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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

[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), 15-55.  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), 401-405. doi: 10.2307/2048084.  Google Scholar

[5]

U. Frisch, "Wave Propagation in Random Media, Probabilistic Methods in Applied Mathematics," I, 75-198, Academic press, New-York 1968.  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 "Nonlinear Partial Differential Equations and Their Applications" (Ed. by N. Kenmochi, M. Ôtani and S. Zheng ), GAKUTO Internat. Ser. Math. Sci. Appl., 20, Gakkotosho, Tokyo, (2004), 505-516.  Google Scholar

[8]

M. Ôtani, $L^\infty$-energy method, basic tools and usage, in "Differential Equations, Chaos and Variational Problems: Progress in Nonlinear Differential Equations and Their Applications" (Ed. by Vasile Staicu), 75, Birkhauser (2007), 357-376. 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, Springer 1983. Google Scholar

[10]

H. B. Stewart, Generation of analytic semi-groups by strongly elliptic operators, Trans.A.M.S., 199 (1974), 141-162.  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), 15-55.  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), 401-405. doi: 10.2307/2048084.  Google Scholar

[5]

U. Frisch, "Wave Propagation in Random Media, Probabilistic Methods in Applied Mathematics," I, 75-198, Academic press, New-York 1968.  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 "Nonlinear Partial Differential Equations and Their Applications" (Ed. by N. Kenmochi, M. Ôtani and S. Zheng ), GAKUTO Internat. Ser. Math. Sci. Appl., 20, Gakkotosho, Tokyo, (2004), 505-516.  Google Scholar

[8]

M. Ôtani, $L^\infty$-energy method, basic tools and usage, in "Differential Equations, Chaos and Variational Problems: Progress in Nonlinear Differential Equations and Their Applications" (Ed. by Vasile Staicu), 75, Birkhauser (2007), 357-376. 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, Springer 1983. Google Scholar

[10]

H. B. Stewart, Generation of analytic semi-groups by strongly elliptic operators, Trans.A.M.S., 199 (1974), 141-162.  Google Scholar

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