# American Institute of Mathematical Sciences

September  2016, 36(9): 5163-5181. doi: 10.3934/dcds.2016024

## The $C$-regularized semigroup method for partial differential equations with delays

 1 Laboratory of Information & Control Technology, Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China 2 College of Mathematics & Information Science, Henan Normal University, Xinxiang 453007, China 3 State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems & Control, Zhejiang University, Hangzhou, Zhejiang 310027

Received  January 2015 Revised  January 2016 Published  May 2016

This paper is devoted to study the abstract functional differential equation (FDE) of the following form $$\dot{u}(t)=Au(t)+\Phi u_t,$$ where $A$ generates a $C$-regularized semigroup, which is the generalization of $C_0$-semigroup and can be applied to deal with many important differential operators that the $C_0$-semigroup can not be used to. We first show that the $C$-well-posedness of a FDE is equivalent to the $\mathscr{C}$-well-posedness of an abstract Cauchy problem in a product Banach space, where the operator $\mathscr{C}$ is related with the operator $C$ and will be defined in the following text. Then, by making use of a perturbation result of $C$-regularized semigroup, a sufficient condition is provided for the $C$-well-posedness of FDEs. Moreover, an illustrative application to partial differential equation (PDE) with delay is given in the last section.
Citation: Xin Yu, Guojie Zheng, Chao Xu. The $C$-regularized semigroup method for partial differential equations with delays. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5163-5181. doi: 10.3934/dcds.2016024
##### References:
 [1] W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327-352. doi: 10.1007/BF02774144. [2] A. Bátkai and S. Piazzera, Semigroups and linear differential equations with delay, J. Math. Anal. Appl., 264 (2001), 1-20. doi: 10.1006/jmaa.2001.6705. [3] A. Bátkai and S. Piazzera, Semigroups for Delay Equations, A. K. Peters, Wellesley, 2005. [4] P. N. Chen and H. S. Qin, Controllability of linear systems in Banach spaces, Syst. Control Lett., 45 (2002), 155-161. doi: 10.1016/S0167-6911(01)00177-3. [5] E. B. Davies and M. M. H. Pang, The Cauchy problem and a generalization of the Hille-Yosida theorem, Proc. London Math. Soc., 55 (1987), 181-208. doi: 10.1112/plms/s3-55.1.181. [6] R. deLaubenfels and E. Families, Functional Calculi and Evolution Equations, Springer-Verlag, 1994. [7] R. deLaubenfels, Matrices of operators and regularized semigroups, Math. Z., 212 (1993), 619-629. doi: 10.1007/BF02571680. [8] R. deLaubenfels, $C$-semigroups and the Cauchy problem, J. Funct. Anal., 111 (1993), 44-61. doi: 10.1006/jfan.1993.1003. [9] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. [10] B. Z. Guo, J. M. Wang and S. P. Yung, On the $C_0$-semigroup generation and exponential stability resulting from a shear force feedback on a rotating beam, Syst. Control Lett., 54 (2005), 557-574. doi: 10.1016/j.sysconle.2004.10.006. [11] J. K. Hale, Functional Differential Equations, Appl. Math. Sci., Vol. 3, Springer-Verlag, 1971. [12] M. Hieber, Laplace transforms and $\alpha$-times integrated semigroups, Forum Math., 3 (1991), 595-612. doi: 10.1515/form.1991.3.595. [13] M. Hieber, Integrated semigroups and differential operators on $L^p$ spaces, Math. Ann., 291 (1991), 1-16. doi: 10.1007/BF01445187. [14] L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math., 104 (1960), 93-140. doi: 10.1007/BF02547187. [15] F. T. Iha and C. F. Schubert, The spectrum of partial differential operators on $L^p(R^n)$, Trans. Amer. Math. Soc., 152 (1970), 215-226. [16] C. Kaiser, Integrated semigroups and linear partial differential equations with delay, J. Math. Anal. Appl., 292 (2004), 328-339. doi: 10.1016/j.jmaa.2003.10.031. [17] H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180. doi: 10.1016/0022-1236(89)90116-X. [18] C. C. Kuo, On perturbation of $\alpha$-times integrated $C$-semigroups, Taiwanese J. Math., 14 (2010), 1979-1992. [19] Y. S. Lei and Q. Zheng, The application of $C$-semigroups to differential operators in $L^p(R^n)$, J. Math. Anal. Appl., 188 (1994), 809-818. doi: 10.1006/jmaa.1994.1464. [20] Y. S. Lei, W. H. Yi and Q. Zheng, Semigroups of operators and polynomials of generators of bounded strongly continuous groups, Proc. London Math. Soc., 69 (1994), 144-170. doi: 10.1112/plms/s3-69.1.144. [21] Y. S. Lei and Q. Zheng, Exponentially bounded $C$-semigroups and integrated semigroups with nondensely defined generators II: Perturbation (in Chinese), Acta Math. Sci., 13 (1993), 428-434. [22] K. S. Liu and Z. Y. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM J. Control Optim., 36 (1998), 1086-1098. doi: 10.1137/S0363012996310703. [23] I. V. Mel'nikova and A. Filinkov, Abstract Cauchy Problems: Three Approaches, Chapman & Hall, London, 2001. doi: 10.1201/9781420035490. [24] I. V. Mel'nikova and A. Filinkov, Integrated semigroups and $C$-semigroups, well-posedness and regularization of differential-operator problems, Russian Math. Surveys, 49 (1994), 115-155. doi: 10.1070/RM1994v049n06ABEH002449. [25] F. Neubrander, Integrated semigroups and their applications to the abstract Cauchy problem, Pac. J. Math., 135 (1988), 111-155. doi: 10.2140/pjm.1988.135.111. [26] M. Schechter, Spectra of Partial Differential Operators, $2^{nd}$, North Holland, Elsevier, 1986. [27] X. L. Song and J. G. Peng, Lipschitzian semigroups and abstract functional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 2346-2355. doi: 10.1016/j.na.2009.10.035. [28] N. Tanaka, On perturbation theory for exponentially bounded $C$-semigroups, Semigroup Forum, 41 (1990), 215-236. doi: 10.1007/BF02573392. [29] N. Tanaka and I. Miyadera, Exponential bounded $C$-semigroups and intgrated semigroups, Tokyo, J. Math., 12 (1989), 99-115. doi: 10.3836/tjm/1270133551. [30] G. S. Wang and L. J. Wang, The Bang-Bang principle of time optimal controls for the heat equation with internal controls. Syst. Control Lett., 56 (2007), 709-713. doi: 10.1016/j.sysconle.2007.06.001. [31] G. Webb, Functional differential equations and nonlinear semigroups in $L^p$-spaces, J. Diff. Eq., 20 (1976), 71-89. doi: 10.1016/0022-0396(76)90097-8. [32] G. Weiss, Optimal control of systems with a unitary semigroup and with colocated control and observation, Syst. Control Lett., 48 (2003), 329-340. doi: 10.1016/S0167-6911(02)00276-1. [33] J. Wu, Theory and Application of Partial Functional Differential Equations, Appl. Math. Sci., Vol. 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [34] Q. Zheng and M. Li, Regularized Semigroups and Non-Elliptic Differential Operators, Sciense Press, Beijing, 2014.

