Initial-boundary value problems for elliptic-parabolic equations in unbounded domains with conditions at infinity

Abstract

The article investigates a initial-boundary value problems for second-order elliptic-parabolic equations given in unbounded domains with respect to the spatial variables. In the class of equations under consideration, in addition to linear ones, there are nonlinear ones with variable nonlinearity exponents. The existence and uniqueness of weak solutions of the studied problems are established under additional conditions on behavior of solutions and the growth of input data at infinity. A priori estimates of weak solutions of the studied problems are obtained. The study uses an analogue of the Saint-Venant principle known from mechanics and the monotonicity method.