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In this article, we establish the Picard-Lindelof theorem and approximating results for dynamic equations on time scale. We present a simple proof for the existence and uniqueness of the solution. The proof is produced by using convergence…
Given global Lipschitz continuity and differentiability of high enough order on the coefficients in It\^{o}'s equation, differentiability of associated semigroups, existence of twice differentiable solutions to Kolmogorov equations and weak…
This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance:…
In this paper we establish the orbital stability of periodic waves related to the logarithmic Korteweg-de Vries equation. Our motivation is inspired in the recent work \cite{carles}, in which the authors established the well-posedness and…
In this paper, we study the asymptotic behavior of solutions to a scalar fractional delay differential equations around the equilibrium points. More precise, we provide conditions on the coefficients under which a linear fractional delay…
In this paper we consider two classes of backward stochastic differential equations. Firstly, under a Lipschitz-type condition on the generator of the equation, which can also be unbounded, we give sufficient conditions for the existence of…
We study the doubly nonlinear PDE $$ |\partial_t u|^{p-2}\,\partial_t u-\textrm{div}(|\nabla u|^{p-2}\nabla u)=0. $$ This equation arises in the study of extremals of Poincar\'e inequalities in Sobolev spaces. We prove spatial Lipschitz…
This work provides stability results in the spatial sup norm for hyperbolic-parabolic loops in one spatial dimension. The results are obtained by an application of the small-gain stability analysis. Two particular cases are selected for the…
In this paper we obtain the continuity of attractors for nonlinear parabolic equations with nonlinear boundary conditions when the boundary of the domain varies very rapidly as a parameter $\epsilon$ goes to zero. We consider the case where…
We prove well posedness and stability in $\mathbf{L}^1$ for a class of mixed hyperbolic-parabolic non linear and non local equations in a bounded domain with no flow along the boundary. While the treatment of boundary conditions for the…
In [2] we introduced a method combining together an observability inequality and a spectral decomposition to get a logarithmic stability estimate for the inverse problem of determining both the potential and the damping coefficient in a…
In this manuscript, we investigate a fractional stochastic neutral differential equation with time delay, which includes both deterministic and stochastic components. Our primary objective is to rigorously prove the existence of a unique…
We consider a class of self-adjoint Sturm-Liouville problems with rational functions of the spectral parameter in the boundary conditions. The uniform stability for direct and inverse spectral problems is proved for the first time for…
We show that there are no general stability results for the logarithmic Sobolev inequality in terms of the Wasserstein distances and $L^{p}(d\gamma)$ distance for $p>1$. To this end, we construct a sequence of centered probability measures…
We provide a self-contained analysis, based entirely on pde methods, of the exponentially long time behavior of solutions to linear uniformly parabolic equations which are small perturbations of a transport equation with vector field having…
This work studies the stabilization for a periodic parabolic system under perturbations in the system conductivity. A perturbed system does not have any periodic solution in general. However, we will prove that the perturbed system can…
Estimation of solution norms and stability for time-dependent nonlinear systems is ubiquitous in numerous engineering, natural science and control problems. Yet, practically valuable results are rare in this area. This paper develops a…
We study the inverse problem of identifying a periodic potential perturbation of the Dirichlet Laplacian acting in an infinite cylindrical domain, whose cross section is assumed to be bounded. We prove log-log stable determination of the…
This paper is concerned with the hypercoercivity property of solutions to the Cauchy problem on the linear Boltzmann equation with a confining potential force. We obtain the exponential time rate of solutions converging to the steady state…
In this paper, we establish several improved Caffarelli-Kohn-Nirenberg and Hardy-type inequalities. Our main results are divided into two parts. In the first part, we consider the following Caffarelli-Kohn-Nirenberg inequality:…