Related papers: Intermittence and nonlinear parabolic stochastic p…

We consider non-negative, weak solutions to the doubly nonlinear parabolic equation $$ \partial_t u^q-\mbox{div}(|Du|^{p-2}Du)=0 $$ in the super-critical fast diffusion regime $0<p-1<q<\frac{N(p-1)}{(N-p)_+}$. We show that when solutions…

We consider a class of semilinear stochastic evolution equations driven by an additive cylindrical stable noise.We investigate structural properties of the solutions like Markov, irreducibility, stochastic continuity, Feller and strong…

Let $\xi$ denote space-time white noise, and consider the following stochastic partial differential equations: (i) $\dot{u}=\frac{1}{2} u" + u\xi$, started identically at one; and (ii) $\dot{Z}=\frac12 Z" + \xi$, started identically at…

We consider stochastic partial differential equations under minimal assumptions: the coefficients are merely bounded and measurable and satisfy the stochastic parabolicity condition. In particular, the diffusion term is allowed to be…

This paper investigates the well-posedness and small-noise asymptotics of a class of stochastic partial differential equations defined on a bounded domain of $\mathbb{R}^d$, where the diffusion coefficient depends nonlinearly and…

We consider nonnegative solutions of a parabolic equation in a cylinder $D \timesI$, where $D$ is a noncompact domain of a Riemannian manifold and $I =(0,T)$ with $0 < T \le \infty$ or $I=(-\infty,0)$. Under the assumption [SSP] (i.e., the…

We examine in this article the one-dimensional, non-local, singular SPDE \begin{equation*} \partial_t u \;=\; -\, (-\Delta)^{1/2} u \,-\, \sinh(\gamma u) \,+\, \xi\;, \end{equation*} where $\gamma\in \mathbb{R}$, $(-\Delta)^{1/2}$ is the…

In this paper we study the existence and summability of the solutions to the following parabolic-elliptic system of partial differential equations with discontinuous coefficients: \begin{equation*} \begin{cases} u_t -…

We study integrability conditions for existence and nonexistence of a local-in-time integral solution of fractional semilinear heat equations with rather general growing nonlinearities in uniformly local $L^p$ spaces. Our main results about…

We consider the large-time behavior of sign-changing solutions of the inhomogeneous equation $u_t-\Delta u=|x|^\alpha |u|^{p}+\zeta(t)\,{\mathbf w}(x)$ in $(0,\infty)\times\mathbb{R}^N$, where $N\geq 3$, $p>1$, $\alpha>-2$, $\z, {\mathbf…

By using the analytic tools of Dirichlet forms, we initiate a study of some non-linear parabolic equations on Sierpinski gasket, motivated by modellings of fluid flows along a fractal (which can be considered as a simplified rough porous…

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 first part of this paper, the existence of infinitely many $L^p$-standing wave solutions for the nonlinear Helmholtz equation $$ -\Delta u -\lambda u=Q(x)|u|^{p-2}u\quad\text{ in }\mathbb{R}^N $$ is proven for $N\geq 2$ and…

We consider the nonlinear stochastic heat equation in one dimension. Under some conditions on the nonlinearity, we show that the "peaks" of the solution are rare, almost fractal like. We also provide an upper bound on the length of the…

Consider the following equation $$\partial_t u_t(x)=\frac{1}{2}\partial _{xx}u_t(x)+\lambda \sigma(u_t(x))\dot{W}(t,\,x)$$ on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution…

We consider the asymptotic behaviour of finite energy solutions to the one-dimensional defocusing nonlinear wave equation $-u_{tt} + u_{xx} = |u|^{p-1} u$, where $p > 1$. Standard energy methods guarantee global existence, but do not…

This book is an introduction to the theory of stochastic partial differential equations (SPDEs), using the random field approach pioneered by J.B. Walsh (1986). It consists of two blocks: the core matter (Chapters 1 to 6) and the appendices…

Consider the following stochastic heat equation, \begin{align*} \frac{\partial u_t(x)}{\partial t}=-\nu(-\Delta)^{\alpha/2} u_t(x)+\sigma(u_t(x))\dot{F}(t,\,x), \quad t>0, \; x \in R^d. \end{align*} Here $-\nu(-\Delta)^{\alpha/2}$ is the…

This article studies the continuity of bounded nonnegative weak solutions to inhomogeneous doubly nonlinear parabolic equations. A model equation is \begin{equation*}\partial_t u-\operatorname{div}(u^{m-1}|Du|^{p-2}Du)=f\qquad…

Consider a parabolic SPDE \[ \partial_t u = \Delta u + \sigma(u)\eta, \] on $(0\,,\infty)\times\mathbb{R}^d$, where $\eta$ is a centered, generalized Gaussian noise with $\text{Cov}[\eta(t\,,x)\,,\eta(s\,,y)]=\delta_0(t-s)\Lambda(x-y)$ for…