On Dissipative Nonlinear Evolutional Pseudo-Differential Equations

TL;DR

This paper develops uniform decomposition techniques to analyze the local and global well-posedness of dissipative nonlinear pseudo-differential equations in modulation and Sobolev spaces, extending solutions under certain conditions.

Contribution

It introduces a novel uniform decomposition method in physical and frequency spaces for studying dissipative pseudo-differential equations in modulation spaces.

Findings

Established local well-posedness in modulation and Sobolev spaces.

Extended local solutions to global solutions in L^2 and H^s spaces.

Provided conditions for global existence based on nonlinearities.

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.