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

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.