On Dissipative Nonlinear Evolutional Pseudo-Differential Equations

Abstract

First, using the uniform decomposition in both physical and frequency spaces, we obtain an equivalent norm on modulation spaces. Secondly, we consider the Cauchy problem for the dissipative evolutionary pseudo-differential equation \partial_t u + A(x,D) u = F\big((\partial^\alpha_x u)_{|\alpha|\leq \kappa}\big), \ \ u(0,x)= u_0(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 $M^s_{p,q}$ and in Sobolev spaces $H^s$. Moreover, the local solution can be extended to a global one in $L^2$ and in $H^s$ ($s>\kappa+d/2$) for certain nonlinearities.