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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

 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

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.
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 and Continuous Dynamical Systems, 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, Theory, Methods and Applications, 71 (2009), 1709-1727. 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), 545-572. [3] O . Diekmann, S. A. Van Gils, S. M. Verduyn Lunel and H. O. walther, "Delay Equations, Functional, Complex and Nonlinear Analysis," 110, Springer-Verlag, New York, 1995. [4] K. J. Engel and R. Nagel, "One-Parameter Semigroups of Linear Evolution Equations," 194, Springer-Verlag, Berlin, 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, Theory, Methods and Applications, 64 (2006), 1690-1709. 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, Theory, Methods and Applications, 70 (2009), 4008-4020. 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), 11-41. [8] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," 1473, Springer-Verlag, Berlin, 1991. [9] T. Naito, J. S. Shin and S. Murakami, On solution semigroups of general functional differential equations, Nonlinear Analysis, 30 (1997), 4565-4576. 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), 323-337. [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), 29-38. [12] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," 44, Springer-Verlag, New York, 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), 397-409. 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), 129-143. doi: 10.2307/1998809. [15] J. Wu, "Theory and Applications of Partial Functional Differential Equations," 119, Springer-Verlag, Berlin, 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, Theory, Methods and Applications, 71 (2009), 1709-1727. 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), 545-572. [3] O . Diekmann, S. A. Van Gils, S. M. Verduyn Lunel and H. O. walther, "Delay Equations, Functional, Complex and Nonlinear Analysis," 110, Springer-Verlag, New York, 1995. [4] K. J. Engel and R. Nagel, "One-Parameter Semigroups of Linear Evolution Equations," 194, Springer-Verlag, Berlin, 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, Theory, Methods and Applications, 64 (2006), 1690-1709. 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, Theory, Methods and Applications, 70 (2009), 4008-4020. 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), 11-41. [8] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," 1473, Springer-Verlag, Berlin, 1991. [9] T. Naito, J. S. Shin and S. Murakami, On solution semigroups of general functional differential equations, Nonlinear Analysis, 30 (1997), 4565-4576. 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), 323-337. [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), 29-38. [12] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," 44, Springer-Verlag, New York, 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), 397-409. 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), 129-143. doi: 10.2307/1998809. [15] J. Wu, "Theory and Applications of Partial Functional Differential Equations," 119, Springer-Verlag, Berlin, 1996.
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