On spatial Gevrey regularity for some strongly dissipative second order evolution equations

TL;DR

This paper proves optimal Gevrey regularity results for solutions of certain strongly dissipative second order evolution equations involving a positive self-adjoint operator, including the damped wave equation, in a Hilbert space setting.

Contribution

It establishes the precise Gevrey regularity class for solutions of a class of dissipative evolution equations, showing optimality of the regularity exponent.

Findings

Solutions belong to specific Gevrey spaces for all positive times.

Optimality of the Gevrey regularity exponent is demonstrated.

Results apply to the damped wave equation with explicit regularity classes.

Abstract

Let A be a positive self-adjoint linear operator acting on a real Hilbert space H and $\alpha$, c be positive constants. We show that all solutions of the evolution equation u + Au + cA $\alpha$ u = 0 with u(0) $\in$ D(A 1 2), u (0) $\in$ H belong for all t > 0 to the Gevrey space G(A, $\sigma$) with $\sigma$ = min{ 1 $\alpha$ , 1 1--$\alpha$ }. This result is optimal in the sense that $\sigma$ can not be reduced in general. For the damped wave equation (SDW) $\alpha$ corresponding to the case where A = --$\Delta$ with domain D(A) = {w $\in$ H 1 0 ($\Omega$), $\Delta$w $\in$ L 2 ($\Omega$)} with $\Omega$ any open subset of R N and (u(0), u (0)) $\in$ H 1 0 ($\Omega$)xL 2 ($\Omega$), the unique solution u of (SDW) $\alpha$ satisfies $\forall$t > 0, u(t) $\in$ G s ($\Omega$) with s = min{ 1 2$\alpha$ , 1 2(1--$\alpha$) }, and this result is also optimal. Mathematics Subject Classification 2010 (MSC2010): 35L10, 35B65, 47A60.