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Existence and uniqueness results for semilinear stochastic partial differential equations. (English) Zbl 0942.60058
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. R. Manthey, Stochastics Stochastics Rep. 57, No. 3-4, 213-234 (1996; Zbl 0887.60070)] in which related methods are employed.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35R60 PDEs with randomness, stochastic partial differential equations
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