zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Global existence and exponential stability of small solutions to nonlinear viscoelasticity. (English) Zbl 0779.35066
Summary: The global existence of smooth solutions of the equations of nonlinear hyperbolic system of second order with third order viscosity is shown for small and smooth initial data in a bounded domain of n-dimensional Euclidean space with smooth boundary. Dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions as t tending to is described. Time periodic solutions are also studied. As an application of our main theorem, nonlinear viscoelasticity, strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated.

MSC:
35L55Higher order hyperbolic systems
35L60Nonlinear first-order hyperbolic equations
35B40Asymptotic behavior of solutions of PDE
74D99Materials of strain-rate type and history type, other materials with memory
35A05General existence and uniqueness theorems (PDE) (MSC2000)
35B65Smoothness and regularity of solutions of PDE
References:
[1]Andrews, G.: On the existence of solutions to the equationu tt=uxxt+σ(u x)x. J. Diff. Eq.35, 200–231 (1980) · Zbl 0415.35018 · doi:10.1016/0022-0396(80)90040-6
[2]Andrews, G., Ball, J.M.: Asymptotic behaviour and changes in phase in one-dimensional nonlinear viscoelasticity. J. Diff. Eq.44, 306–341 (1982) · Zbl 0501.35011 · doi:10.1016/0022-0396(82)90019-5
[3]Ang, D.D., Dinh, A.P.N.: On the strongly damped wave equation:u ttuu t+f(u)=0. SIAM J. Math. Anal.19, 1409–1418 (1988) · Zbl 0685.35071 · doi:10.1137/0519103
[4]Arima, R., Hasegawa, Y.: On global solutions for mixed problem of semilinear differential equation. Proc. Jpn Acad.39, 721–725 (1963) · Zbl 0173.11804 · doi:10.3792/pja/1195522891
[5]Aviles, P., Sandefur, J.: Nonlinear second order equations with applications to partial differential equations. J. Diff. Eq.58, 404–427 (1985) · Zbl 0572.34004 · doi:10.1016/0022-0396(85)90008-7
[6]Cleménts, J.: Existence theorems for a quasilinear evolution equation. SIAM J. Appl. Math.26, 745–752 (1974) · Zbl 0284.35048 · doi:10.1137/0126066
[7]Cleménts, J.: On the existence and uniqueness of solutions of the equation u u (/x i )σ i (u x i )-Δ N u t =f . Canad. Math. Bull.18, 181–187 (1975) · Zbl 0312.35017 · doi:10.4153/CMB-1975-036-1
[8]Dafermos, C.M.: The mixed initial-boundary value problem for the equations of nonlinear one-dimensional visco-elasticity. J. Diff. Eq.6, 71–86 (1969) · Zbl 0218.73054 · doi:10.1016/0022-0396(69)90118-1
[9]Davis, P.: A quasi-linear hyperbolic and related third order equation. J. Math. Anal. Appl.51, 596–606 (1975) · Zbl 0312.35018 · doi:10.1016/0022-247X(75)90110-9
[10]Ebihara, Y.: Some evolution equations with the quasi-linear strong dissipation. J. Math. Pures et Appl.58, 229–245 (1979)
[11]Engler, H.: Strong solutions for strongly damped quasilinear wave equations. Contemp. Math.64, 219–237 (1987)
[12]Friedman, A., Necas, J.: Systems of nonlinear wave equations with nonlinear viscosity. Pacific J. Math.135, 29–55 (1988)
[13]Greenberg, J.M., MacCamy, R.C., Mizel, J.J.: On the existence, uniqueness, and stability of the equation σ’(ux)uxx-λuxxt=ρouu. J. Math. Mech.17, 707–728 (1968)
[14]Greenberg, J.M.: On the existence, uniqueness, and stability of the equation ρoXtt=E(Xx)Xxx+λXxxt. J. Math. Anal. Appl.25, 575–591 (1969) · Zbl 0192.44803 · doi:10.1016/0022-247X(69)90257-1
[15]Kato, T.: Abstract differential equations and nonlinear mixed problem. Scuola Normale Superiore, Lezioni Fermiane, Pisa (1985)
[16]Matsumura, A.: Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first-order dissipation. Publ. RIMS, Kyoto Univ.13, 349–379 (1977) · Zbl 0371.35030 · doi:10.2977/prims/1195189813
[17]Mizohata, K., Ukai, S.: The global existence of small amplitude solutions to the nonlinear acoustic wave equation. Preprint in 1991, Department of Information Sci., Tokyo Inst. of Tech.
[18]Pecher, H.: On global regular solutions of third order partial differential equations. J. Math. Anal. Appl.73, 278–299 (1980) · Zbl 0429.35057 · doi:10.1016/0022-247X(80)90033-5
[19]Potier-Ferry, M.: On the mathematical foundation of elastic stability, I. Arch. Radional Mech. Anal.78, 55–72 (1982)
[20]Rabinowitz, P.: Periodic solutions of nonlinear partial differential equations. Commun. Pure Appl. Math.,20, 145–205 (1967); II,-om ibid Rabinowitz, P.: Periodic solutions of nonlinear partial differential equations. Commun. Pure Appl. Math.22, 15–39 (1969) · Zbl 0152.10003 · doi:10.1002/cpa.3160200105
[21]Shibata, Y.: On the Neumann problem for some linear hyperbolic systems of 2nd order with coefficients in Sobolev spaces. Tsukuba J. Math.13, 283–352 (1989)
[22]Shibata, Y., Kikuchi, M.: On the mixed problem for some quasilinear hyperbolic system with fully nonlinear boundary condition. J. Diff. Eq.80, 154–197 (1989) · Zbl 0689.35055 · doi:10.1016/0022-0396(89)90099-5
[23]Webb, G.F.: Existence and asymptotic behavior for a strongly damped nonlinear wave equation. Canada J. Math.32, 631–643 (1980) · Zbl 0432.35046 · doi:10.4153/CJM-1980-049-5
[24]Yamada, Y.: Some remarks on the equationy tt(y x)yxxxtx=f. Osaka J. Math.17, 303–323 (1980)