Optimal regularity and stability analysis in the \alpha-Norm for a class of partial functional differential equations with infinite delay

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April 2011, 30(1): 115-135. doi: 10.3934/dcds.2011.30.115

Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay

Abdelhai Elazzouzi ^1, and Aziz Ouhinou ^2,

1.	Université Cadi Ayyad, Faculté des Sciences Semlalia, Département de Mathématiques, B.P.2390 Marrakech, Morocco
2.	African Institute for Mathematical Sciences (AIMS), 6 Melrose Road, Muizenberg 7945, South Africa

Received January 2010 Revised July 2010 Published February 2011

Abstract
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This work aims to investigate the regularity and the stability of the solutions for a class of partial functional differential equations with infinite delay. Here we suppose that the undelayed part generates an analytic semigroup and the delayed part is continuous with respect to fractional powers of the generator. First, we give a new characterization for the infinitesimal generator of the solution semigroup, which allows us to give necessary and sufficient conditions for the regularity of solutions. Second, we investigate the stability of the semigroup solution. We proved that one of the fundamental and wildly used assumption, in the computing of eigenvalues and eigenvectors, is an immediate consequence of the already considered ones. Finally, we discuss the asymptotic behavior of solutions.

Keywords: Analytic semigroup, infinite delay, essential spectrum, mild and strict solution, fractional power of operators, stability., uniform fading memory space.

Mathematics Subject Classification: Primary: 34K06,34K20; Secondary: 35B3.

Citation: Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115

References:

[1]	M. Adimy, A. Elazzouzi and K. Ezzinbi, Reduction principe and dynamic behaviors for a class of partial functional differential equations,, Nonlinear Analysis, 71 (2009), 1709. doi: 10.1016/j.na.2009.01.008.
[2]	R. Benkhalti and K. Ezzinbi, Existence and stability in the $\alpha$-norm for some partial functinal differential equations with infinite delay,, Differential and Integral Equations, 19 (2006), 545.
[3]	O . Diekmann, S. A. Van Gils, S. M. Verduyn Lunel and H. O. walther, "Delay Equations, Functional, Complex and Nonlinear Analysis,", \textbf{110}, 110 (1995).
[4]	K. J. Engel and R. Nagel, "One-Parameter Semigroups of Linear Evolution Equations,", \textbf{194}, 194 (2000).
[5]	K. Ezzinbi and A. Ouhinou, Necessary and sufficient conditions for the regularity and stability for some partial functional differential equations with infinite delay,, Nonlienar Analysis, 64 (2006), 1690. doi: 10.1016/j.na.2005.07.017.
[6]	K. Ezzinbi and A. Ouhinou, Stability and asymptotic behavior of solutions for some linear partial functional differential equations in critical cases,, Nonlienar Analysis, 70 (2009), 4008. doi: 10.1016/j.na.2008.08.010.
[7]	J. Hale and J. Kato, Phase space for retarded equations with unbounded delay,, Funkcial Ekvac, 21 (1978), 11.
[8]	Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", \textbf{1473}, 1473 (1991).
[9]	T. Naito, J. S. Shin and S. Murakami, On solution semigroups of general functional differential equations,, Nonlinear Analysis, 30 (1997), 4565. doi: 10.1016/S0362-546X(97)00315-5.
[10]	T. Naito, J. S. Shin and S. Murakami, On stability of solutions in linear autonomous functional differential equations,, Funkcialaj Ekvacioj, 43 (2000), 323.
[11]	T. Naito, J. S. Shin and S. Murakami, The generator of the solution semigroup for the general linear functional differential equation,, Bull. Univ.Electro-Communications, 11 (1998), 29.
[12]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", \textbf{44}, 44 (1983).
[13]	C. C. Travis and G. F. Webb, Partial differential equations with deviating arguments in the time variable,, Journal of Mathematical Analysis and Applications, 56 (1976), 397. doi: 10.1016/0022-247X(76)90052-4.
[14]	C. C. Travis and G. F. Webb, Existence, stability and compactness in the $\alpha-$norm for partial functional differential equations,, Transactions of the American Mathematical Society, 240 (1978), 129. doi: 10.2307/1998809.
[15]	J. Wu, "Theory and Applications of Partial Functional Differential Equations,", \textbf{119}, 119 (1996).

show all references

References:

[1]	M. Adimy, A. Elazzouzi and K. Ezzinbi, Reduction principe and dynamic behaviors for a class of partial functional differential equations,, Nonlinear Analysis, 71 (2009), 1709. doi: 10.1016/j.na.2009.01.008.
[2]	R. Benkhalti and K. Ezzinbi, Existence and stability in the $\alpha$-norm for some partial functinal differential equations with infinite delay,, Differential and Integral Equations, 19 (2006), 545.
[3]	O . Diekmann, S. A. Van Gils, S. M. Verduyn Lunel and H. O. walther, "Delay Equations, Functional, Complex and Nonlinear Analysis,", \textbf{110}, 110 (1995).
[4]	K. J. Engel and R. Nagel, "One-Parameter Semigroups of Linear Evolution Equations,", \textbf{194}, 194 (2000).
[5]	K. Ezzinbi and A. Ouhinou, Necessary and sufficient conditions for the regularity and stability for some partial functional differential equations with infinite delay,, Nonlienar Analysis, 64 (2006), 1690. doi: 10.1016/j.na.2005.07.017.
[6]	K. Ezzinbi and A. Ouhinou, Stability and asymptotic behavior of solutions for some linear partial functional differential equations in critical cases,, Nonlienar Analysis, 70 (2009), 4008. doi: 10.1016/j.na.2008.08.010.
[7]	J. Hale and J. Kato, Phase space for retarded equations with unbounded delay,, Funkcial Ekvac, 21 (1978), 11.
[8]	Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", \textbf{1473}, 1473 (1991).
[9]	T. Naito, J. S. Shin and S. Murakami, On solution semigroups of general functional differential equations,, Nonlinear Analysis, 30 (1997), 4565. doi: 10.1016/S0362-546X(97)00315-5.
[10]	T. Naito, J. S. Shin and S. Murakami, On stability of solutions in linear autonomous functional differential equations,, Funkcialaj Ekvacioj, 43 (2000), 323.
[11]	T. Naito, J. S. Shin and S. Murakami, The generator of the solution semigroup for the general linear functional differential equation,, Bull. Univ.Electro-Communications, 11 (1998), 29.
[12]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", \textbf{44}, 44 (1983).
[13]	C. C. Travis and G. F. Webb, Partial differential equations with deviating arguments in the time variable,, Journal of Mathematical Analysis and Applications, 56 (1976), 397. doi: 10.1016/0022-247X(76)90052-4.
[14]	C. C. Travis and G. F. Webb, Existence, stability and compactness in the $\alpha-$norm for partial functional differential equations,, Transactions of the American Mathematical Society, 240 (1978), 129. doi: 10.2307/1998809.
[15]	J. Wu, "Theory and Applications of Partial Functional Differential Equations,", \textbf{119}, 119 (1996).

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