Related papers: Hypercontractivity, supercontractivity, ultrabound…

We study, in the periodic setting, the well-posedness of the Cauchy problem associated to the operator $P(t, D_{x}, D_{t}) = D_{t} - a_{2}(t) \Delta_{x} + \sum_{j = 1}^{N} a_{1, j}(t) D_{x_{j}} + a_{0}(t)$, with $T> 0$, $t \in [0, T]$ and…

We consider the Cauchy problem for nonlinear Schrodinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we…

This paper considers a general framework for the study of the existence of quasi-variational and variational solutions to a class of nonlinear evolution systems in convex sets of Banach spaces describing constraints on a linear combination…

This work is concerned with an optimal control problem governed by a non-smooth quasilinear elliptic equation with a nonlinear coefficient in the principal part that is locally Lipschitz continuous and directionally but not G\^ateaux…

We consider the Cauchy problem for doubly non-linear degenerate parabolic equations on Riemannian manifolds of infinite volume, or in $\R^N$. The equation contains a weight function as a capacitary coefficient which we assume to decay at…

We investigate a smoothing property for strongly-continuous operator semigroups, akin to ultracontractivity in parabolic evolution equations. Specifically, we establish the stability of this property under certain relatively bounded…

We consider the following Cauchy problem for weakly coupled systems of semi-linear damped elastic waves with a power source non-linearity in three-dimensions: \begin{equation*} U_{tt}-a^2\Delta U-\big(b^2-a^2\big)\nabla\text{div }…

We face the well-posedness of linear transport Cauchy problems $$\begin{cases}\dfrac{\partial u}{\partial t} + b\cdot\nabla u + c\,u = f&(0,T)\times{\mathbb R}^n\\u(0,\cdot)=u_0\in L^\infty&{\mathbb R}^n\end{cases}$$ under borderline…

We study a nonlinear equation in the half-space $\{x_1>0\}$ with a Hardy potential, specifically \[-\Delta u -\frac{\mu}{x_1^2}u+u^p=0\quad\text{in}\quad \mathbb R^n_+,\] where $p>1$ and $-\infty<\mu<1/4$. The admissible boundary behavior…

In this paper, we would like to study the linear Cauchy problems for semi-linear $\sigma$-evolution models with mixing a parabolic like damping term corresponding to $\sigma_1 \in [0,\sigma/2)$ and a $\sigma$-evolution like damping…

The Cauchy problem for the von Neumann hierarchy of nonlinear equations is investigated. One describes the evolution of all possible states of quantum many-particle systems by the correlation operators. A solution of such nonlinear…

We study the well-posedness of Cauchy problems on the upper half space $\mathbb{R}^{n+1}_+$ associated to higher order systems $\partial_t u =(-1)^{m+1}\mbox{div}_m A\nabla ^m u$ with bounded measurable and uniformly elliptic coefficients.…

We provide general product formulas for the solutions of non-autonomous abstract Cauchy problems. The main technical tool is the application of evolution semigroup methods, allowing the direct application of existing results on autonomous…

We prove that solutions to Cauchy problems related to the $p$-parabolic equations are stable with respect to the nonlinearity exponent $p$. More specifically, solutions with a fixed initial trace converge in an $L^q$-space to a solution of…

We deal with the existence of solutions having L2 regularity for a class of non autonomous evolution equations. Associated with the equation, a general non local condition is studied. The technique we used combines a finite dimensional…

In this paper we propose a nonconforming finite element method for the solution of the ill-posed elliptic Cauchy problem. We prove error estimates using continuous dependence estimates in the $L^2$-norm. The effect of perturbations in data…

In the paper we derive two formulas representing solutions of Cauchy problem for two Schr\"{o}dinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally…

This article studies a class of semilinear scalar field equations on the real line with variable coefficients in the linear terms. These coefficients are not necessarily small perturbations of a constant. We prove that under suitable…

We consider the Cauchy problem for the Schr\"odinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the Euclidean plane is the existence of a rich…

In this paper we consider the Schr{\"o}dinger equation with nonlinear derivative term. Our goal is to initiate the study of this equation with non vanishing boundary conditions. We obtain the local well posedness for the Cauchy problem on…