Banach Journal of Mathematical Analysis

$L^p$-maximal regularity of degenerate delay equations with periodic conditions

Shangquan Bu

Full-text: Open access

Abstract

Under suitable assumptions on the delay operator $F$, we give necessary and sufficient conditions for the inhomogeneous abstract degenerate delay equations: $ (Mu)'(t)=Au(t)+Fu_{t}+f(t), \ (t\in \T)$ to have $L^p$-maximal regularity.

Article information

Source
Banach J. Math. Anal., Volume 8, Number 2 (2014), 49-59.

Dates
First available in Project Euclid: 4 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1396640050

Digital Object Identifier
doi:10.15352/bjma/1396640050

Mathematical Reviews number (MathSciNet)
MR3189537

Zentralblatt MATH identifier
1301.47056

Subjects
Primary: 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]
Secondary: 34G10: Linear equations [See also 47D06, 47D09] 34K30: Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx] 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.

Keywords
Maximal regularity degenerate equations delay equations

Citation

Bu, Shangquan. $L^p$-maximal regularity of degenerate delay equations with periodic conditions. Banach J. Math. Anal. 8 (2014), no. 2, 49--59. doi:10.15352/bjma/1396640050. https://projecteuclid.org/euclid.bjma/1396640050


Export citation

References

  • W. Arendt and S. Bu, The operator-valued Marcinkiewicz multiplier theorems and maximal regularity, Math. Z. 240 (2002), 311–343.
  • W. Arendt and S. Bu, Operator-valued Fourier multipliers on peoriodic Besov spaces and applications, Proc. Edinb. Math. Soc. (2) 47 (2004), no.1, 15–33.
  • J. Bourgain, Vector-valued singular integrals and the $H^1$-BMO duality, in: Probability Theory and Harmonic Analysis (Cleveland, OH, 1983), Monogr. Festbooks Pure Appl. Math. 98, dekker, New York, 1988, 1–19.
  • S. Bu, Well-posedness of second order degenerate differential equations in vector-valued function spaces, Studia Math. 214 (2013), no. 1, 1–16.
  • S. Bu and Y. Fang, Periodic solutions of delay equations in Besov spaces and Triebel-Lizorkin spaces. Taiwanese J. Math. 13 (2009), no. 3, 1063–1076.
  • S. Bu and Y. Fang, Maximal regularity for integro-differential equations on periodic Triebel-Lisorkin spaces, Taiwanese J. Math. 12 (2009), no. 2, 281–292.
  • S. Bu and J. Kim, Operator-valued Fourier multipliers on peoriodic Triebel spaces, Acta Math. Sinica, English Series 17 (2004), 15–25.
  • A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Pure and Appl. Math., 215, Dekker, New York, Basel, Hong Kong, 1999.
  • J.K. Hale, Functional Differential Equations, Appl. Math. Sci., Vol.3, Springer-Verlag, (1971).
  • V. Keyantuo and C. Lizama, Fourier multipliers and integro-differential equations in Banach spaces, J. London Math. Soc. 69 (2004), 737–750.
  • V. Keyantuo and C. Lizama, Maximal regularity for a class of integro-differential equations with infinite delay in Banach spaces, Studia Math.168 (2005), 25–50.
  • J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II. Spinger, Berlin, 1996.
  • C. Lizama, Fourier multipliers and periodic solutions of delay equations in Banach spaces, J. Math. Anal. Appl. 324 (2006), 921–933.
  • C. Lizama and V. Poblete, Periodic solutions of fractional differential equations with delay, J. Evol. Equ. 11 (2011), no. 1, 57–70.
  • C. Lizama and V. Poblete, Maximal regularity of delay equations in Banach spaces, Studia Math. 175 (2006), 91–102.
  • C. Lizama and R. Ponce, Maximal regularity for degenerate differential equations with infinite delay in periodic vector-valued function spaces, Proc. Edinb. Math. Soc. (to appear).
  • V. Poblete, Maximal regularity of second-order equations with delay, J. Differential Equations 246 (2009) no. 1, 261–276.
  • G. Webb, Functional differential equations and nonlinear semigroups in $L^{p}$-spaces, J. Differential Equations 29 (1976), 71–89.
  • L. Weis, Operator-valued Fourier multipliers and maximal $L_{p}$-regularity, Math. Ann. 319 (2001), 735–758.