Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

4 Issues per year


IMPACT FACTOR 2017: 0.658

CiteScore 2017: 1.05

SCImago Journal Rank (SJR) 2017: 1.291
Source Normalized Impact per Paper (SNIP) 2017: 0.893

Mathematical Citation Quotient (MCQ) 2017: 0.76

Online
ISSN
1609-9389
See all formats and pricing
More options …
Ahead of print

Issues

Nonlinear Evolution Equations with Exponentially Decaying Memory: Existence via Time Discretisation, Uniqueness, and Stability

André EikmeierORCID iD: http://orcid.org/0000-0002-0270-6491 / Etienne EmmrichORCID iD: http://orcid.org/0000-0001-9869-0334 / Hans-Christian Kreusler
Published Online: 2018-11-21 | DOI: https://doi.org/10.1515/cmam-2018-0268

Abstract

The initial value problem for an evolution equation of type v+Av+BKv=f is studied, where A:VAVA is a monotone, coercive operator and where B:VBVB induces an inner product. The Banach space VA is not required to be embedded in VB or vice versa. The operator K incorporates a Volterra integral operator in time of convolution type with an exponentially decaying kernel. Existence of a global-in-time solution is shown by proving convergence of a suitable time discretisation. Moreover, uniqueness as well as stability results are proved. Appropriate integration-by-parts formulae are a key ingredient for the analysis.

Keywords: Nonlinear Evolution Equation; Monotone Operator; Volterra Operator; Exponentially Decaying Memory; Existence; Uniqueness; Stability; Time Discretisation; Convergence

MSC 2010: 47J35; 45K05; 34K30; 35K90; 35R09; 65J08; 65M12

Dedicated to Professor Rolf D. Grigorieff on the occasion of his 80th birthday

References

  • [1]

    V. Barbu, P. Colli, G. Gilardi and M. Grasselli, Existence, uniqueness, and longtime behavior for a nonlinear Volterra integrodifferential equation, Differential Integral Equations 13 (2000), no. 10–12, 1233–1262. Google Scholar

  • [2]

    A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Res. Notes Math. 10, A K Peters, Wellesley, 2005. Google Scholar

  • [3]

    S. Bonaccorsi and G. Desch, Volterra equations in Banach spaces with completely monotone kernels, NoDEA Nonlinear Differential Equations Appl. 20 (2013), no. 3, 557–594. Google Scholar

  • [4]

    H. Brézis, Analyse Fonctionnelle: Théorie et Applications, Dunod, Paris, 1999. Google Scholar

  • [5]

    M. P. Calvo, E. Cuesta and C. Palencia, Runge–Kutta convolution quadrature methods for well-posed equations with memory, Numer. Math. 107 (2007), no. 4, 589–614. Google Scholar

  • [6]

    P. Cannarsa and D. Sforza, Global solutions of abstract semilinear parabolic equations with memory terms, NoDEA Nonlinear Differential Equations Appl. 10 (2003), no. 4, 399–430. Google Scholar

  • [7]

    C. Chen and T. Shih, Finite Element Methods for Integrodifferential Equations, Ser. Appl. Math. 9, World Scientific, River Edge, 1998. Google Scholar

  • [8]

    C. Chen, V. Thomée and L. B. Wahlbin, Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel, Math. Comp. 58 (1992), no. 198, 587–602. Google Scholar

  • [9]

    C. Corduneanu, Functional Equations with Causal Operators, Stab. Control Theory Methods Appl. 16, Taylor & Francis, London, 2002. Google Scholar

  • [10]

    M. G. Crandall, S.-O. Londen and J. A. Nohel, An abstract nonlinear Volterra integrodifferential equation, J. Math. Anal. Appl. 64 (1978), no. 3, 701–735. Google Scholar

  • [11]

    M. G. Crandall and J. A. Nohel, An abstract functional differential equation and a related nonlinear Volterra equation, Israel J. Math. 29 (1978), no. 4, 313–328. Google Scholar

