A priori estimates for the solution of the first boundary-value problem for a class of second-order parabolic systems

We consider two classes of second-order parabolic matrix-vector systems (with solutions u element of Mmx1, m≥2) that can be reduced to a single second-order parabolic equation for a scalar function v=, where p element of Mmx1 is a fixed stochastic constant vector. We consider the first boundary-value problem for a scalar second-order parabolic equation (with unbounded coefficients) in a domain unbounded with respect to x under the assumption of strong absorption at infinity. We obtain an a priori estimate for solutions of the first boundary-value problem in the generalized Tikhonov-Taecklind classes. (The problem under investigation has at most one solution in these classes.)