Evolution of the radius of analyticity for mKdV-type equations

In this paper, we obtain new lower bounds for the evolution of the radius of analyticity of solutions to two initial value problems (IVPs) with initial data belonging to the class of analytic functions upper H sigma comma s left parenthesis double struck upper R right parenthesis Hσ,s(R) H σ , s ( ℝ ) defined via a hyperbolic cosine weight. First, we consider the IVP for the modified Korteweg–de Vries (mKdV) equation. For this problem, we prove that the evolution of the radius of analyticity sigma left parenthesis upper T right parenthesis σ(T) σ ( T ) of the solution admits an algebraic lower bound cT minus 12 cT- 12 c T − 1 2 for some c greater than 0 c > 0 c > 0 and given arbitrarily large upper T greater than 0 T > 0 T > 0 . Next, we analyse the IVP for the mKdV equation with generalized dispersion (mKdVm) and a damping term. For this problem, we guarantee the local well-posedness in upper H sigma comma s left parenthesis double struck upper R right parenthesis Hσ,s(R) H σ , s ( ℝ ) and demonstrate that the local solution can be extended globally in time and admits constant lower bounds for the radius of analyticity sigma left parenthesis t right parenthesis