Related papers: On global solutions to the semidiscrete stochastic…

We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension $d \geq 3$ with bounded spatial extent is globally nonlinearly stable under sufficiently small compactly-supported perturbations, where…

We use the semi-discrete method, originally proposed in Halidias (2012), Semi-discrete approximations for stochastic differential equations and applications, International Journal of Computer Mathematics, 89(6), to reproduce qualitative…

We consider one-dimensional stochastic heat equation with nonlinear drift, $\displaystyle \partial_t u=\frac{1}{2}\Delta u+b(u)u+\sigma(u)\dot{W}(t,x)$, where $b:\mathbb{R}_{+}\to \mathbb{R}$ is a continuous function and…

The aim of this paper is to study the $d$-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and it has the covariance of a fractional Brownian motion with Hurst parameter $% H\in (0,1)$ in…

It is well known that the global well-posedness of the Navier-Stokes equations with temperature-dependent coefficients is a challenging problem, especially in multi-dimensional space. In this paper, we study the 3D Navier-Stokes equations…

We study stochastic heat equations in the sense of Walsh defined by fractal Laplacians on Cantor-like sets. For this purpose, we first investigate the corresponding heat kernels. Then, we prove existence and uniqueness of mild solutions to…

In this paper, we study a nonlinear one spatial dimensional stochastic heat equations driven by Gaussian noise: $\frac{\partial u }{\partial t}=\frac{\partial^2 u }{\partial x^2}+\sigma(u )\dot{W} $, where $\dot{W} $ is white in time and…

We establish nonuniqueness of solutions for Cauchy problems of semilinear heat equations with a wide class of nonlinearities. Specifically, we consider \[ \begin{cases} \partial_tu-\Delta u=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), &…

We consider a system of interacting diffusions on the integer lattice. By letting the mesh size go to zero and by using a suitable scaling, we show that the system converges (in a strong sense) to a solution of the stochastic heat equation…

For the fundamental solutions of heat-type equations of order $n$ we give a general stochastic representation in terms of damped oscillations with generalized gamma distributed parameters. By composing the pseudo-process $X_n$ related to…

We propose and analyze a space-time virtual element method for the discretization of the heat equation in a space-time cylinder, based on a standard Petrov-Galerkin formulation. Local discrete functions are solutions to a heat equation…

This paper is devoted to global well-posedness, self-similarity and symmetries of solutions for a superdiffusive heat equation with superlinear and gradient nonlinear terms with initial data in new homogeneous Besov-Morrey type spaces.…

We study small-ball probabilities for the stochastic heat equation with multiplicative noise in the moderate-deviations regime. We prove the existence of a small-ball constant and related it to other known quantities in the literature.…

We derive a nice representation for point symmetry transformations of the (1+1)-dimensional linear heat equation and properly interpret them. This allows us to prove that the pseudogroup of these transformations has exactly two connected…

We propose a simple method to obtain semigroup representation of solutions to the heat equation using a local $L^2$ condition with prescribed growth and a boundedness condition within tempered distributions. This applies to many functional…

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…

In this paper, we consider the 2-D dissipative incompressible porous media (IPM) equation in both supercritical and subcritical cases. The dissipative IPM equation admits a class of special solutions of the form $\rho(x_1,x_2,t)=f(x_2,t)$,…

We consider singular quasilinear stochastic partial differential equations (SPDEs) studied in \cite{FHSX}, which are defined in paracontrolled sense. The main aim of the present article is to establish the global-in-time solvability for a…

In the paper, we study spatially distributed particle systems whose time evolution is governed by vanishing diffusion in space $\mathbb{R}^d$, $d\ge 1$, and by size-continuous fragmentation and coagulation processes with unbounded rates. We…

In this paper, we consider the global solvability and energy conservation for initial value problem of nonlinear semi-discrete wave equation of Kirchhoff type, which is a discretized model of Kirchhoff equation.