On dissipative nonlinear evolutional pseudo-differential equations

Abstract

Using the local time-frequency analysis techniques, we obtain an equivalent norm on modulation spaces. Secondly, applying this equivalent norm, we consider the Cauchy problem for the dissipative evolutionary pseudo-differential equation∂tu+A(x,D)u=F((∂xαu)|α|⩽κ),u(0,x)=u0(x), where A(x,D) is a dissipative pseudo-differential operator and F(z) is a multi-polynomial. We will develop the uniform decomposition techniques in both physical and frequency spaces to study its local well posedness in modulation spaces Mp,qs and in Sobolev spaces Hs. Moreover, the local solution can be extended to a global one in L2 and in Hs (s>κ+d/2) for certain nonlinearities.