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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 $$x \in R\} = 1$$.

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