Decay Rate on the Radius of Spatial Analyticity to Solutions for the Modified Camassa–Holm Equation

The initial value problem associated with the modified Camassa–Holm equation for initial data u 0 ( x ) that is analytic on the line and having uniform radius of spatial analyticity σ 0 is considered. We have shown the persistence of the radius of spatial analyticity till some time δ . Thus, for large time t ≥ δ , we proved that the radius of spatial analyticity σ ( t ) of solutions is bounded from below by c | t | −(1/2 β ) for some 0 < β ≤ 1. Our strategy mainly relies on the standard contraction mapping principle, an approximate conservation law in the modified Gevrey space H σ ,1 , linear estimates, Strichartz estimate, interpolation, transference principle, and one‐dimensional Sobolev embedding.