The initial boundary value problem for the nonlinear parabolic equation in unbounded domain

Abstract

Abstract. In this paper we consider the mixed problem for the equation utt + A1u + A2(ut) + g(ut) = f (x, t) in unbounded domain, where A1 is a linear elliptic operator of the fourth order and A2 is a nonlinear elliptic operator of the second order. Under natural assumptions on the equation coefficients and f we proof existence of a solution. This result contains, as a special case, some of known before theorems of existence. Essentially, in difference up to previous results we prove theorems of existence without the additional assumption on behavior of solution at infinity. Mathematics Subject Classification (2000). Primary 35K55, Secondary 35K60.