  • [12]

    E. Cuesta and C. Palencia, A numerical method for an integro-differential equation with memory in Banach spaces: Qualitative properties, SIAM J. Numer. Anal. 41 (2003), no. 4, 1232–1241. Google Scholar

  • [13]

    J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Springer, Berlin, 1977. Google Scholar

  • [14]

    C. M. Dafermos and J. A. Nohel, Energy methods for nonlinear hyperbolic Volterra integrodifferential equations, Comm. Partial Differential Equations 4 (1979), no. 3, 219–278. Google Scholar

  • [15]

    W. Desch, R. Grimmer and W. Schappacher, Well-posedness and wave propagation for a class of integrodifferential equations in Banach space, J. Differential Equations 74 (1988), no. 2, 391–411. Google Scholar

  • [16]

    J. Diestel and J. J. Uhl, Jr., Vector Measures, American Mathematical Society, Providence, 1977. Google Scholar

  • [17]

    D. A. Edwards, Non-Fickian diffusion in thin polymer films, J. Polym. Sci. Part B Polym. Phys. 34 (1996), 981–997. Google Scholar

  • [18]

    D. A. Edwards and D. S. Cohen, A mathematical model for a dissolving polymer, AIChE Journal 41 (1995), no. 11, 2345–2355. Google Scholar

  • [19]

    E. Emmrich and M. Thalhammer, Doubly nonlinear evolution equations of second order: existence and fully discrete approximation, J. Differential Equations 251 (2011), no. 1, 82–118. Google Scholar

  • [20]

    S. Fedotov and A. Iomin, Probabilistic approach to a proliferation and migration dichotomy in tumor cell invasion, Phys. Rev. E (3) 77 (2008), no. 3, Article ID 031911. Google Scholar

  • [21]

    J. A. Ferreira, E. Gudiño and P. de Oliveira, A second order approximation for quasilinear non-Fickian diffusion models, Comput. Methods Appl. Math. 13 (2013), no. 4, 471–493. Google Scholar

  • [22]

    H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie, Berlin, 1974. Google Scholar

  • [23]

    G. Gilardi and U. Stefanelli, Time-discretization and global solution for a doubly nonlinear Volterra equation, J. Differential Equations 228 (2006), no. 2, 707–736. Google Scholar

  • [24]

    G. Gilardi and U. Stefanelli, Existence for a doubly nonlinear Volterra equation, J. Math. Anal. Appl. 333 (2007), no. 2, 839–862. Google Scholar

  • [25]

    M. Grasselli and A. Lorenzi, Abstract nonlinear Volterra integrodifferential equations with nonsmooth kernels, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 2 (1991), no. 1, 43–53. Google Scholar

  • [26]

    R. Grimmer and M. Zeman, Nonlinear Volterra integro-differential equations in a Banach space, Israel J. Math. 42 (1982), no. 1–2, 162–176. Google Scholar

  • [27]

    G. Gripenberg, On a nonlinear Volterra integral equation in a Banach space, J. Math. Anal. Appl. 66 (1978), no. 1, 207–219. Google Scholar

  • [28]

    G. Gripenberg, An abstract nonlinear Volterra equation, Israel J. Math. 34 (1979), no. 3, 198–212. Google Scholar

  • [29]

    G. Gripenberg, Nonlinear Volterra equations of parabolic type due to singular kernels, J. Differential Equations 112 (1994), no. 1, 154–169. Google Scholar

  • [30]

    G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia Math. Appl. 34, Cambridge University Press, Cambridge, 1990. Google Scholar

  • [31]

    S. Larsson, V. Thomée and L. B. Wahlbin, Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method, Math. Comp. 67 (1998), no. 221, 45–71. Google Scholar

  • [32]

    J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer, New York, 1972. Google Scholar

  • [33]

    S.-O. Londen, On an integrodifferential Volterra equation with a maximal monotone mapping, J. Differential Equations 27 (1978), no. 3, 405–420. Google Scholar

