching type parabolic problems involving a negative power of the unknown in the equation.
keywords: free boundary interpolation inequalities Quenching type parabolic equations instantaneous shrinking of the support. local energy methods
A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity
Pascal Bégout Jesús Ildefonso Díaz
We prove the compactness of the support of the solution of some stationary Schrödinger equations with a singular nonlinear order term. We present here a sharper version of some energy methods previously used in the literature.
keywords: solutions with compact support. Schrödinger equation Energy method
Finite extinction time property for a delayed linear problem on a manifold without boundary
Alfonso C. Casal Jesús Ildefonso Díaz José M. Vegas
We prove that the mere presence of a delayed term is able to connect the initial state u0 on a manifold without boundary
keywords: Finite extinction time dynamic boundary conditions delayed feedback control linear parabolic equations
Complete recuperation after the blow up time for semilinear problems
Alfonso C. Casal Jesús Ildefonso Díaz José Manuel Vegas
We consider explosive solutions $y^{0}(t)$, $t\in \lbrack 0,T_{y^{0}}),$ of some ordinary differential equations \begin{equation*} P(T_{y^{0}}): \begin{array}{lc} \frac{dy}{dt}(t)=f(y(t)),y(0)=y_{0}, & \end{array} \end{equation*} where $f:$ $\mathbb{R}^{d}\rightarrow \mathbb{R}^{d}$ is a locally Lipschitz superlinear function and $d\geq 1$. In this work we analyze the following question of controlability: given $\epsilon >0$, a continuous deformation $y(t)$ de $y^{0}(t)$, built as a solution of the perturbed control problem obtained by replacing $f(y(t))$ by $f(y(t))+u(t),$ for a suitable control $u$, such that $y(t)=y^{0}(t)$ for any $t\in \lbrack 0,T_{y^{0}}-\epsilon ]$ and such that $y(t)$ also blows up in $t=T_{y_{0}}$ but in such a way that $y(t)$ could be extended beyond $T_{y_{0}}$ as a function $y\in L_{loc}^{1}(0,+\infty :\mathbb{R}^{d})$?
keywords: Solutions beyond Blow-up Time Nonlinear Variation of Constants Formula. Semilinear Problems

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