American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay
  • DCDS-B Home
  • This Issue
  • Next Article
    A criterion for the existence of relaxation oscillations with applications to predator-prey systems and an epidemic model
April  2020, 25(4): 1279-1298. doi: 10.3934/dcdsb.2019220

On regularity of stochastic convolutions of functional linear differential equations with memory

Kai Liu a,b,

a. 

School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China
b. 

Department of Mathematical Sciences, School of Physical Sciences, The University of Liverpool, Liverpool, L69 7ZL, UK

The author is grateful to the Tianjin Thousand Talents Plan for its financial support

Received  April 2019 Published  April 2020 Early access  September 2019

In this work, we consider the regularity property of stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded operator coefficients. We first establish some useful estimates on fundamental solutions which are time delay versions of those on $ C_0 $-semigroups. To this end, we develop a scheme of constructing the resolvent operators for the integrodifferential equations of Volterra type since the equation under investigation is of this type in each subinterval describing the segment of its solution. Then we apply these estimates to stochastic convolutions of our equations to obtain the desired regularity property.

Keywords: Regularity property, fundamental solution, stochastic convolution.
Mathematics Subject Classification: Primary: 60H15, 60G15, 60H05.
Citation: Kai Liu. On regularity of stochastic convolutions of functional linear differential equations with memory. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1279-1298. doi: 10.3934/dcdsb.2019220
References:
[1]

B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208.  doi: 10.1007/BF01596912.  Google Scholar

[2] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Second Edition, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[3]

G. Di Blasio, K. Kunisch and E. Sinestrari, Stability for abstract linear functional differential equations, Israel J. Math., 50 (1985), 231-263.  doi: 10.1007/BF02761404.  Google Scholar

[4]

J. Jeong, Stabilizability of retarded functional differential equation in Hilbert space, Osaka J. Math., 28 (1991), 347-365.   Google Scholar

[5]

J. Jeong, S. I. Nakagiri and H. Tanabe, Structural operators and semigroups associated with functional differential equations in Hilbert spaces, Osaka J. Math., 30 (1993), 365-395.   Google Scholar

[6]

J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.  doi: 10.1090/qam/295683.  Google Scholar

[7]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, , Applied Mathematical Sciences, Vol. 44. Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[8]

J. Prüss, On resolvent operators for linear integrodifferential equations of Volterra type, J. Integral Equations, 5 (1983), 211-236.   Google Scholar

[9]

E. Sinestrari, On a class of retarded partial differential equations of Volterra type, Math. Z., 186 (1984), 223-246.  doi: 10.1007/BF01161806.  Google Scholar

[10]

E. Sinestrari, A noncompact differentiable semigroup arising from an abstract delay equation, C. R. Math. Rep. Acad. Sci. Canada., 6 (1984), 43-48.   Google Scholar

[11]

H. Tanabe, On fundamental solution of differential equation with time delay in Banach space, Proc. Japan Acad., 64 (1988), 131-134.  doi: 10.3792/pjaa.64.131.  Google Scholar

[12]

H. Tanabe, Fundamental solutions for linear retarded functional differential equations in Banach spaces, Funkcialaj Ekvacioj, 35 (1992), 149-177.   Google Scholar

show all references

References:
[1]

B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208.  doi: 10.1007/BF01596912.  Google Scholar

[2] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Second Edition, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[3]

G. Di Blasio, K. Kunisch and E. Sinestrari, Stability for abstract linear functional differential equations, Israel J. Math., 50 (1985), 231-263.  doi: 10.1007/BF02761404.  Google Scholar

[4]

J. Jeong, Stabilizability of retarded functional differential equation in Hilbert space, Osaka J. Math., 28 (1991), 347-365.   Google Scholar

[5]

J. Jeong, S. I. Nakagiri and H. Tanabe, Structural operators and semigroups associated with functional differential equations in Hilbert spaces, Osaka J. Math., 30 (1993), 365-395.   Google Scholar

[6]

J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.  doi: 10.1090/qam/295683.  Google Scholar

[7]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, , Applied Mathematical Sciences, Vol. 44. Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[8]

