Analyticity of Semigroups Generated by Singular Differential Matrix Operators

ABSTRACT

In this paper we prove the analyticity of the semigroups generated by some singular differential matrix operators of the form in the Banach space with suitable boundary conditions. To illustrate the work an example is discussed.

In this paper we prove the analyticity of the semigroups generated by some singular differential matrix operators of the form in the Banach space with suitable boundary conditions. To illustrate the work an example is discussed.

KEYWORDS

Dissipative Operators, Positive Operators, Spectrum, Analytic Semigroups, Evolution Equations

Dissipative Operators, Positive Operators, Spectrum, Analytic Semigroups, Evolution Equations

Cite this paper

nullO. Ahmed and A. Saddi, "Analyticity of Semigroups Generated by Singular Differential Matrix Operators,"*Applied Mathematics*, Vol. 1 No. 4, 2010, pp. 283-287. doi: 10.4236/am.2010.14036.

nullO. Ahmed and A. Saddi, "Analyticity of Semigroups Generated by Singular Differential Matrix Operators,"

References

[1] R. Nagel, “One-Parameter Semigroups of Positive Operators,” Lecture Notes in Math, Springer-Verlag, 1986.

[2] K. J. Engel and R. Nagel, “One-Parameter Semigroups for Linear Evolution Equations,” Springer-Verlag, 2000.

[3] A. Pazy, “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Applied Math Sciences 44, Springer, 1983.

[4] K. Ito and F. Kappel, “Evolution Equations and Approximations,” Series on Advances in Mathematics for Applied Sciences, Vol. 61, 2002.

[5] E. M. Ouhabaz, “Analysis of Heat Equations on Domains,” Princeton University Press, New Jersey, 2005.

[6] K. J. Engel, “Operator Matrices and Systems of Evolution Equations.” (Preprint).

[7] N. H. Mahmoud, “Partial Differential Equations with Matricial Coefficients and Generalised Translation Operators,” Transactions of the Americain Mathematical Society, Vol. 352, No. 8, 2000, pp. 3687-3706.

[8] N. H. Mahmoud, “Heat Equations Associated with Matrix Singular Differential Operators and Spectral Theory,” Integral Transforms and Special Functions, Vol. 15, No. 3, 2004, pp. 251-266.

[9] A. Saddi and O. A. M. S. 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.

[10] G. Metafune, “Analyticity for Some Degenerate One-Dimentional Evolution Equations,” Studia Mathematica, Vol. 127, No. 3, 1998, pp. 251-276.

[11] Z. S. Agranovich and V. A. Marchenko, “The Inverse Problem of Scattering Theory,” Kharkov State University, Gordon and Breach, NewYork and London, 1963.

[12] R. B. Bapat and T. E. S. Raghavan, “Nonegative Matrices and Applications,” Cambridge University Press, Cambridge, 1997.

[13] R. Dautray and J.-L. Lions, “Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques,” Tome 3, Série Scientifique, Masson, 1985.

[1] R. Nagel, “One-Parameter Semigroups of Positive Operators,” Lecture Notes in Math, Springer-Verlag, 1986.

[2] K. J. Engel and R. Nagel, “One-Parameter Semigroups for Linear Evolution Equations,” Springer-Verlag, 2000.

[3] A. Pazy, “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Applied Math Sciences 44, Springer, 1983.

[4] K. Ito and F. Kappel, “Evolution Equations and Approximations,” Series on Advances in Mathematics for Applied Sciences, Vol. 61, 2002.

[5] E. M. Ouhabaz, “Analysis of Heat Equations on Domains,” Princeton University Press, New Jersey, 2005.

[6] K. J. Engel, “Operator Matrices and Systems of Evolution Equations.” (Preprint).

[7] N. H. Mahmoud, “Partial Differential Equations with Matricial Coefficients and Generalised Translation Operators,” Transactions of the Americain Mathematical Society, Vol. 352, No. 8, 2000, pp. 3687-3706.

[8] N. H. Mahmoud, “Heat Equations Associated with Matrix Singular Differential Operators and Spectral Theory,” Integral Transforms and Special Functions, Vol. 15, No. 3, 2004, pp. 251-266.

[9] A. Saddi and O. A. M. S. 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.

[10] G. Metafune, “Analyticity for Some Degenerate One-Dimentional Evolution Equations,” Studia Mathematica, Vol. 127, No. 3, 1998, pp. 251-276.

[11] Z. S. Agranovich and V. A. Marchenko, “The Inverse Problem of Scattering Theory,” Kharkov State University, Gordon and Breach, NewYork and London, 1963.

[12] R. B. Bapat and T. E. S. Raghavan, “Nonegative Matrices and Applications,” Cambridge University Press, Cambridge, 1997.

[13] R. Dautray and J.-L. Lions, “Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques,” Tome 3, Série Scientifique, Masson, 1985.