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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:…
We study the two-dimensional two-component Coulomb gas in the canonical ensemble and at inverse temperature $\beta>2$. In this regime, the partition function diverges and the interaction needs to be cut off at a length scale $\lambda\in…
We investigate the existence and multiplicity of positive solutions to the following problem driven by the superposition of the Laplacian and the fractional Laplacian with Hardy potential \begin{equation*} \left\{ \begin{aligned} -\Delta u…
The largest Lyapunov exponent $\lambda^+$ for a dilute gas with short range interactions in equilibrium is studied by a mapping to a clock model, in which every particle carries a watch, with a discrete time that is advanced at collisions.…
We introduce, and numerically study, a system of $N$ symplectically and globally coupled standard maps localized in a $d=1$ lattice array. The global coupling is modulated through a factor $r^{-\alpha}$, being $r$ the distance between maps.…
This paper considers a one-dimensional generalized Allen-Cahn equation of the form \[ u_t = \varepsilon^2 (D(u)u_x)_x - f(u), \] where $\varepsilon>0$ is constant, $D=D(u)$ is a positive, uniformly bounded below diffusivity coefficient that…
We establish the exact quenched asymptotic growth of the solution to the parabolic Anderson model (PAM) in the hyperbolic space with a regular, stationary, time-independent Gaussian potential. More precisely, we show that with probability…
We calculate the spectrum of Lyapunov exponents for a point particle moving in a random array of fixed hard disk or hard sphere scatterers, i.e. the disordered Lorentz gas, in a generic nonequilibrium situation. In a large system which is…
We are concerned with the existence and asymptotic properties of solutions to the following fourth-order Schr\"{o}dinger equation \begin{equation}\label{1} {\Delta}^{2}u+\mu \Delta u-{\lambda}u={|u|}^{p-2}u, ~~~~x \in \R^{N}\\…
A mathematical model describing the capture of nonlinear systems into the autoresonance by a combined parametric and external periodic slowly varying perturbation is considered. The autoresonance phenomenon is associated with solutions…
Inspired by Lin-Pan-Wang (Comm. Pure Appl. Math., 65(6): 833-888, 2012), we continue to study the corresponding time-independent case of the Keller-Rubinstein-Sternberg problem. To be precise, we explore the asymptotic behavior of…
Consider an infinite system \[\partial_tu_t(x)=(\mathscr{L}u_t)(x)+ \sigma\bigl(u_t(x)\bigr)\partial_tB_t(x)\] of interacting It\^{o} diffusions, started at a nonnegative deterministic bounded initial profile. We study local and global…
We carry out a systematical study of the size scaling of Liouvillian gap in boundary-dissipated one-dimensional quasiperiodic and disorder systems. By treating the boundary-dissipation operators as a perturbation, we derive an analytical…
In this paper we deal with infinite-dimensional nonlinear forward complete dynamical systems which are subject to external disturbances. We first extend the well-known Datko lemma to the framework of the considered class of systems. Thanks…
In this paper, we study the large-time behavior of solutions to a class of partially dissipative linear hyperbolic systems with applications in velocity-jump processes in several dimensions. Given integers $n,d\ge 1$, let $\mathbf…
In this paper, we study the following Kirchhoff type problem:% $$ \left\{\aligned&-\bigg(\alpha\int_{\bbr^3}|\nabla u|^2dx+1\bigg)\Delta u+(\lambda a(x)+a_0)u=|u|^{p-2}u&\text{ in }\bbr^3,\\%…
In this work, we study the problem of finding the asymptotic growth rate of the number of of $d$-dimensional arrays with side length $n$ over a given alphabet which avoid a list of one-dimensional "forbidden" words along all cardinal…
We are concerned with a system governing the evolution of the pressureless compressible Euler equations with Riesz interaction and damping in $\mathbb{R}^{d}$ ($d\geq1$), where the interaction force is given by…
We study the regularity of the Lyapunov exponent for quasi-periodic cocycles $(T_\omega, A)$ where $T_\omega$ is an irrational rotation $x\to x+ 2\pi\omega$ on $\SS^1$ and $A\in {\cal C}^l(\SS^1, SL(2,\mathbb{R}))$, $0\le l\le \infty$. For…
We study the asymptotic behavior as $p\to\infty$ of the Gelfand problem \[ -\Delta_{p} u=\lambda\,e^{u}\ \textrm{in}\ \Omega\subset\mathbb{R}^n,\quad u=0 \ \textrm{on}\ \partial\Omega. \] Under an appropriate rescaling on $u$ and $\lambda$,…