The dynamics of solutions of the Cauchy problem for a parabolic type stochastic equation with power nonlinearities (the stochastic term is linear) (Q2771530)

Almost sure upper and lower bounds are constructed for the solution \(u(t,x)\) of the following stochastic equation NEWLINE\[NEWLINEdu(t,x)=(a(u^{\sigma+1}_{xx}+bu^{\beta}))dt+cud W(t),\quad t\in[0,T),\;x\in R^1,NEWLINE\]NEWLINE with the initial condition \(u(0,x)=u_0(x)\), where \(W(t)\) is a standard Brownian motion, \(a,b,c,\beta\) and \(\sigma\) are positive numbers, \(1\leq \beta \leq \sigma +1\), \(u_0(x)\) is non-negative bounded function vanishing on \(x\in (-\infty,l)\) and \(x\in (l,\infty)\) for some \(l>0\). In special cases, when solutions can be obtained in explicit form, an asymptotic behaviour of \(u(t,x)\) as \(t\to \infty\) is investigated.