Well-posedness of inhomogeneous nonlinear wave equations in ${\mathbb{R}}^3$

Abstract

This paper is devoted to the well-posedness of the inhomogeneous nonlinear wave equations. By combining Strichartz estimates with the contraction mapping principle, we establish local and global well-posedness in the function spaces ${\dot{H}}^1({\mathbb{R}}^3)\times L^2({\mathbb{R}}^3)$ and ${\dot{H}}^{s+1}({\mathbb{R}}^3)\times {\dot{H}}^s({\mathbb{R}}^3)$. The analysis is carried out in the energy-subcritical regime. As a consequence, our results extend and improve upon previous results in the literature for general nonlinear wave equations.