show all references

##### References:
 [1] W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327-352. doi: 10.1007/BF02774144. [2] A. Bátkai and S. Piazzera, Semigroups and linear differential equations with delay, J. Math. Anal. Appl., 264 (2001), 1-20. doi: 10.1006/jmaa.2001.6705. [3] A. Bátkai and S. Piazzera, Semigroups for Delay Equations, A. K. Peters, Wellesley, 2005. [4] P. N. Chen and H. S. Qin, Controllability of linear systems in Banach spaces, Syst. Control Lett., 45 (2002), 155-161. doi: 10.1016/S0167-6911(01)00177-3. [5] E. B. Davies and M. M. H. Pang, The Cauchy problem and a generalization of the Hille-Yosida theorem, Proc. London Math. Soc., 55 (1987), 181-208. doi: 10.1112/plms/s3-55.1.181. [6] R. deLaubenfels and E. Families, Functional Calculi and Evolution Equations, Springer-Verlag, 1994. [7] R. deLaubenfels, Matrices of operators and regularized semigroups, Math. Z., 212 (1993), 619-629. doi: 10.1007/BF02571680. [8] R. deLaubenfels, $C$-semigroups and the Cauchy problem, J. Funct. Anal., 111 (1993), 44-61. doi: 10.1006/jfan.1993.1003. [9] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. [10] B. Z. Guo, J. M. Wang and S. P. Yung, On the $C_0$-semigroup generation and exponential stability resulting from a shear force feedback on a rotating beam, Syst. Control Lett., 54 (2005), 557-574. doi: 10.1016/j.sysconle.2004.10.006. [11] J. K. Hale, Functional Differential Equations, Appl. Math. Sci., Vol. 3, Springer-Verlag, 1971. [12] M. Hieber, Laplace transforms and $\alpha$-times integrated semigroups, Forum Math., 3 (1991), 595-612. doi: 10.1515/form.1991.3.595. [13] M. Hieber, Integrated semigroups and differential operators on $L^p$ spaces, Math. Ann., 291 (1991), 1-16. doi: 10.1007/BF01445187. [14] L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math., 104 (1960), 93-140. doi: 10.1007/BF02547187. [15] F. T. Iha and C. F. Schubert, The spectrum of partial differential operators on $L^p(R^n)$, Trans. Amer. Math. Soc., 152 (1970), 215-226. [16] C. Kaiser, Integrated semigroups and linear partial differential equations with delay, J. Math. Anal. Appl., 292 (2004), 328-339. doi: 10.1016/j.jmaa.2003.10.031. [17] H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180. doi: 10.1016/0022-1236(89)90116-X. [18] C. C. Kuo, On perturbation of $\alpha$-times integrated $C$-semigroups, Taiwanese J. Math., 14 (2010), 1979-1992. [19] Y. S. Lei and Q. Zheng, The application of $C$-semigroups to differential operators in $L^p(R^n)$, J. Math. Anal. Appl., 188 (1994), 809-818. doi: 10.1006/jmaa.1994.1464. [20] Y. S. Lei, W. H. Yi and Q. Zheng, Semigroups of operators and polynomials of generators of bounded strongly continuous groups, Proc. London Math. Soc., 69 (1994), 144-170. doi: 10.1112/plms/s3-69.1.144. [21] Y. S. Lei and Q. Zheng, Exponentially bounded $C$-semigroups and integrated semigroups with nondensely defined generators II: Perturbation (in Chinese), Acta Math. Sci., 13 (1993), 428-434. [22] K. S. Liu and Z. Y. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM J. Control Optim., 36 (1998), 1086-1098. doi: 10.1137/S0363012996310703. [23] I. V. Mel'nikova and A. Filinkov, Abstract Cauchy Problems: Three Approaches, Chapman & Hall, London, 2001. doi: 10.1201/9781420035490. [24] I. V. Mel'nikova and A. Filinkov, Integrated semigroups and $C$-semigroups, well-posedness and regularization of differential-operator problems, Russian Math. Surveys, 49 (1994), 115-155. doi: 10.1070/RM1994v049n06ABEH002449. [25] F. Neubrander, Integrated semigroups and their applications to the abstract Cauchy problem, Pac. J. Math., 135 (1988), 111-155. doi: 10.2140/pjm.1988.135.111. [26] M. Schechter, Spectra of Partial Differential Operators, $2^{nd}$, North