Related papers: Comparison principle for stochastic heat equation …

We study existence and uniqueness of weak solutions to (F) $\partial\_t u+ (-\Delta)^\alphau+h(t, u)=0 $ in $(0,\infty)\times\R^N$,with initial condition $u(0,\cdot)=\nu$ in $\R^N$, where $N\ge2$, the operator $(-\Delta)^\alpha$is the…

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)) +…

In [HHL+17] the authors showed existence and uniqueness of solutions to the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and rougher than white in space (in particular, its covariance…

We consider a system of $d$ non-linear stochastic heat equations in spatial dimension $k \geq 1$, whose solution is an $\R^d$-valued random field $u= \{u(t\,,x),\, (t,x) \in \R_+ \times \R^k\}$. The $d$-dimensional driving noise is white in…

In this paper, we investigate pointwise time analyticity of solutions to fractional heat equations in the settings of $\mathbb{R}^d$ and a complete Riemannian manifold $\mathrm{M}$. On one hand, in $\mathbb{R}^d$, we prove that any solution…

We study the stochastic heat equation (SHE) $\partial_t u = \frac12 \Delta u + \beta u \xi$ driven by a multiplicative L\'evy noise $\xi$ with positive jumps and amplitude $\beta>0$, in arbitrary dimension $d\geq 1$. We prove the existence…

In this paper, we consider the fractional heat equation $u_{t}=\triangle^{\alpha/2}u+f(u)$ with Dirichlet boundary conditions on the ball $B_{R}\subset \mathbb{R}^{d}$, where $\triangle^{\alpha/2}$ is the fractional Laplacian,…

Let $u=\{u(t,x);t \in [0,T], x \in {\mathbb{R}}^{d}\}$ be the process solution of the stochastic heat equation $u_{t}=\Delta u+ \dot F, u(0,\cdot)=0$ driven by a Gaussian noise $\dot F$, which is white in time and has spatial covariance…

Sharp Besov regularities in time and space variables are investigated for $\left(u(t,x),\; t\in [0,T],\; x\in \mathbb{R}\right)$, the mild solution to the stochastic heat equation driven by space-time white noise. Existence, H\"{o}lder…

We consider the problem of existence of a solution $u$ to $\partial_t u-\partial_{xx} u = 0$ in $(0,T)\times\mathbb{R}_+$ subject to the boundary condition $-u_x(t,0)+g(u(t,0))=\mu$ on $(0,T)$ where $\mu$ is a measure on $(0,T)$ and $g$ a…

Various effects of the noise intensity upon the solution $u(t,x)$ of the stochastic heat equation with Dirichlet boundary conditions on $[0,1]$ are investigated. We show that for small noise intensity, the $p$-th moment of $\sup_{x \in…

Consider the solution $\mathcal{Z}(t,x)$ of the one-dimensional stochastic heat equation, with a multiplicative spacetime white noise, and with the delta initial data $\mathcal{Z}(0,x) = \delta(x)$. For any real $p>0$, we obtained detailed…

We consider nonlinear parabolic SPDEs of the form $\partial_t u=\sL u + \sigma(u)\dot w$, where $\dot w$ denotes space-time white noise, $\sigma:\R\to\R$ is [globally] Lipschitz continuous, and $\sL$ is the $L^2$-generator of a L\'evy…

We study the time-fractional stochastic heat equation driven by time-space white noise with space dimension $d\in\mathbb{N}=\{1,2,...\}$ and the fractional time-derivative is the Caputo derivative of order $\alpha \in (0,2)$. We consider…

In this article, we consider the stochastic wave and heat equations driven by a Gaussian noise which is spatially homogeneous and behaves in time like a fractional Brownian motion with Hurst index $H>1/2$. The solutions of these equations…

This article considers the stochastic partial differential equation \[ \left\{ \begin{array}{l} u_t = \frac{1}{2} u_{xx} + u^\gamma \xi u(0,.) = u_0 \end{array}\right. \] \noindent where $\xi$ is a space / time white noise Gaussian random…

We consider a class of stochastic heat equations driven by truncated $\alpha$-stable white noises for $1<\alpha<2$ with noise coefficients that are continuous but not necessarily Lipschitz and satisfy globally linear growth conditions. We…

We study the compact support property for solutions of the following stochastic partial differential equations: $$\partial_t u = a^{ij}u_{x^ix^j}(t,x)+b^{i}u_{x^i}(t,x)+cu+h(t,x,u(t,x))\dot{F}(t,x),\quad (t,x)\in…

We consider the solution of $\partial_t u=\partial_x^2 u+\partial_x\partial_t B,\,(x,t)\in R\times(0,\infty)$, subject to $u(x,0)=0,\,x\in R$, where $B$ is a Brownian sheet. We show that $u$ also satisfies $\partial_x^2 u…

We present uniqueness and existence in weighted Sobolev spaces of the equation $$ u_t=(au_{xx}+bu_x+cu)+ \xi |u|^{1+\lambda} {\dot{B}}, \quad\,\, t>0, \, x\in (0,1) $$ with initial data $u(0,\cdot)=u_0$ and zero boundary data. Here…