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Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. (English) Zbl 0801.60050
The author considers the following equation \[ \partial u(t,x)/ \partial t = \Delta u(t,x) + b \bigl( u(t,x) \bigr) + a \bigl( u(t,x) \bigr) \dot W (t,x),\;t \geq 0,\;x \in R, \quad u (0,x) = f(x), \tag{1} \] where \(\dot W(t,x)\) is a space-time white noise. Let \[ C_{\text{tem}} = \Bigl\{ f \in C(R) \biggl| \sup_{x\in R} \biggl | \exp \bigl\{ - \lambda | x | \bigr\} f(x) \biggr |\biggr. < \infty \quad \text{for every} \quad \lambda > 0 \Bigr\}, \] \(C^ +_{\text{tem}}\) be the totality of nonnegative elements of \(C_{\text{tem}}\), \(C^ +_{\text{c}}\) be the totality of nonnegative, continuous functions with compact support.
The main results can be formulated as follows: Let \(a(u)\), \(b(u)\) be continuous functions, such that \(| a(u) | + | b(u) | \leq C(1 + | u |)\), \(u \in R\).
(i) If \(a(0) = 0\), \(b(0) \geq 0\), then for every \(f \in C^ +_{\text{tem} }\) there exists a \(C^ +_{\text{tem}}\)-valued solution \(u(t,x)\) of problem (1).
(ii) Assume that for each \(K>0\) there exists a constant \(a_ K>0\) such that \(| a(u) | \geq a_ K u^{1/2}\) for \(0 \leq u \leq K\), and that for some \(C>0\): \(| b(u) | \leq C | u |\) for \(u \in R\). Then if \(f \in C^ +_{\text{c}}\), \(P\{u(t,x) \in C^ +_{\text{c}}\) for every \(t>0\} = 1\) holds for every \(C^ +_{\text{tem}}\)- valued solution of (1).
(iii) Let \(a(u)\), \(b(u)\) be Lipschitz continuous, and \(u_ 1(t,x)\), \(u_ 2(t,x)\) be two \(C_{\text{tem}}\)-valued solutions of (1) with the initial conditions \(u_ 1(0) = f_ 1 \in C_{\text{tem}}\) and \(u_ 2(0) = f_ 2 \in C_{\text{tem}}\). Suppose that \(f_ 1 \geq f_ 2\) and \(f_ 1(x) > f_ 2(x)\) for some \(x \in R\). Then \(P\{u_ 1(t,x) > u_ 2(t,x)\) for every \(t>0\) and every \(x \in R\} = 1\).

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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