On the radius of spatial analyticity for semilinear symmetric hyperbolic systems

Abstract

Abstract. We study the problem of propagation of analytic regularity for semilinear symmetric hyperbolic systems. We adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius (=width) ε 0 , the same happens for the solution u(t, ·) for a certain radius ε(t), as long as the solution exists. Our focus is on precise lower bounds on the spatial radius of analyticity ε(t) as t grows.We also get similar results for the Schrödinger equation with a real-analytic electromagnetic potential.

On the radius of spatial analyticity for the 1d Dirac–Klein–Gordon equations

“…This type of question was introduced in an abstract setting of nonlinear evolutionary PDE by Kato and Masuda [12], who showed in particular that for the Korteweg-de Vries equation (KdV) the radius of analyticity σ(t) can decay to zero at most at a super-exponential rate. A similar rate of decay for semilinear symmetric hyperbolic systems has been proved recently by Cappiello, D'Ancona and Nicola [7]. An algebraic rate of decay for KdV was shown by Bona and Kalisch [3].…”

On the radius of spatial analyticity for cubic nonlinear Schrödinger equations

“…For large times on the other hand we use the idea introduced in [17] (see also [16]) to show that σ(t ) can decay no faster than 1/|t | as |t | → ∞. For studies on related issues for nonlinear partial differential equations see for instance [2,3,10,11,12,13,15].…”

Some remarks on the radius of spatial analyticity for the Euler equations

“…also [5,6,8,9,12]). In our recent paper [7] we developed a method for the estimate of the radius of analyticity for semilinear symmetrizable hyperbolic systems based on inductive estimates in standard Sobolev spaces. The purpose of this note is to adapt this method to the Euler equations on T d and to prove that…”