Weak solutions of hyperbolic-parabolic Volterra equations

Author:
Gustaf Gripenberg

Journal:
Trans. Amer. Math. Soc. **343** (1994), 675-694

MSC:
Primary 45K05; Secondary 35D05, 35K60, 45D05, 73F15

DOI:
https://doi.org/10.1090/S0002-9947-1994-1216335-0

MathSciNet review:
1216335

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Abstract | References | Similar Articles | Additional Information

Abstract: The existence of a global weak solution, satisfying certain a priori -bounds, of the equation is established. The kernel *k* is locally integrable and log-convex, and has only one local minimum which is positive.

**[1]**D. Brandon and W. J. Hrusa,*Global existence of smooth shearing motions of a nonlinear viscoelastic fluid*, J. Integral Equations Appl.**2**(1990), 333-351. MR**1094473 (92e:45015)****[2]**C. M. Dafermos,*Development of singularities in the motion of materials with fading memory*, Arch. Rational Mech. Anal.**91**(1986), 193-205. MR**806001 (87a:73033)****[3]**C. M. Dafermos and J. A. Nohel,*Energy methods for nonlinear hyperbolic Volterra integrodifferential equations*, Comm. Partial Differential Equations**4**(1979), 219-278. MR**522712 (80b:45018)****[4]**W. Desch and R. Grimmer,*Smoothing properties of linear Volterra integrodifferential equations*, SIAM J. Math. Anal.**20**(1989), 116-132. MR**977492 (89m:45014)****[5]**H. Engler,*Weak solutions of a class of quasilinear hyperbolic integro-differential equations describing viscoelastic materials*, Arch. Rational Mech. Anal.**113**(1991), 1-38. MR**1079180 (91k:35052)****[6]**Y. Fujita,*Integrodifferential equation which interpolates the heat equation and the wave equation*. Osaka J. Math.**27**(1990), 309-321. MR**1066629 (91i:45007)****[7]**Y. Fujita,*Integrodifferential equation which interpolates the heat equation and the wave equation*. II, Osaka J. Math.**27**(1990), 797-804. MR**1088183 (92a:45032)****[8]**G. Gripenberg,*On Volterra equations of the first kind*, Integral Equations Operator Theory**3/4**(1980), 473-488. MR**595747 (82c:45007)****[9]**-,*Nonexistence of smooth solutions for shearing flows in a nonlinear viscoelastic fluid*, SIAM J. Math. Anal.**13**(1982), 954-961. MR**674764 (83m:76007)****[10]**-,*Volterra integro-differential equations with accretive nonlinearity*, J. Differential Equations**60**(1985), 57-79. MR**808257 (86m:45017)****[11]**-,*Global existence of solutions of Volterra integrodifferential equations of parabolic type*, J. Differential Equations**102**(1993), 382-390. MR**1216735 (94e:45007)****[12]**-,*Nonlinear Volterra equations of parabolic type due to singular kernels*, J. Differential Equations (to appear). MR**1287555 (95c:45004)****[13]**-,*Compensated compactness and one-dimensional elastodynamics*, manuscript.**[14]**G. Gripenberg, S-O. Londen, and O. Staffans,*Volterra integral and functional equations*, Cambridge University Press, Cambridge, 1990. MR**1050319 (91c:45003)****[15]**H. Hattori,*Breakdown of smooth solutions in dissipative nonlinear hyperbolic equations*, Quart. Appl. Math.**40**(1982), 113-127. MR**666668 (83j:35109)****[16]**W. J. Hrusa,*Global existence and asymptotic stability for semilinear hyperbolic Volterra equation with large initial data*, SIAM J. Math. Anal.**16**(1985), 110-134. MR**772871 (86h:45026)****[17]**W. J. Hrusa and J. A. Nohel,*The Cauchy problem in one-dimensional nonlinear viscoelasticity*, J. Differential Equations**59**(1985), 388-412. MR**807854 (87a:73030)****[18]**W. J. Hrusa, J. A. Nohel, and M. Renardy,*Initial value problems in viscoelasticity*, Appl. Mech. Rev.**41**(1988), 371-378.**[19]**W. J. Hrusa and M. Renardy,*On wave propagation in linear viscoelasticity*, Quart. Appl. Math.**43**(1985), 237-254. MR**793532 (86j:45022)****[20]**-,*On a class of quasilinear partial integrodifferential equations with singular kernels*, J. Differential Equations**64**(1986), 195-220. MR**851911 (88c:45010)****[21]**-,*A model equation for viscoelasticity with a strongly singular kernel*, SIAM J. Math. Anal.**19**(1988), 257-269. MR**930025 (89d:35159)****[22]**S-O. Londen,*An existence result on a Volterra equation in a Banach space*, Trans. Amer. Math. Soc.**235**(1978), 285-304. MR**0473770 (57:13432)****[23]**-,*On an integrodifferential Volterra equation with a maximal monotone mapping*, J. Differential Equations**27**(1978), 405-420. MR**0499976 (58:17711)****[24]**-,*Some existence results for a nonlinear hyperbolic integrodifferential equation with singular kernel*, J. Integral Equations Appl.**3**(1991), 3-30. MR**1094929 (92a:45035)****[25]**R. C. MacCamy,*A model for one-dimensional nonlinear viscoelasticity*, Quart. Appl. Math.**35**(1977), 21-33. MR**0478939 (57:18395)****[26]**J. A. Nohel and M. Renardy,*Development of singularities in nonlinear viscoelasticity*, Amorphous Polymers and Non-Newtonian Fluids (C. Dafermos, J. L. Ericksen, and D. Kinderlehrer, eds.), IMA Math. Appl., Vol. 6, Springer-Verlag, Berlin and New York, 1987, pp. 139-152. MR**902188 (88m:73014)****[27]**J. A. Nohel, R. C. Rogers, and A. E. Tzavaras,*Weak solutions for a nonlinear system in viscoelasticity*, Comm. Partial Differential Equations**13**(1988), 97-127. MR**914816 (89h:35063)****[28]**J. Prüss,*Positivity and regularity of hyperbolic Volterra equations in Banach spaces*, Math. Ann.**279**(1987), 317-344. MR**919509 (89h:45004)****[29]**J. Prüss,*Quasilinear parabolic Volterra equations in spaces of integrable functions*, Semigroup Theory and Evolution Equations: The Second International Conference (Ph. Clement, E. Mitidieri, and B. de Pagter, eds.), Lecture Notes in Pure and Appl. Math., vol. 135, Dekker, New York, 1991, pp. 401-420. MR**1164666 (93j:35176)****[30]**M. Renardy,*Coercive estimates and existence of solutions for a model of one-dimensional viscoelasticity with a non-integrable memory function*, J. Integral Equations Appl.**1**(1988), 7-16. MR**955160 (89k:45009)****[31]**M. Renardy, W. J. Hrusa, and J. A. Nohel,*Mathematical problems in viscoelasticity*, Longman, London, 1987. MR**919738 (89b:35134)****[32]**O. Staffans,*On a nonlinear hyperbolic Volterra equation*, SIAM J. Math. Anal.**11**(1980), 793-812. MR**586908 (81j:45018)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1216335-0

Keywords:
Volterra equation,
weak solution,
-bound,
viscoelasticity,
parabolic,
hyperbolic

Article copyright:
© Copyright 1994
American Mathematical Society