Nonsmooth data error estimates for exponential Runge-Kutta methods and applications to split exponential integrators

We derive rigorous error bounds for exponential Runge-Kutta discretizations of parabolic equations with nonsmooth initial data. Our analysis is carried out in the framework of abstract semi-linear evolution equations, allowing for operators with non-dense domains. The results provide a foundation for establishing error estimates for nonsmooth data in prototypical problems such as the Allen-Cahn and Burgers equations. Furthermore, we apply these estimates to the analysis of split exponential integrators, yielding convergence results expressed explicitly in terms of the prescribed data.