Cauchy problem in function spaces with asymptotic expansions with respect to time variable

TL;DR

This paper investigates nonlinear Cauchy problems within function spaces characterized by asymptotic expansions in time, demonstrating that solutions inherit the summability properties of the given functions, thus advancing asymptotic analysis techniques.

Contribution

It establishes that solutions to nonlinear Cauchy problems in Borel summable or multisummable spaces also possess these summability properties, extending the understanding of asymptotic solutions.

Findings

Solutions inherit the summability properties of the data functions.

The study confirms solutions are in the same asymptotic function spaces as the given functions.

Provides a framework for analyzing nonlinear PDEs with asymptotic expansions.

Abstract

A system of nonlinear Cauchy problem $\partial_t u_i=f_i(t,x, U, \nabla_xU )$ $u_i(0,x)= u_{i,0}(x)$ is studied in function spaces with asymptotic expansion with respect to $t$. To be specific, it is discussed in Borel summable or multisummable function space.It is recognized that these functions are important classes in asymptotic analysis. We study equations under the condition $\{f_i(t,x, U, P)\}_{i=1}^m$ are in these function spaces with respect to $t$ and show $\{u_i(t,x)\}_{i=1}^m$ have also the same summability.