On global solvability and asymptotic behaviour of a mixed problem for a nonlinear degenerate Kirchhoff model in moving domains

In this paper we prove the existence, uniqueness and asymptotic behaviour of global regular solutions of the mixed problem for the Kirchhoff nonlinear model given by the hyperbolic-parabolic equation (rho(1)u(t))(t) + rho(2)u(t) - M (t, integral (beta(t))(alpha(t)) \u(x)\(2)dx) u(xx) = f in (Q) over cap, where (Q) over cap = {(x, t) is an element of R-2\ alpha(t) < x < beta(t)(-), 0 < t < infinity} is a noncylindrical domain of R-2 and beta(.), alpha(.) are positive functions such that lim t --> infinity (beta(t) - alpha(t)) +infinity; The real function M(., .) is such that M(t, lambda) greater than or equal to m(0) > 0 For All(t, lambda) is an element of [0, infinity] x [0, infinity], while rho(1)(.), rho(2)(.) are given functions which satisfy some appropriate conditions.