Existence and uniqueness results for semilinear stochastic partial differential equations
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 addition to articles referred to in the paper, also works by Manthey and his collaborators [see e.g. \textit{R. Manthey}, Stochastics Stochastics Rep. 57, No. 3-4, 213-234 (1996; Zbl 0887.60070)] in which related methods are employed.
stochastic partial differential equations
scientific article; zbMATH DE number 1409174
scientific article