Large time behavior of solutions for a Cauchy problem on nonlinear conservation laws with large initial data in the whole space

TL;DR

This paper investigates the long-term behavior of solutions to nonlinear conservation laws with large initial data, establishing global existence and decay estimates using advanced analytical techniques.

Contribution

It provides new results on the global existence and decay rates for solutions with large initial data in nonlinear conservation laws.

Findings

Proved global existence of solutions with large initial data.

Established optimal decay estimates over time.

Applied Green's function, Fourier analysis, and pseudo-differential operators.

Abstract

We consider the Cauchy problem on a nonlinear conversation law with large initial data. By Green's function methods, energy methods, Fourier analysis, frequency decomposition, pseudo-differential operators, we obtain the global existence and the optimal decay estimate of $t$.