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March  2013, 2(1): 101-117. doi: 10.3934/eect.2013.2.101

Analyticity and regularity for a class of second order evolution equations

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, France

2. 

Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo, 169-8555, Japan

Received  November 2012 Revised  December 2012 Published  January 2013

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 $u_0\in C_0(\Omega)\cap C^1(\overline {\Omega})$.
Citation: Alain Haraux, Mitsuharu Ôtani. Analyticity and regularity for a class of second order evolution equations. Evolution Equations & Control Theory, 2013, 2 (1) : 101-117. doi: 10.3934/eect.2013.2.101
References:
[1]

T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,", Oxford Lecture Series in Mathematics and its Applications, 13 (1998). Google Scholar

[2]

G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping,, Quart. Appl. Math., 39 (): 433. Google Scholar

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M. Ôtani, $L^\infty$-energy method and its applications,, in, 20 (2004), 505. Google Scholar

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M. Ôtani, $L^\infty$-energy method, basic tools and usage,, in, 75 (2007), 357. doi: 10.1007/978-3-7643-8482-1_27. Google Scholar

[9]

A. Pazy, "Semi-Groups of Linear Operators and Applications to PDE,", Applied Mathematical Science 44, (1983). Google Scholar

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H. B. Stewart, Generation of analytic semi-groups by strongly elliptic operators,, Trans.A.M.S., 199 (1974), 141. Google Scholar

show all references

References:
[1]

T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,", Oxford Lecture Series in Mathematics and its Applications, 13 (1998). Google Scholar

[2]

G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping,, Quart. Appl. Math., 39 (): 433. Google Scholar

[3]

S. Chen and R. Triggiani, Proof of extension of two conjectures on structural damping for elastic systems: The case $\frac{1}{2} \le\alpha\le 1$,, Pacific. J. Math., 136 (1989), 15. Google Scholar

[4]

S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems. The case $\alpha \in (0, \frac{1}{2})$,, Proc. American Math. Soc., 110 (1989), 401. doi: 10.2307/2048084. Google Scholar

[5]

U. Frisch, "Wave Propagation in Random Media, Probabilistic Methods in Applied Mathematics,", I, I (1968), 75. Google Scholar

[6]

K. Masuda, Manuscript for seminar at Kyoto University,, 1970., (). Google Scholar

[7]

M. Ôtani, $L^\infty$-energy method and its applications,, in, 20 (2004), 505. Google Scholar

[8]

M. Ôtani, $L^\infty$-energy method, basic tools and usage,, in, 75 (2007), 357. doi: 10.1007/978-3-7643-8482-1_27. Google Scholar

[9]

A. Pazy, "Semi-Groups of Linear Operators and Applications to PDE,", Applied Mathematical Science 44, (1983). Google Scholar

[10]

H. B. Stewart, Generation of analytic semi-groups by strongly elliptic operators,, Trans.A.M.S., 199 (1974), 141. Google Scholar

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