Analysis of the sine-Gordon equation with a nonlinear $\delta$-potential

This paper is devoted to the analysis of the following nonlinear wave equation $$u_{tt}-u_{xx}+(1+q{\delta }_0(x))\sin u=0,$$ where ${\delta }_0={\delta }_0(x)$ is the Dirac delta function centered at the origin and $q\in \mathbb{R}$ is a constant. Equations of this form arise in the study of propagating solitons in the presence of a localized inhomogeneity. It is proved that the Cauchy problem for this equation is globally well-posed in the energy space $H_{\sin }^{}\times L^2$. A complete characterization of stationary waves in the energy space, based on the parameter $q$, is also provided. Finally, a criterion to determine the stability or instability of the stationary waves, which depends upon the sign of the parameter $q$, is established.