Related papers: New Stability and Exact Observability Conditions f…

We bound the modulus of continuity of solutions to quasilinear parabolic equations in one space variable in terms of the initial modulus of continuity and elapsed time. In particular we characterize those equations for which the Lipschitz…

This paper investigates the critical quintic wave equation in a 3D bounded domain subject to locally distributed Kelvin-Voigt damping. The study tackles two major mathematical challenges: the severe loss of derivatives induced by the…

We propose a novel framework for stabilization, with an estimate of the region of attraction, of quasilinear parabolic partial differential equations (PDEs) that exhibit finite-time blow-up phenomena when null boundary inputs are imposed.…

In this work we develop a new numerical approach for recovering a spatially dependent source component in a standard parabolic equation from partial interior measurements. We establish novel conditional Lipschitz stability and H\"{o}lder…

We introduce a novel framework for the analysis of linear wave equations on nonstationary asymptotically flat spacetimes, under the assumptions of mode stability and absence of zero energy resonances for a stationary model operator. Our…

We establish a priori regularity estimates for viscosity solutions of degenerate fully nonlinear elliptic equations with integrable right-hand sides. When the nonhomogeneous term belongs to $L^p$ with $p>n$, we prove optimal interior…

For any subcritical index of regularity $s>3/2$, we prove the almost global well posedness for the 2-dimensional semilinear wave equation with the cubic nonlinearity in the derivatives, when the initial data are small in the Sobolev space…

We establish new bounds of the Sobolev norms of solutions of semilinear wave equations for data lying in the Hs, s<1, closure of compactly supported data inside a ball of radius R, with R a fixed and positive number. In order to do that we…

The stability of periodic traveling wave solutions to dispersive PDEs with respect to `arbitrary' perturbations is still widely open. The focus is put here on stability with respect to perturbations of the same period as the wave, for…

We establish the full asymptotic stability of solitary waves for the focusing cubic Schr\"odinger equation on the line under small even perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances…

In this paper, we investigate the stability of the linear wave equation where one part of the boundary, which is seen as a lower-dimensional Riemannian manifold, is governed by a coupled wave equation, while the other part is subject to a…

We study finite-time blowup for a nonlinear wave equation for maps from the Minkowski space $\mathbb{R}^{1+d}$ into the 1-sphere $\mathbb{S}^1$, whose nonlinearity exhibits a null-form structure. We construct, for every dimension $d \geq…

Systems of N = 1, 2, . . . first-order hyperbolic conservation laws feature N undamped waves propagating at finite speeds. On their own hand, multi-step Finite Difference and lattice Boltzmann schemes with q = N + 1, N + 2, . . . unknowns…

In this paper we provide bound estimates for the two fastest wave speeds emerging from the solution of the Riemann problem for three well-known hyperbolic systems, namely the Euler equations of gas dynamics, the shallow water equations and…

In this article, we first present the construction of Gibbs measures associated to nonlinear Schr\"odinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial…

In recent papers (arXiv:2407.16507, arXiv:2408.05158) we presented results suggesting the existence of a new class of time-periodic solutions to the defocusing cubic wave equation on a one-dimensional interval with Dirichlet boundary…

We consider the null controllability problem for the wave equation, and analyse a stabilized finite element method formulated on a global, unstructured spacetime mesh. We prove error estimates for the approximate control given by the…

We examine $L^p$-viscosity solutions to fully nonlinear elliptic equations with bounded-measurable ingredients. By considering $p_0<p<d$, we focus on gradient-regularity estimates stemming from nonlinear potentials. We find conditions for…

The stability and convergence analysis of high-order numerical approximations for the one- and two-dimensional nonlocal wave equations on unbounded spatial domains are considered. We first use the quadrature-based finite difference schemes…

We address the exact boundary controllability of the semilinear wave equation $\partial_{tt}y-\Delta y + f(y)=0$ posed over a bounded domain $\Omega$ of $\mathbb{R}^d$. Assuming that $f$ is continuous and satisfies the condition…