Asymptotics of solutions for sub critical non-convective type equations

摘要

We study the Cauchy problem for non-linear dissipative evolution equations[GRAPHICS]where L is the linear pseudodifferential operator Lu = (F) over bar xi-->x(L(xi)u(xi)) and the non-linearity is a quadratic pseudodifferential operator[GRAPHICS]udropF(x-->xi)u is the Fourier transformation. We consider non-convective type non-linearity, that is we suppose that a(t, 0, y) = 0. Let the initial data u(0) is an element of H-rho,H-0 boolean AND H-0,H-rho, rho > 1/2, are sufficiently small and have a non-zero total mass integral u(0)(x)dx=0, where H-n,H-m = {phi is an element of L-2(m)(n)phi(x)(L2.)