Holland, Elsevier, 1986. [27] X. L. Song and J. G. Peng, Lipschitzian semigroups and abstract functional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 2346-2355. doi: 10.1016/j.na.2009.10.035. [28] N. Tanaka, On perturbation theory for exponentially bounded $C$-semigroups, Semigroup Forum, 41 (1990), 215-236. doi: 10.1007/BF02573392. [29] N. Tanaka and I. Miyadera, Exponential bounded $C$-semigroups and intgrated semigroups, Tokyo, J. Math., 12 (1989), 99-115. doi: 10.3836/tjm/1270133551. [30] G. S. Wang and L. J. Wang, The Bang-Bang principle of time optimal controls for the heat equation with internal controls. Syst. Control Lett., 56 (2007), 709-713. doi: 10.1016/j.sysconle.2007.06.001. [31] G. Webb, Functional differential equations and nonlinear semigroups in $L^p$-spaces, J. Diff. Eq., 20 (1976), 71-89. doi: 10.1016/0022-0396(76)90097-8. [32] G. Weiss, Optimal control of systems with a unitary semigroup and with colocated control and observation, Syst. Control Lett., 48 (2003), 329-340. doi: 10.1016/S0167-6911(02)00276-1. [33] J. Wu, Theory and Application of Partial Functional Differential Equations, Appl. Math. Sci., Vol. 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [34] Q. Zheng and M. Li, Regularized Semigroups and Non-Elliptic Differential Operators, Sciense Press, Beijing, 2014.
 [1] G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327 [2] Hermen Jan Hupkes, Emmanuelle Augeraud-Véron. Well-posedness of initial value problems for functional differential and algebraic equations of mixed type. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 737-765. doi: 10.3934/dcds.2011.30.737 [3] Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007 [4] Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036 [5] Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241 [6] A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469 [7] Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527 [8] Can Li, Weihua Deng, Lijing Zhao. Well-posedness and numerical algorithm for the tempered fractional differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1989-2015. doi: 10.3934/dcdsb.2019026 [9] András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks and Heterogeneous Media, 2012, 7 (1) : 43-58. doi: 10.3934/nhm.2012.7.43 [10] Avner Friedman, Harsh Vardhan Jain. A partial differential equation model of metastasized prostatic cancer. Mathematical Biosciences & Engineering, 2013, 10 (3) : 591-608. doi: 10.3934/mbe.2013.10.591 [11] George Avalos, Pelin G. Geredeli, Justin T. Webster. Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1267-1295. doi: 10.3934/dcdsb.2018151 [12] Jerry Bona, Hongqiu Chen. Well-posedness for regularized nonlinear dispersive wave equations. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1253-1275. doi: 10.3934/dcds.2009.23.1253 [13] Hongwei Wang, Amin Esfahani. Well-posedness and asymptotic behavior of the dissipative Ostrovsky equation. Evolution Equations and Control Theory, 2019, 8 (4) : 709-735. doi: 10.3934/eect.2019035 [14] Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393 [15] Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic and Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029 [16] Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763 [17] Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321 [18] Ricardo A. Pastrán, Oscar G. Riaño. Sharp well-posedness for the Chen-Lee equation. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2179-2202. doi: 10.3934/cpaa.2016033 [19] Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure and Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673 [20] Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387

2021 Impact Factor: 1.588