  • [34]

    C. Lubich, Convolution quadrature and discretized operational calculus. I, Numer. Math. 52 (1988), no. 2, 129–145. Google Scholar

  • [35]

    C. Lubich, Convolution quadrature and discretized operational calculus. II, Numer. Math. 52 (1988), no. 4, 413–425. Google Scholar

  • [36]

    C. Lubich, Convolution quadrature revisited, BIT 44 (2004), no. 3, 503–514. Google Scholar

  • [37]

    C. Lubich, I. H. Sloan and V. Thomée, Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term, Math. Comp. 65 (1996), no. 213, 1–17. Google Scholar

  • [38]

    R. C. MacCamy, A model for one-dimensional, nonlinear viscoelasticity, Quart. Appl. Math. 35 (1977/78), no. 1, 21–33. Google Scholar

  • [39]

    R. C. MacCamy, An integro-differential equation with application in heat flow, Quart. Appl. Math. 35 (1977/78), no. 1, 1–19. Google Scholar

  • [40]

    R. C. MacCamy and J. S. W. Wong, Stability theorems for some functional equations, Trans. Amer. Math. Soc. 164 (1972), 1–37. Google Scholar

  • [41]

    W. McLean and V. Thomée, Numerical solution of an evolution equation with a positive-type memory term, J. Aust. Math. Soc. Ser. B 35 (1993), no. 1, 23–70. Google Scholar

  • [42]

    A. Mehrabian and Y. Abousleiman, General solutions to poroviscoelastic model of hydrocephalic human brain tissue, J. Theoret. Biol. 291 (2011), 105–118. Google Scholar

  • [43]

    R. K. Miller, An integro-differential equation for rigid heat conductors with memory, J. Math. Anal. Appl. 66 (1978), no. 2, 313–332. Google Scholar

  • [44]

    J. Prüß, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993. Google Scholar

  • [45]

    T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2005. Google Scholar

  • [46]

    S. Shaw and J. R. Whiteman, Some partial differential Volterra equation problems arising in viscoelasticity, Proceedings of Equadiff 9, Masaryk University, Brno (1998), 183–200. Google Scholar

  • [47]

    I. H. Sloan and V. Thomée, Time discretization of an integro-differential equation of parabolic type, SIAM J. Numer. Anal. 23 (1986), no. 5, 1052–1061. Google Scholar

  • [48]

    U. Stefanelli, On some nonlocal evolution equations in Banach spaces, J. Evol. Equ. 4 (2004), no. 1, 1–26. Google Scholar

  • [49]

    W. A. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math. 19 (1966), 543–551. Google Scholar

  • [50]

    G. F. Webb, Functional differential equations and nonlinear semigroups in Lp-spaces, J. Differential Equations 20 (1976), no. 1, 71–89. Google Scholar

  • [51]

    G. F. Webb, Volterra integral equations and nonlinear semigroups, Nonlinear Anal. 1 (1976/77), no. 4, 415–427. Google Scholar

  • [52]

    G. F. Webb, An abstract semilinear Volterra integrodifferential equation, Proc. Amer. Math. Soc. 69 (1978), no. 2, 255–260. Google Scholar

  • [53]

    R. Zacher, Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients, J. Math. Anal. Appl. 348 (2008), no. 1, 137–149. Google Scholar

  • [54]

    R. Zacher, Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces, Funkcial. Ekvac. 52 (2009), no. 1, 1–18. Google Scholar

  • [55]

    E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B, Springer, New York, 1990. Google Scholar

About the article

Received: 2018-06-20

Accepted: 2018-10-20

Published Online: 2018-11-21


This work has been supported by the Deutsche Forschungsgemeinschaft through Collaborative Research Center 910 “Control of self-organizing nonlinear systems: Theoretical methods and concepts of application”.


Citation Information: Computational Methods in Applied Mathematics, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0268.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in