Existence and uniqueness results for semilinear stochastic partial differential equations
type: Item
instance of: scholarly article
title: Existence and uniqueness results for semilinear stochastic partial differential equations
zbMATH Open document ID: 0942.60058
DOI: 10.1016/S0304-4149(97)00103-8
author: István Gyöngy
published in: Stochastic Processes and their Applications
publication date: 2000-03-01T00:00:00Z
review text: A semilinear stochastic parabolic equation \[ \frac {\partial u}{\partial t} = \frac {\partial ^{2}u}{\partial x^{2}} + f(t,x,u) + \frac {\partial }{\partial x}g(t,x,u) + \sigma (t,x,u)\frac {\partial ^{2}W}{\partial t\partial x}, \quad t\in [0,T],\;x\in [0,1], \tag{1} \] with a Dirichlet boundary condition \(u(t,0)=u(t,1)=0\) and an initial datum \(u(0,\cdot)=u_{0}\) is studied. By \((\partial ^{2}/\partial t\partial x)W\) a space-time white noise is denoted, and \(f,g,\sigma : [0,T]\times [0,1]\times \mathbb R\longrightarrow \mathbb R\) are Borel functions. (Note that with the special choice \(f=0\) and \(g(t,x,r)= \frac 12 r^2\) one gets a stochastic Burgers equation which has been paid a considerable attention recently.) The function \(\sigma \) is assumed to be bounded and globally Lipschitz continuous in the third variable; the functions \(f\), \(g\) are locally Lipschitz in the sense that \(|f(t,x,p)-f(t,x,q)|\leq L(1+|p|+|q|) |p-q|\) for any \(0\leq t \leq T\), \(0\leq x\leq 1\), \(p,q\in \mathbb R\) and analogously for \(g\). Further it is supposed that \(f\) satisfies the linear growth condition and \(g(t,x,r)=g_{1}(t,x,r)+g_{2}(t,r)\), \(g_1\) being of a linear growth and \(g_2\) obeying a quadratic growth condition. It is shown that if \(u_{0}\) is an \({\mathcal F}_{0}\)-measurable \(L^{p}([0,1])\)-valued random variable for \(p\geq 2\), then there exists a unique solution (in the sense of Walsh) \(u\) of the equation (1) and \(u\) is an \(L^{p}\)-valued process with continuous paths. The solution \(u\) has a modification continuous in \((t,x)\) provided \(u_{0}\) takes values in the space of continuous functions. Moreover, it is proven that the assumption of local Lipschitz continuity of the coefficient \(f\) is superfluous if \(|\sigma |\geq \lambda >0\) on \([0,T]\times [0,1]\times \mathbb R\). Proofs of these important results are based on a new comparison theorem that is also established. A reader interested in similar problems may consult, in addi…
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
zbMATH DE Number: 1409174
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: stochastic Burgers equations
zbMATH Keywords: space-time white noise
zbMATH Keywords: comparison theorems
MaRDI profile type: Publication
cites work: White noise driven parabolic SPDEs with measurable drift
cites work: The stochastic Burgers equation
cites work: Q4057385
cites work: Stochastic Burgers' equation
cites work: Stochastic burgers equation with correlated noise
cites work: White noise driven SPDEs with reflection
cites work: Random motion of strings and related stochastic evolution equations
cites work: Q5665949
cites work: On nondegenerate quasilinear stochastic partial differential equations
cites work: Existence of strong solutions for Itô's stochastic equations via approximations
cites work: The partial differential equation ut + uux = μxx
cites work: On $L_p $-Theory of Stochastic Partial Differential Equations in the Whole Space
cites work: On Itô’s Stochastic Integral Equations
cites work: Q3910361
cites work: Q5890172
cites work: Stochastic partial differential equations and filtering of diffusion processes
cites work: Q3997285
cites work: Q3747436
full work available at URL: s0304-4149(97)00103-8
OpenAlex ID: W2084202582
description: scientific article; zbMATH DE number 1409174