相关论文: Stochastic Cauchy Initial Value Formulation Of The…
We study two initial value problems of the linear diffusion equation and a nonlinear diffusion equation, when Cauchy data are bounded and oscillate mildly. The latter nonlinear heat equation is the equation of the curvature flow, when the…
We investigate the large time behavior of the hot spots of the solution to the Cauchy problem for the heat equation with a potential $\partial_t u-\Delta u+V(|x|)u=0$, where $V=V(r)$ decays quadratically as $r\to\infty$. In this paper,…
In this article, we establish global-in-time maximal regularity for the Cauchy problem of the classical heat equation $\partial_t u(x,t)-\Delta u(x,t)=f(x,t)$ with $u(x,0)=0$ in a certain $\rm BMO$ setting, which improves the local-in-time…
Consider the following space-time fractional heat equation with Riemann-Liouville derivative of non-homogeneous time-fractional Poisson process \begin{eqnarray*} \partial^\beta_t u(x,t) =-\kappa(-\Delta)^{\alpha/2} u(x,t) +…
In this note we focus our attention on a stochastic heat equation defined on the Heisenberg group $\mathbf{H}^{n}$ of order $n$. This equation is written as $\partial_t u=\frac{1}{2}\Delta u+u\dot{W}_\alpha$, where $\Delta$ is the…
We investigate the inverse Cauchy and data completion problems for elliptic partial differential equations in a bounded domain $D \subset \mathbb{R}^d$, $d \ge 2$, with a special emphasis on the steady-state heat conduction in anisotropic…
We consider the Cauchy problem for a time fractional semilinear heat equation with initial data belonging to inhomogeneous/homogeneous Besov--Morrey spaces. We present sufficient conditions for the existence of local/global-in-time…
In this article, we consider the stochastic heat equation $du=(\Delta u+f(t,x))dt+ \sum_{k=1}^{\infty} g^{k}(t,x) \delta \beta_t^k, t \in [0,T]$, with random coefficients $f$ and $g^k$, driven by a sequence $(\beta^k)_k$ of i.i.d.…
This paper deals with the long term behavior of the solution to the nonlinear stochastic heat equation $\partial u /\partial t - \frac{1}{2}\Delta u = b(u)\dot{W}$, where $b$ is assumed to be a globally Lipschitz continuous function and the…
We use the theory of regularity structures to develop an It\^o formula for $u$, the solution of the one dimensional stochastic heat equation driven by space-time white noise with periodic boundary conditions. In particular for any smooth…
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 study the Cauchy problem for a semilinear heat equation with initial data non-rarefied at $\infty$. Our interest lies in the discussion of the effect of the non-rarefied factors on the life span of solutions, and some sharp estimates on…
Consider the following nonlinear one-dimensional stochastic fractional heat equation $$\frac{\partial }{\partial t}u(t, x)= -(-\Delta)^{\alpha/2}u(t, x) +\sigma(t,x,u(t,x)) \dot{W}(t, x), $$ where $-(-\Delta)^{\alpha/2}$ is the fractional…
In this paper, we consider the one-dimensional stochastic heat equation driven by a space time white noise. In two different scenarios: {\it (i)} initial condition $u_0=1$ and general nonlinear coefficient $\sigma$ and {\it (ii)}: initial…
In this paper we study Cauchy problem for thermoelastic plate equations with friction or structural damping in $\mathbb{R}^n$, $n\geq1$, where the heat conduction is modeled by Fourier's law. We explain some qualitative properties of…
In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping, i.e.$\frac{2}{1+t}\partial_t v$ and a cubic convolution $(|x|^{-\gamma}*v^2)v$ with $\gamma\in (0,n)$, where $v=v(x,t)$…
The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_t u(t,z)=\Delta u(t,z)+\xi(z) u(t,z)$ on $(0,\infty)\times {\mathbb Z}^d$ with random potential $(\xi(z) \colon z\in {\mathbb Z}^d)$. We consider…
We consider the stochastic heat equation (SHE) on the torus $\mathbb{T}=[0,1]$, driven by space-time white noise $\dot W$, with an initial condition $u_0$ that is nonnegative and not identically zero: \begin{equation*} \frac{\partial…
We consider 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 $\nu>0$, $\beta\in (0,1)$, $\alpha\in (0,2]$,…
The aim of this short note is to obtain the existence, uniqueness and moment upper bounds of the solution to a stochastic heat equation with measure initial data, without using the iteration method in Chen and Dalang(2015), Chen and…