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Global well-posedness and stability of a partial integro-differential equation with applications to viscoelasticity

  • Chapter
Nonlinear Evolution Equations and Related Topics

Abstract

In this paper we consider the equation

$$ {u_{{tt}}}(t,x) = \int_{0}^{t} {a(t - s){u_{{txx}}}(s.x)ds + \frac{d}{{dt}}\int_{0}^{t} {b(t - s)(g{{({u_{x}}(s,x))}_{x}}ds + f(t,x),} } $$
((1.1))

t > 0, x ∈ (0, 1), with boundary conditions

$$ u(t,0) = u(t,1) = 0,t > 0, $$
((1.2))

and initial values

$$ u(0,x) = {u^{0}}(x),{u_{t}}(0,x) = {u^{1}}(x). $$
((1.3))

Work supported by the grant A1019002 of GA AV ČR.

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Author information

Authors and Affiliations

  1. Institute of Mathematics, Helsinki University of Technology, Espoo, FIN-02150, Finland

    S.-O. Londen

  2. Institute of Mathematics, Czech Academy of Sciences Žitná 25, 15 67 Praha 1, Czech Republic

    H. Petzeltová

  3. FB Mathematik und Informatik, Martin-Luther-Universität Halle-Wittenberg, Theodor-Lieser-Str. 5, Halle, D-60120, Germany

    J. Prüss

Authors
  1. S.-O. Londen
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  2. H. Petzeltová
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  3. J. Prüss
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Editor information

Editors and Affiliations

  1. Department of Applied Analysis, University of Ulm, 89069, Ulm, Germany

    Wolfgang Arendt

  2. Analyse Numérique, Université Pierre et Marie Curie, B.C. 187, 4, place Jussieu, 75252, Paris Cedex 05, France

    Haïm Brézis

  3. Department of Mathematics Hill Center, Busch Campus, Rutgers University, 110 Frelinghuysen RD, Piscataway, NJ, 08854, USA

    Haïm Brézis

  4. Campus de Ker Lann, Antenne de Bretagne de l’ENS Cachan, 35170, Bruz, France

    Michel Pierre

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Londen, SO., Petzeltová, H., Prüss, J. (2003). Global well-posedness and stability of a partial integro-differential equation with applications to viscoelasticity. In: Arendt, W., Brézis, H., Pierre, M. (eds) Nonlinear Evolution Equations and Related Topics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7924-8_9

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  • DOI: https://doi.org/10.1007/978-3-0348-7924-8_9

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-7643-7107-4

  • Online ISBN: 978-3-0348-7924-8

  • eBook Packages: Springer Book Archive

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