J. Prüss, On resolvent operators for linear integrodifferential equations of Volterra type, J. Integral Equations, 5 (1983), 211-236.   Google Scholar

[9]

E. Sinestrari, On a class of retarded partial differential equations of Volterra type, Math. Z., 186 (1984), 223-246.  doi: 10.1007/BF01161806.  Google Scholar

[10]

E. Sinestrari, A noncompact differentiable semigroup arising from an abstract delay equation, C. R. Math. Rep. Acad. Sci. Canada., 6 (1984), 43-48.   Google Scholar

[11]

H. Tanabe, On fundamental solution of differential equation with time delay in Banach space, Proc. Japan Acad., 64 (1988), 131-134.  doi: 10.3792/pjaa.64.131.  Google Scholar

[12]

H. Tanabe, Fundamental solutions for linear retarded functional differential equations in Banach spaces, Funkcialaj Ekvacioj, 35 (1992), 149-177.   Google Scholar

[1]

Yalçin Sarol, Frederi Viens. Time regularity of the evolution solution to fractional stochastic heat equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 895-910. doi: 10.3934/dcdsb.2006.6.895
[2]

Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101
[3]

Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146
[4]

Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099
[5]

István Győri, László Horváth. On the fundamental solution and its application in a large class of differential systems determined by Volterra type operators with delay. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1665-1702. doi: 10.3934/dcds.2020089
[6]

Reinhard Farwig, Ronald B. Guenther, Enrique A. Thomann, Šárka Nečasová. The fundamental solution of linearized nonstationary Navier-Stokes equations of motion around a rotating and translating body. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 511-529. doi: 10.3934/dcds.2014.34.511
[7]

Guillaume Warnault. Regularity of the extremal solution for a biharmonic problem with general nonlinearity. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1709-1723. doi: 10.3934/cpaa.2009.8.1709
[8]

Hua Qiu. Regularity criteria of smooth solution to the incompressible viscoelastic flow. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2873-2888. doi: 10.3934/cpaa.2013.12.2873
[9]

Liang Zhao. New developments in using stochastic recipe for multi-compartment model: Inter-compartment traveling route, residence time, and exponential convolution expansion. Mathematical Biosciences & Engineering, 2009, 6 (3) : 663-682. doi: 10.3934/mbe.2009.6.663
[10]

Zdzisław Brzeźniak, Paul André Razafimandimby. Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1051-1077. doi: 10.3934/dcdsb.2016.21.1051
[11]

Peng Gao. Unique continuation property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2493-2510. doi: 10.3934/dcdsb.2018262
[12]

Zhaoyang Qiu, Yixuan Wang. Martingale solution for stochastic active liquid crystal system. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2227-2268. doi: 10.3934/dcds.2020360
[13]

Miroslava Růžičková, Irada Dzhalladova, Jitka Laitochová, Josef Diblík. Solution to a stochastic pursuit model using moment equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 473-485. doi: 10.3934/dcdsb.2018032
[14]

Priyanjana M. N. Dharmawardane. Decay property of regularity-loss type for quasi-linear hyperbolic systems of viscoelasticity. Conference Publications, 2013, 2013 (special) : 197-206. doi: 10.3934/proc.2013.2013.197
[15]

Jacinto Marabel Romo. A closed-form solution for outperformance options with stochastic correlation and stochastic volatility. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1185-1209. doi: 10.3934/jimo.2015.11.1185
[16]

Piernicola Bettiol, Hélène Frankowska. Lipschitz regularity of solution map of control systems with multiple state constraints. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 1-26. doi: 10.3934/dcds.2012.32.1
[17]

H. M. Yin. Optimal regularity of solution to a degenerate elliptic system arising in electromagnetic fields. Communications on Pure & Applied Analysis, 2002, 1 (1) : 127-134. doi: 10.3934/cpaa.2002.1.127
[18]

Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227
[19]

Nan Chen, Cheng Wang, Steven Wise. Global-in-time Gevrey regularity solution for a class of bistable gradient flows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1689-1711. doi: 10.3934/dcdsb.2016018
[20]

Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (173)
  • HTML views (246)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences