The Dynamical Models of Sobolev Type with Showalter–Sidorov Condition and Additive “Noise”

The concept of “white noise”, initially established in finite-dimensional spaces, has been transfered to infinite-dimensional spaces. The goal of this transition is to develop the theory of stochastic Sobolev type equations and to elaborate applications of practical value. The derivative of Nelson–Gliklikh is entered to reach this goal, as well as the spaces of “noises” are developed. The equations of Sobolev type with relatively bounded operators are considered in the spaces of differentiable “noises”. Besides, the existence and uniqueness of their classical solutions are proved. A stochastic equation of Barenblatt–Zheltov–Kochina is considered as an application in bounded domain with homogeneous boundary condition of Dirichlet and initial condition of Showalter–Sidorov.