Estimates for the solution of the Cauchy problem for a stochastic differential equation of parabolic type with power-law nonlinearities (a strong source) (Q1874967)

The following Cauchy problem is considered: \[ du(t,x) = (a(u^{\sigma +1})_{xx} + bu^{\beta})dt + cudw(t), t\in [0,T], x\in R^1,\quad u(0,x) = u_0(x) \] where \( w(t)\) is a standard Wiener process, \( a, b, c, \sigma \) are positive numbers, \( \beta > \sigma +1 \), and \( u_0(x) \) is a function satisfying some special assumptions. The author constructs processes that bound the solution of the above Cauchy problem from above and below with probability \( \geq 1 - \alpha\), \( 0 < \alpha < 1 \) for all \( t \in [0,T] \) and \( x \in R^1 \).