Existence of solutions to the semilinear damped wave equation with non-$$L^2$$ slowly decaying data: polynomial nonlinearity case
Abstract
We study the Cauchy problem of the semilinear damped wave equation with polynomial nonlinearity, and establish the local and global existence of the solution for slowly decaying initial data. By employing $$L^{p}$$ – $$L^{q}$$ estimates for the linear problem and a fractional Leibniz rule in suitable homogeneous Besov spaces, we show the existence of the solution for initial data that may not belong to $$L^{2}$$ at the spatial infinity in general. The main novelty of our result is to construct the solution for initial data having the decay like $$L^r$$ with $$r>2$$ . A crucial point in our argument is to control the derivative loss from the high frequency part by appropriately choosing the function space.
Fields of Study & Subjects
Advanced Mathematical Physics Problems
Stability and Controllability of Differential Equations
Mathematical Analysis and Transform Methods
Publication details
- Journal / source
- Journal of Evolution Equations
- Publisher
- Springer Science and Business Media LLC
- Volume
- 26
- Issue
- 1
- Publication date
- 22 February 2026
- Language
- EN