SOLUTIONS OF ANISOTROPIC PARABOLIC EQUATIONS WITH DOUBLE NON-LINEARITY IN UNBOUNDED DOMAINS

This work is devoted to some class of parabolic equations of high order with double nonlinearity which can be represented by a model equation partial derivative/partial derivative t(vertical bar u vertical bar(k-2)u) = Sigma(n )(alpha=1)(-1)(m alpha-1)partial derivative(m alpha)/partial derivative x(alpha)(m alpha) [vertical bar partial derivative(m alpha)u/partial derivative x(alpha)(m alpha)vertical bar(p alpha-2) partial derivative(m alpha)u/partial derivative x(alpha)(m alpha)], m(1), ..., m(n )is an element of N(, )p(n )>= ... >= p(1 )> k, k > 1. For the solution of the first mixed problem in a cylindrical domain D = (0, infinity) x x Omega, Omega subset of R-n, n >= 2, with homogeneous Dirichlet boundary condition and finite initial function the highest rate of decay established as t -> infinity. Earlier upper estimates were obtained by the authors for anisotropic equation of the second order and prove their accuracy.