Analyticity of Semigroups generated by Degenerate Mixed Differential Operators

Adel Saddi
DOI: 10.4236/apm.2011.13010                

Abstract

In this paper we are interested in studying the dissipativity of degenerate mixed differential operators involving an interface point. We show that, under particular interface conditions, such operators generate analytic semigroups on an appropriate Hilbert space . To illustrate the results an example is discussed.

Keywords

Adjoint, Interface, Dissipative Operators, Analytic Semigroups

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	K. Ito and F. Kappel, “Evolution Equations and Approxima-tions,” Series on Advances in Mathematics for Applied Sci-ences, World Scientific Publishing Company, River Edge, Vol. 61, 2002.
[2]	C. A. Boyes, “Acoustic Waveguides,” Applica-tion to Oceanic Sciences, Wiley, New York, 1984.
[3]	A. Pazy, “Semigroups of Linear Operators an Applications to Partial Differential Equations,” Applied Math Sciences, Springer, New York, Vol. 44, 1983.
[4]	N. H. Mahmoud, “Partial Differential Equations with Matricial Coefficients and Generalized Translation Operators,” Transactions of the American Mathematical Society, Vol. 352, No. 8, 2000, pp. 3687-3706.
[5]	N. H. Mahmoud, “Heat Equations Associated with Matrix Singular Differential Operators and Spectral The-ory,” Integral Transforms and Special Functions, Vol. 15, No. 3, 2004. pp. 251-266. doi:10.1080/10652460310001600591
[6]	J. Weidmann, “Spectral Theory of Ordinary Differential Operators,” Lecture Notes in Mathematics, Springer, Berlin, Vol. 1258, 1987.
[7]	K. J. Engel and R. Nagel, “One-Parameter Semi-groups for Linear Evolution Equations,” Springer-Verlag, New York, 2000.
[8]	R. Nagel, “One-Parameter Semigroups of Positive Operators,” Lecture Notes in Mathematics, Springer- Verlag, Berlin, Vol. 1184, 1986.
[9]	A. Saddi and O. A. Mahmoud Sid Ahmed, “Analyticity of Semigroups Generated by a Class of Differential Operators with Matrix Coefficients and Interface,” Semigroup Forum, Vol. 71, No. 1, 2005, pp. 1-17. doi:10.1007/s00233-004-0173-6
[10]	T. G. Bhaskar and R. Kumar, “Analyticity of Semigroup Generated by a Class of Differential Operators with Interface,” Nonlinear Analysis, Vol. 39, No. 6, 2000, pp. 779-791. doi:10.1016/S0362-546X(98)00237-5
[11]	H. O. Fattorini, “The Cauchy Problem,” Addison Wesley, Massachusetts, Vol. 18, 1983.
[12]	H. Chebli, “Analyse Hilbertienne,” Centre de Publication Universitaire, Tunis, 2001.
[13]	T. Kato, “Pertur-bation Theory for Linear Operators,” Springer-Verlag, Berlin, 1966.