Related papers: Lp Solutions for Stochastic Evolution Equation wit…

In this paper, we discuss an initial boundary value problem for the stochastic wave equation involving the nonlinear damping term $|u_t|^{q-2}u_t$ and a source term of the type $|u|^{p-2}u$. We firstly establish the local existence and…

This work is concerned with the probabilistic representation of solutions to the $p$-Laplace evolution equation $\frac{\partial u}{\partial t}={\rm div}(|\nabla u|^{p-2}\nabla u)$ in $(0,\infty)\times\mathbb{R}^d$, $u(0,x)=u_0(x),$…

This paper establishes a complete homogenization theory for the one-dimensional parabolic equation with long-range correlated random potential: \[ \partial_t u_\varepsilon(t,x) = \frac{1}{2} \partial_{xx} u_\varepsilon(t,x) +…

We study the character of dependence on the data and the nonlinear structure of the equation for the solutions of the homogeneous Dirichlet problem for the evolution $p(x,t)$-Laplacian with the nonlinear source \[…

We consider solutions $u(t)$ to the 3d NLS equation $i\partial_t u + \Delta u + |u|^2u=0$ such that $\|xu(t)\|_{L^2} = \infty$ and $u(t)$ is nonradial. Denoting by $M[u]$ and $E[u]$, the mass and energy, respectively, of a solution $u$, and…

The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial…

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…

Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: $$\dot{u}=A(t)u+F(t,u)+b(t), \quad t\ge 0; \quad u(0)=u_0. \qquad (*)$$ Here $\dot{u}:=\frac {du}{dt}$, $u=u(t)\in H$,…

We prove existence of variational solutions for a class of doubly nonlinear nonlocal evolution equations whose prototype is the double phase equation \begin{align*} \partial_t u^m &+ \text{P.V.}\int_{\mathbb{R}^N}…

In this article, we study the following fractional $p$-Laplacian equation with critical growth singular nonlinearity \begin{equation*} \quad (-\De_{p})^s u = \la u^{-q} + u^{\alpha}, u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n…

We consider the periodic fractional nonlinear Schr\"{o}dinger equation $$ iu_t -(-\Delta)^{\frac{s}{2}} u + \mathcal{N}(|u|)u=0, \quad x\in \mathbb{T}^N,\, \, t \in \mathbb R, \, \, s>0, $$ where the nonlinearity term is expressed in two…

We establish existence, uniqueness as well as quantitative estimates for solutions to the fractional nonlinear diffusion equation, $\partial_t u +{\mathcal L}_{s,p} (u)=0$, where ${\mathcal L}_{s,p}=(-\Delta)_p^s$ is the standard fractional…

This paper is devoted to the analysis of the problem of stabilization of fractional (in time) partial differential equations. We consider the following equation $$ \partial^{\alpha,\eta}_{t} u(t)=\mathcal{A}u(t)-\frac{\eta}{\Gamma…

We investigate the strict positivity and the compact support property of solutions to the one-dimensional nonlinear stochastic heat equation: $$\partial_t u(t,x) = \frac{1}{2}\partial^2_x u(t,x) + \sigma(u(t,x))\dot{W}(t,x), \quad (t,x)\in…

We present existence, uniqueness, and sharp regularity results of solution to the stochastic partial differential equation (SPDE) \begin{align} \label{abs eqn} du=(a^{ij}(\omega,t)u_{x^ix^j}+f)dt + (\sigma^{ik}(\omega,t)u_{x^i}+g^k)dw^k_t,…

In this article, we show the existence of a unique entropy solution to the following problem: \begin{equation} \begin{split} (-\Delta)_{p,\alpha}^su&= f(x)h(u)+g(x) ~\text{in}~\Omega,\\ u&>0~\text{in}~\Omega,\\ u&=…

In this paper, we establish associativity, spatial ergodicity and a central limit theorem for certain nonnegative solutions to the stochastic heat equation $\partial_t u=\frac12\partial_x^2 u+ u^\gamma \xi$ with $\gamma\in (0, 1)$. When…

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 $$\alpha \in (1/2, 1)\cup(1, 2);$$ $\textit{B}$…

Pathwise non-uniqueness is established for non-negative solutions of the parabolic stochastic pde $$\frac{\partial X}{\partial t}=\frac{\Delta}{2}X+X^p\dot W+\psi,\ X_0\equiv 0$$ where $\dot W$ is a white noise, $\psi\ge 0$ is smooth,…

We consider the following stochastic space-time fractional diffusion equation with vanishing initial condition:$$ \partial^{\beta} u(t, x)=- \left(-\Delta\right)^{\alpha / 2} u(t, x)+ I_{0+}^{\gamma}\left[\dot{W}(t, x)\right],\quad…