相关论文: Stochastic wave equation with H\"older noise coeff…

In this paper, we obtain several asymptotic profiles of solutions to the Cauchy problem for structurally damped wave equations $\partial_{t}^{2} u - \Delta u + \nu (-\Delta)^{\sigma} \partial_{t} u=0$, where $\nu >0$ and $0< \sigma \le1$.…

We consider a class of wave equations with constant damping and polynomial nonlinearities that are perturbed by small, multiplicative, space-time white noise. The equations are defined on a one-dimensional bounded interval with Dirichlet…

In this paper, we consider a system of $k$ second order non-linear stochastic partial differential equations with spatial dimension $d \geq 1$, driven by a $q$-dimensional Gaussian noise, which is white in time and with some spatially…

We consider non-linear time-fractional stochastic heat type equation $$\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\sigma(u)\stackrel{\cdot}{W}(t,x)]$$ in $(d+1)$ dimensions, where $\nu>0, \beta\in (0,1)$,…

We consider the semilinear equation $$ \epsilon^{2s} (-\Delta)^s u + V(x)u - u^p = 0, \quad u>0, \quad u\in H^{2s}(\R^N) $$ where $0<s<1,\ 1<p<\frac{N+2s}{N-2s}$, $ V(x)$ is a sufficiently smooth potential with $\inf_\R V(x)> 0$, and…

We consider the variational wave equation in one-dimensional space with stochastic forcing by an additive noise. Blow-up of local smooth solutions is established, and global existence is proved in the class of weak martingale solutions.

We consider the stochastic fractional heat equation $\partial_{t}u=\triangle^{\alpha/2}u+\lambda\sigma(u)\dot{w}$ on $[0,L]$ with Dirichlet boundary conditions, where $\dot{w}$ denotes the space-time white noise. For any $\lambda>0$, we…

In this article, we investigate the semiclassical version of the wave equation for the discrete Schr\"{o}dinger operator, $\mathcal{H}_{\hbar,V}:=-\hbar^{-2}\mathcal{L}_{\hbar}+V$ on the lattice $\hbar\mathbb{Z}^{n},$ where…

Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple $V\subseteq H\subseteq V^*$ $$ \left\{ \begin{align} &dX_t=A(t,X_t)dt+B(t,X_t)dW_t,\ t\in (0,T]\\\\& X_0=x\in H,…

We consider the Cauchy problem for the nonlinear Schr\"odinger equation on $\mathbb{R}^2$, $iu_t + u_{xx} + u_{yy} + \lambda|u|^\sigma u =0$, $\lambda\in \mathbb{R}$, $\sigma>0$. We introduce new functional spaces over which the initial…

We study the one-dimensional stochastic partial differential equation \begin{equation*} \frac{\partial u}{\partial t}(t,x) = -\kappa \frac{\partial^4 u}{\partial x^4}(t,x) + \rho \frac{\partial^2 u}{\partial x^2}(t,x) + b(u(t,x)) +…

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: \[ \left(\partial^\beta+\frac{\nu}{2}(-\Delta)^{\alpha/2}\right)u(t,x) =…

In this article, we consider fractional stochastic wave equations on $\mathbb R$ driven by a multiplicative Gaussian noise which is white/colored in time and has the covariance of a fractional Brownian motion with Hurst parameter…

This paper addresses the Cauchy problem for wave equations with scale-invariant time-dependent damping and nonlinear time-derivative terms, modeled as $$\partial_{t}^2u- \Delta u +\frac{\mu}{1+t}\partial_tu= f(\partial_tu), \quad x\in…

We investigate the stochastic heat equation driven by space-time white noise defined on an abstract Hilbert space, assuming that the drift and diffusion coefficients are both merely H\"older continuous. Random field SPDEs are covered as…

In this paper, we extend Walsh's stochastic integral with respect to a Gaussian noise, white in time and with some homogeneous spatial correlation, in order to be able to integrate some random measure-valued processes. This extension turns…

We prove the existence and uniqueness of mild solution for the stochastic partial differential equation $$\left(\partial^\alpha - \textit{B} \right) u(t,x)= u(t,x) \cdot \dot{W}(t,x),$$ where $$\alpha \in (1/2, 1)\cup(1, 2);$$ $\textit{B}$…

This paper is concerned with the analysis of a one dimensional wave equation $z_{tt}-z_{xx}=0$ on $[0,1]$ with a Dirichlet condition at $x=0$ and a damping acting at $x=1$ which takes the form $(z_t(t,1),-z_x(t,1))\in\Sigma$ for every…

A parameter estimation problem is considered for a one-dimensional stochastic wave equation driven by additive space-time Gaussian white noise. The estimator is of spectral type and utilizes a finite number of the spatial Fourier…

We consider standing waves of the nonlinear Schr\"odinger equation $i\partial_t u = -\Delta_\alpha u + |u|^{p-1}u$ in the defocusing case in dimensions $N=2$ and $N=3$. Here, $-\Delta_\alpha$ denotes the Laplacian with a point interaction.…