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August  2022, 21(8): 2739-2773. doi: 10.3934/cpaa.2022071

On spectral and fractional powers of damped wave equations

1. 

Departamento de Matemática, Universidade Federal de São Carlos, 13565-905 São Carlos SP, Brazil

2. 

Universidade Federal da Paraíba, Departamento de Matemática, 58051-900 João Pessoa PB, Brazil

* Corresponding author

Received  November 2021 Published  August 2022 Early access  April 2022

Fund Project: The first author is partially supported by FAPESP grant # 2017/09406-0 and # 2017/17502-0, Brazil.
The second author is supported by CNPq/Finance Code # 303039/2021-3, Brazil.
The third author is partially supported by FAPESP grant # 2019/26841-8, Brazil

In this paper we explore the theory of fractional powers of positive operators, more precisely, we use the Balakrishnan formula to obtain parabolic approximations of (damped) wave equations in bounded smooth domains in $ \mathbb{R}^N $. We also explicitly calculate the fractional powers of wave operators in terms of the fractional Laplacian in bounded smooth domains and characterize the spectrum of these operators.

Citation: Maykel Belluzi, Flank D. M. Bezerra, Marcelo J. D. Nascimento. On spectral and fractional powers of damped wave equations. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2739-2773. doi: 10.3934/cpaa.2022071
References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems, Volume Ⅰ: Abstract Linear Theory, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.

[2]

A. V. Balakrishnan, Fractional powers of closed operators and the semigroup generated by them, Pacific J. Math., 10 (1960), 419-437. 

[3]

F. D. M. BezerraA. N. CarvalhoJ. W. Cholewa and M. J. D. Nascimento, Parabolic approximation of damped wave equations via fractional powers: fast growing nonlinearities and continuity of the dynamics, J. Math. Anal. Appl., 450 (2017), 377-405.  doi: 10.1016/j.jmaa.2017.01.024.

[4]

F. D. M. BezerraA. N. Carvalho and M. J. D. Nascimento, Fractional approximations of abstract semilinear parabolic problems, Discrete Contin. Dyn. Syst. B, 25 (2020), 4221-4255.  doi: 10.3934/dcdsb.2020095.

[5]

F. D. M. Bezerra and L. A. Santos, Fractional powers approach of operators for abstract evolution equations of third order in time, J. Differ. Equ., 269 (2020), 5661-5679.  doi: 10.1016/j.jde.2020.04.020.

[6]

A. Bonito and J. E. Pasciak, Numerical approximation of fractional powers of regularly accretive operators, IMA J. Numer. Anal., 37 (2017), 124-1273.  doi: 10.1093/imanum/drw067.

[7]

F. E. Brower, On the spectral theory of elliptic differential operators, I. Math. Ann., 142 (1961), 22-130.  doi: 10.1007/BF01343363.

[8]

C. M. Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, 187, North-Holland Publishing Co., Amsterdam, 2001.

[9]

A. N. Carvalho and J. W. Cholewa, Local well posedness for strongly damped wave equations with critical nonlinearities, Bull. Aust. Math. Soc., 66 (2002), 443-463.  doi: 10.1017/S0004972700040296.

[10]

S. Chen and R. Triggiani, Proof of extension of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55. 

[11]

S. Chen and R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Differ. Equ., 88 (1990), 279-293.  doi: 10.1016/0022-0396(90)90100-4.

[12]

K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80. 

[13]

D. Fujiwara, Concrete characterizations of domains of fractional powers of some elliptic differential power of some elliptic differential operators of the second order, Proc. Acad. Japan, 43 (1967), 82-86.  doi: 10.1007/BF00967108.

[14]

P. R. Halmos, A Hilbert Space Problem Book, Second edition, Springer-Verlag, New York-Berlin, 1982.

[15]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, 1981.

[16]

T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad., 36 (1960), 94-96.  doi: 10.3792/pja/1195524082.

[17]

T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Anal. Math., 6 (1958), 261-322.  doi: 10.1007/BF02790238.

[18]

I. Lasiecka and R. Triggiani, Domain of Fractional Powers of Matrix-valued Operators: A general Approach, Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, Birkhauser, 250 (2015), 297-311.  doi: 10.1007/978-3-319-18494-4_20.

[19]

Q. Lü and E. Zuazua, On the lack of controllability of fractional in time ODE and PDE, Math. Control Signals Syst., 28 (2016), 21 pp. doi: 10.1007/s00498-016-0162-9.

[20]

S. Micu and E. Zuazua, On the Controllability of a Fractional Order Parabolic Equation, SIAM J. Control Optim., 44 (2006), 1950-1972.  doi: 10.1137/S036301290444263X.

[21]

R. Nagel, Towards a "matrix theory" for unbounded operator matrices, Math. Z., 201 (1989), 57-68.  doi: 10.1007/BF01161994.

