Decay of solutions of non-homogenous hyperbolic equations

TL;DR

This paper investigates the conditions under which solutions to non-homogeneous hyperbolic equations decay over time, establishing criteria for their convergence to zero in the L2 norm and uniqueness of solutions that fade as time approaches infinity.

Contribution

It provides a characterization of decay conditions for solutions and proves the uniqueness of solutions that diminish over time for non-homogeneous hyperbolic equations.

Findings

Solutions decay to zero in L2 norm under specific conditions.

The right-hand side of the equation determines initial conditions uniquely.

There exists a unique solution that fades as time approaches infinity.

Abstract

We consider conditions for the decay in time of solutions of non-homogenous hyperbolic equations. It is proven that solutions of the equations go to $0$ in $L^2$ at infinity if and only if an equation's right-hand side uniquely determines the initial conditions in a certain way. We also obtain that a hyperbolic equation has a unique solution that fades when $t\to\infty$.