Related papers: Stochastic CGL equations without linear dispersion…

We prove the monotonicity of positive solutions to the problem $-\Delta u = f(u)$ in $\mathbb{R}^N_+ := \{(x',x_N)\in\mathbb{R}^N \mid x_N>0 \}$ under zero Dirichlet boundary condition with a possible singular nonlinearity $f$. In some…

A time-fractional Fokker-Planck initial-boundary value problem is considered, with differential operator $u_t-\nabla\cdot(\partial_t^{1-\alpha}\kappa_\alpha\nabla u-\textbf{F}\partial_t^{1-\alpha}u)$, where $0<\alpha <1$. The forcing…

In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we…

In this paper, we are interested in the regularity of weak solutions $u\colon\Omega_T\to\mathbb{R}$ to parabolic equations of the type \begin{equation*} \partial_t u - \mathrm{div} \nabla \mathcal{F}(x,t,Du) = f\qquad\mbox{in $\Omega_T$},…

We consider the fluctuations of a time-integrated particle current around an atypical value in a generic stochastic Markov process involving classical particles with two-site interaction and hardcore repulsion on a finite one-dimensional…

We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the following integro-differential equation $$\partial\_t u(t, x) = \left(a(x) -- \int\_{\Omega} k(x, y)u(t, y) dy\right ) u(t, x) +…

We prove global existence and uniqueness of strong solutions to the logarithmic porous medium type equation with fractional diffusion $$ \partial_tu+(-\Delta)^{1/2}\log(1+u)=0, $$ posed for $x\in \mathbb{R}$, with nonnegative initial data…

Consider a parabolic stochastic PDE of the form $\partial_t u=\frac{1}{2}\Delta u + \sigma(u)\eta$, where $u=u(t\,,x)$ for $t\ge0$ and $x\in\mathbb{R}^d$, $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz continuous and non random, and…

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline…

We investigate the Cauchy problem for a semilinear parabolic equation driven by a mixed local-nonlocal diffusion operator of the form \[ \partial_t u - (\Delta - (-\Delta)^{\mathsf{s}})u = \mathsf{h}(t)|x|^{-b}|u|^p + t^\varrho…

We consider a zero-range process $\eta^N_t(x)$ with superlinear local jump rate, which in a hydrodynamic-small particle rescaling converges to the porous medium equation $\partial_t u=\frac12\Delta u^\alpha, \alpha>1$. As a main result we…

Goal of this paper is to study classes of Cauchy-Dirichlet problems which include parabolic equations of the type $$u_t -\Delta u= a(x,t)f(u)\quad\hbox{in $\Omega\times(0,T)$}$$ with $\Omega\subset\mathbb{R}^N$ bounded, convex domain and…

In this paper we prove existence and uniqueness results for nonlinear parabolic problems with Dirichlet boundary values whose model is \[ \left\{ \begin{aligned} &b(u)_t-\Delta_{p}u=\mu\;\mbox{in }(0,T)\times\Omega,\\…

The flow characterized by a linear longitudinal velocity field $u_x(x,t)=a(t)x$, where $a(t)={a_0}/({1+a_0t})$, a uniform density $n(t)\propto a(t)$, and a uniform temperature $T(t)$ is analyzed for dilute granular gases by means of a…

We consider the homogenization problem for the stochastic porous-medium type equation $\p_{t} u^\epsilon =\Delta f\left(T\left(\frac{x}{\ep}\right)\om,u^\ep\right)$, with a well-prepared initial datum, where $f(T(y)\om,u)$ is a stationary…

Consider the following stochastic partial differential equation, \begin{equation*} \partial_t u_t(x)= \mathcal{L}u_t(x)+ \xi\sigma (u_t(x)) \dot F(t,x), \end{equation*} where $\xi$ is a positive parameter and $\sigma$ is a globally…

In this paper, we first study the dual fractional parabolic equation \begin{equation*} \partial^\alpha_t u(x,t)+(-\Delta)^s u(x,t) = f(u(x,t))\ \ \mbox{in}\ \ B_1(0)\times\R , \end{equation*} subject to the vanishing exterior condition. We…

We study large time behaviour of solutions of the Cauchy problem for equations of the form $\partial_tu-L u+\lambda u=f(x,u)+g(x,u)\cdot\mu$, where $L$ is the operator associated with a regular lower bounded semi-Dirichlet form…

We are interested in the ``smoothest'' averaging that can be achieved by convolving functions $f \in \ell^2(\mathbb{Z})$ with an averaging function $u$. More precisely, suppose $u:\{-n, \ldots, n\} \to \mathbb{R}$ is a symmetric function…

The stochastic Landau--Lifshitz--Gilbert (LLG) equation describes the behaviour of the magnetization under the influence of the effective field consisting of random fluctuations. We first reformulate the equation into an equation the…