[22]

M. J. D. Nascimento and F. D. M. Bezerra, Non-autonomous approximations governed by the fractional powers of damped wave operators, Electron. J. Differ. Equ., 2019 (2019), 1-19. 

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[24]

M. Schechter, On the Essential Spectrum of an Arbitrary Operator I, J. Math. Anal. Appl., 13 (1966), 205-215. 

show all references

References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems, Volume Ⅰ: Abstract Linear Theory, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.

[2]

A. V. Balakrishnan, Fractional powers of closed operators and the semigroup generated by them, Pacific J. Math., 10 (1960), 419-437. 

[3]

F. D. M. BezerraA. N. CarvalhoJ. W. Cholewa and M. J. D. Nascimento, Parabolic approximation of damped wave equations via fractional powers: fast growing nonlinearities and continuity of the dynamics, J. Math. Anal. Appl., 450 (2017), 377-405.  doi: 10.1016/j.jmaa.2017.01.024.

[4]

F. D. M. BezerraA. N. Carvalho and M. J. D. Nascimento, Fractional approximations of abstract semilinear parabolic problems, Discrete Contin. Dyn. Syst. B, 25 (2020), 4221-4255.  doi: 10.3934/dcdsb.2020095.

[5]

F. D. M. Bezerra and L. A. Santos, Fractional powers approach of operators for abstract evolution equations of third order in time, J. Differ. Equ., 269 (2020), 5661-5679.  doi: 10.1016/j.jde.2020.04.020.

[6]

A. Bonito and J. E. Pasciak, Numerical approximation of fractional powers of regularly accretive operators, IMA J. Numer. Anal., 37 (2017), 124-1273.  doi: 10.1093/imanum/drw067.

[7]

F. E. Brower, On the spectral theory of elliptic differential operators, I. Math. Ann., 142 (1961), 22-130.  doi: 10.1007/BF01343363.

[8]

C. M. Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, 187, North-Holland Publishing Co., Amsterdam, 2001.

[9]

A. N. Carvalho and J. W. Cholewa, Local well posedness for strongly damped wave equations with critical nonlinearities, Bull. Aust. Math. Soc., 66 (2002), 443-463.  doi: 10.1017/S0004972700040296.

[10]

S. Chen and R. Triggiani, Proof of extension of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55. 

[11]

S. Chen and R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Differ. Equ., 88 (1990), 279-293.  doi: 10.1016/0022-0396(90)90100-4.

[12]

K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80. 

[13]

D. Fujiwara, Concrete characterizations of domains of fractional powers of some elliptic differential power of some elliptic differential operators of the second order, Proc. Acad. Japan, 43 (1967), 82-86.  doi: 10.1007/BF00967108.

[14]

P. R. Halmos, A Hilbert Space Problem Book, Second edition, Springer-Verlag, New York-Berlin, 1982.

[15]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, 1981.

[16]

T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad., 36 (1960), 94-96.  doi: 10.3792/pja/1195524082.

[17]

T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Anal. Math., 6 (1958), 261-322.  doi: 10.1007/BF02790238.

[18]

I. Lasiecka and R. Triggiani, Domain of Fractional Powers of Matrix-valued Operators: A general Approach, Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, Birkhauser, 250 (2015), 297-311.  doi: 10.1007/978-3-319-18494-4_20.

[19]

Q. Lü and E. Zuazua, On the lack of controllability of fractional in time ODE and PDE, Math. Control Signals Syst., 28 (2016), 21 pp. doi: 10.1007/s00498-016-0162-9.

[20]

S. Micu and E. Zuazua, On the Controllability of a Fractional Order Parabolic Equation, SIAM J. Control Optim., 44 (2006), 1950-1972.  doi: 10.1137/S036301290444263X.

[21]

R. Nagel, Towards a "matrix theory" for unbounded operator matrices, Math. Z., 201 (1989), 57-68.  doi: 10.1007/BF01161994.

[22]

M. J. D. Nascimento and F. D. M. Bezerra, Non-autonomous approximations governed by the fractional powers of damped wave operators, Electron. J. Differ. Equ., 2019 (2019), 1-19. 

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[24]

M. Schechter, On the Essential Spectrum of an Arbitrary Operator I, J. Math. Anal. Appl., 13 (1966), 205-215. 

Figure 1.  Location of the eigenvalues of $ -A_1 $ and $ -A_2 $
Figure 2.  Location of the eigenvalues of $ -A_1 $ and $ -A_2 $
Figure 3.  Spectrum of $ \varLambda_{(a,0)}^{\alpha} $
Figure 4.  Location of the eigenvalues of $ -\varLambda_{(0,0)}^{\alpha} $
Figure 5.  Location of the eigenvalues of $ -\varLambda_{(1,\frac12)}^\alpha $
Figure 6.  $ \sigma (-\varLambda_{(a,1)}) $: Localization of $ \lambda_n^{\pm} $
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