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

Abstract

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