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 & Continuous Dynamical Systems, 2016, 36 (9) : 5163-5181. doi: 10.3934/dcds.2016024
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show all references

References:
[1]

Israel J. Math., 59 (1987), 327-352. doi: 10.1007/BF02774144.  Google Scholar

[2]

J. Math. Anal. Appl., 264 (2001), 1-20. doi: 10.1006/jmaa.2001.6705.  Google Scholar

[3]

A. K. Peters, Wellesley, 2005.  Google Scholar

[4]

Syst. Control Lett., 45 (2002), 155-161. doi: 10.1016/S0167-6911(01)00177-3.  Google Scholar

[5]

Proc. London Math. Soc., 55 (1987), 181-208. doi: 10.1112/plms/s3-55.1.181.  Google Scholar

[6]

Springer-Verlag, 1994.  Google Scholar

[7]

Math. Z., 212 (1993), 619-629. doi: 10.1007/BF02571680.  Google Scholar

[8]

J. Funct. Anal., 111 (1993), 44-61. doi: 10.1006/jfan.1993.1003.  Google Scholar

[9]

Springer-Verlag, New York, 2000.  Google Scholar

[10]

Syst. Control Lett., 54 (2005), 557-574. doi: 10.1016/j.sysconle.2004.10.006.  Google Scholar

[11]

Appl. Math. Sci., Vol. 3, Springer-Verlag, 1971. Google Scholar

[12]

Forum Math., 3 (1991), 595-612. doi: 10.1515/form.1991.3.595.  Google Scholar

[13]

Math. Ann., 291 (1991), 1-16. doi: 10.1007/BF01445187.  Google Scholar

[14]

Acta Math., 104 (1960), 93-140. doi: 10.1007/BF02547187.  Google Scholar

[15]

Trans. Amer. Math. Soc., 152 (1970), 215-226.  Google Scholar

[16]

J. Math. Anal. Appl., 292 (2004), 328-339. doi: 10.1016/j.jmaa.2003.10.031.  Google Scholar

[17]

J. Funct. Anal., 84 (1989), 160-180. doi: 10.1016/0022-1236(89)90116-X.  Google Scholar

[18]

Taiwanese J. Math., 14 (2010), 1979-1992.  Google Scholar

[19]

J. Math. Anal. Appl., 188 (1994), 809-818. doi: 10.1006/jmaa.1994.1464.  Google Scholar

[20]

Proc. London Math. Soc., 69 (1994), 144-170. doi: 10.1112/plms/s3-69.1.144.  Google Scholar

[21]

Acta Math. Sci., 13 (1993), 428-434.  Google Scholar

[22]

SIAM J. Control Optim., 36 (1998), 1086-1098. doi: 10.1137/S0363012996310703.  Google Scholar

[23]

Chapman & Hall, London, 2001. doi: 10.1201/9781420035490.  Google Scholar

[24]

Russian Math. Surveys, 49 (1994), 115-155. doi: 10.1070/RM1994v049n06ABEH002449.  Google Scholar

[25]

Pac. J. Math., 135 (1988), 111-155. doi: 10.2140/pjm.1988.135.111.  Google Scholar

[26]

$2^{nd}$, North Holland, Elsevier, 1986.  Google Scholar

[27]

Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 2346-2355. doi: 10.1016/j.na.2009.10.035.  Google Scholar

[28]

Semigroup Forum, 41 (1990), 215-236. doi: 10.1007/BF02573392.  Google Scholar

[29]

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Syst. Control Lett., 56 (2007), 709-713. doi: 10.1016/j.sysconle.2007.06.001.  Google Scholar

[31]

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Syst. Control Lett., 48 (2003), 329-340. doi: 10.1016/S0167-6911(02)00276-1.  Google Scholar

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Appl. Math. Sci., Vol. 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

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Sciense Press, Beijing, 2014. Google Scholar

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