New estimates for the mild solution to modified wave equation on the unbounded domain

Abstract

This paper studies the modified wave equation, Love equation, on the unbounded domain $R$. We prove that, given suitable initial data, there exists a unique mild solution that remains bounded in $L^{\infty }\left(R\right)$. Our approach uses a smooth cutoff function to split the Fourier integral into high- and low-frequency parts and incorporates a scaling factor to ensure convergence of the time integrals. New estimates for some kernels are introduced. We also show that, as a key parameter tends to zero, the solution of the modified wave equation converges to the solution of the classical wave equation. To the best of our knowledge, this is the first study of the modified wave equation (Love equation) in the $L^{\infty }$ setting, addressing a significant problem in the analysis of wave equations on unbounded domains.