Global existence and exponential stability of small solutions to nonlinear viscoelasticity.

*(English)*Zbl 0779.35066Summary: 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 \(\infty\) 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:

35L55 | Higher-order hyperbolic systems |

35L60 | First-order nonlinear hyperbolic equations |

35B40 | Asymptotic behavior of solutions to PDEs |

74D99 | Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials) |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

35B65 | Smoothness and regularity of solutions to PDEs |

##### Keywords:

asymptotic exponential decay; time periodic solutions; exponential stability; global existence; smooth solutions; nonlinear hyperbolic system of second order; Dirichlet boundary condition; nonlinear wave equation
PDF
BibTeX
XML
Cite

\textit{S. Kawashima} and \textit{Y. Shibata}, Commun. Math. Phys. 148, No. 1, 189--208 (1992; Zbl 0779.35066)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Andrews, G.: On the existence of solutions to the equationu tt=uxxt+{\(\sigma\)}(u x)x. J. Diff. Eq.35, 200–231 (1980) · Zbl 0415.35018 |

[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 |

[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 |

[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 |

[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 |

[6] | Cleménts, J.: Existence theorems for a quasilinear evolution equation. SIAM J. Appl. Math.26, 745–752 (1974) · Zbl 0284.35048 |

[7] | Cleménts, J.: On the existence and uniqueness of solutions of the equation \(u_u (\partial /\partial x_i )\sigma _i (u_{x_i } ) - \Delta _N u_t = f\) . Canad. Math. Bull.18, 181–187 (1975) · Zbl 0312.35017 |

[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 |

[9] | Davis, P.: A quasi-linear hyperbolic and related third order equation. J. Math. Anal. Appl.51, 596–606 (1975) · Zbl 0312.35018 |

[10] | Ebihara, Y.: Some evolution equations with the quasi-linear strong dissipation. J. Math. Pures et Appl.58, 229–245 (1979) · Zbl 0405.35049 |

[11] | Engler, H.: Strong solutions for strongly damped quasilinear wave equations. Contemp. Math.64, 219–237 (1987) · Zbl 0638.35054 |

[12] | Friedman, A., Necas, J.: Systems of nonlinear wave equations with nonlinear viscosity. Pacific J. Math.135, 29–55 (1988) · Zbl 0685.35070 |

[13] | Greenberg, J.M., MacCamy, R.C., Mizel, J.J.: On the existence, uniqueness, and stability of the equation {\(\sigma\)}’(ux)uxx-{\(\lambda\)}uxxt={\(\rho\)}ouu. J. Math. Mech.17, 707–728 (1968) |

[14] | Greenberg, J.M.: On the existence, uniqueness, and stability of the equation {\(\rho\)}oXtt=E(Xx)Xxx+{\(\lambda\)}Xxxt. J. Math. Anal. Appl.25, 575–591 (1969) · Zbl 0192.44803 |

[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 |

[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. · Zbl 0794.35108 |

[18] | Pecher, H.: On global regular solutions of third order partial differential equations. J. Math. Anal. Appl.73, 278–299 (1980) · Zbl 0429.35057 |

[19] | Potier-Ferry, M.: On the mathematical foundation of elastic stability, I. Arch. Radional Mech. Anal.78, 55–72 (1982) · Zbl 0488.73043 |

[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 |

[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) · Zbl 0706.35082 |

[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 |

[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 |

[24] | Yamada, Y.: Some remarks on the equationy tt(y x)yxxxtx=f. Osaka J. Math.17, 303–323 (1980) · Zbl 0446.35071 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.