Analyticity and regularity for a class of second orderevolution equations

Alain Haraux; Mitsuharu Ôtani

doi:10.3934/eect.2013.2.101

Article Contents

2013, Volume 2, Issue 1: 101-117. Doi: 10.3934/eect.2013.2.101

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Analyticity and regularity for a class of second order evolution equations

Alain Haraux^1, and
Mitsuharu Ôtani^2,

1.
Laboratoire Jacques-Louis Lions, U.M.R C.N.R.S. 7598, Université Pierre et Marie Curie, Boite courrier 187, 75252 Paris Cedex 05
2.
Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo, 169-8555

Received: October 31, 2012

Revised: November 30, 2012

Published: February 2013

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Abstract

The regularity conservation as well as the smoothing effect are studied for the equation $ u''+ Au+ cA^\alpha u' = 0$, where $A$ is a positive selfadjoint operator on a real Hilbert space $H$ and $\alpha\in (0, 1]; \,\, c >0$. When $\alpha\ge {1\over 2}$ the equation generates an analytic semigroup on $D(A^{1/2})\times H $ , and if $\alpha\in (0, {1\over 2})$ a weaker optimal smoothing property is established. Some conservation properties in other norms are also established, as a typical example, the strongly dissipative wave equation $u_{tt} - \Delta u -c\Delta u_t = 0$ with Dirichlet boundary conditions in a bounded domain is given, for which the space $C_0(\Omega)\times C_0(\Omega)$ is conserved for $t>0$, which presents a sharp contrast with the conservative case $u_{tt} - \Delta u = 0$ for which $C_0(\Omega)$-regularity can be lost even starting from an initial state $(u_0, 0)$ with $u_0\in C_0(\Omega)\cap C^1(\overline {\Omega})$.

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Mathematics Subject Classification: 35B65, 35D99, 35L05, 35L90.

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