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Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity

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References

  1. Andrews, G.: On the existence of solutions to the equationu tt =u xxt +σ(u x ) x . J. Differ. Equations35, 200–231 (1980)

    Google Scholar 

  2. Andrews, G., Ball, J.M.: Asymptotic behavior and changes in phase in one-dimensional non-linear viscoelasticity. J. Differ. Equations44, 306–341 (1982)

    Google Scholar 

  3. Brezis, H., Wainger, S.: A note on limiting cases of Sobolev imbeddings. Commun. Partial Differ. Equations5, 773–789 (1980)

    Google Scholar 

  4. Clements, J.: Existence theorems for a quasilinear evolution equation. SIAM J. Appl. Math.26, 745–752 (1974)

    Google Scholar 

  5. Ebihara, Y.: On some nonlinear evolution equations with strong dissipation. J. Differ. Equations45, 332–355 (1982)

    Google Scholar 

  6. Engler, H.: Existence of radially symmetric solutions of strongly damped wave equations. In: Gill, T.L. Zachary, W.W. (eds.), Nonlinear semigroups, partial differential equations and attractors, Proceedings, Washington, D.C. 1985. (Lect. Notes Math., vol. 1248, pp. 40–51) Berlin Heidelberg New York: Springer 1987

    Google Scholar 

  7. Engler, H., Neubrander, F., Sandefur, J.: Strongly damped second order equations. In: Gill, T.L. Zachary, W.W. (eds.), Nonlinear semigroups, partial differential equations and attractors, Proceedings, Washington, D.C. 1985. (Lect. Notes Math., vol. 1248, pp. 52–62) Berlin Heidelberg New York: Springer 1987

    Google Scholar 

  8. Engler, H.: Strong solutions for strongly damped quasilinear wave equations. In: Keen, L. (ed.), The Legacy of Sonya Kovalevskaya, Contemporary Mathematics. AMS, Providence, R.I.64, 219–237 (1987)

    Google Scholar 

  9. Engler, H.: An alternative proof of the Brezis-Wainger inequality, Commun. Partial Differ. Equations.14, 541–544 (1989)

    Google Scholar 

  10. Friedman, A., Nečas, J.: Systems of nonlinear wave equations with nonlinear viscosity. Pac. J. Math.135, 27–56 (1988)

    Google Scholar 

  11. Goldstein, J.A.: Semigroups of linear operators and applications. New York: Oxford University Press 1985

    Google Scholar 

  12. Greenberg, J.M., MacCamy, R.C., Mizel, V.J.: On the existence, uniqueness, and stability of the equation σ′(u x )u xx u xxt 0 u tt . J. Math. Mech.17, 707–728 (1968)

    Google Scholar 

  13. Greenberg, J.M.. On the existence, uniqueness, and stability of the equation ρ0 X tt =E(X x )X xx +X xxt . J. Math. Anal. Appl.25, 575–591 (1969)

    Google Scholar 

  14. Grisvard, P.: Elliptic problems in nonsmooth domains. Boston: Pitman 1985

    Google Scholar 

  15. Knowles, J.K.: On finite antiplane shear for incompressible elastic materials. J. Austral. Math. Soc. Ser. B,19, 400–415 (1975/76)

    Google Scholar 

  16. Ladyženskaya, O.A., Solonnikov, V.A., Ural'tseva, N.N.: Linear and quasilinear equations of parabolic type. AMS, Providence, R.I., 1968

    Google Scholar 

  17. Pecher, H.: On global regular solutions of third order partial differential equations. J. Math. Anal. Appl.73, 278–299 (1980)

    Google Scholar 

  18. Pego, R.L.: Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability. Arch. Rational. Mech. Anal.97, 353–394 (1987)

    Google Scholar 

  19. Ponce, G.: Long time stability of solutions of nonlinear evolution equations. Ph.D. Thesis, New York University, 1982

  20. Renardy, M., Hrusa, W.J., Nohel, J.A.: Mathematical problems in viscoelasticity. New York: Longman Scientific & Technical/John Wiley 1987

    Google Scholar 

  21. Truesdell, C., Noll, W.: The nonlinear field theories of mechanics, Handbuch der Physik, vol. III/1. Berlin Heidelberg New York: Springer 1965

    Google Scholar 

  22. Yamada, N.: Note on certain nonlinear evolution equations of second order. Proc. Japan Acad., Ser. A55, 167–171 (1979)

    Google Scholar 

  23. Yamada, Y.: Some remarks on the equationy tt −σ(y x )y xx y xtx =f. Osaka J. Math.17, 303–323 (1980)

    Google Scholar 

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This work was supported by the National Science Foundation under grant no. DMS 8805192

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Engler, H. Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity. Math Z 202, 251–259 (1989). https://doi.org/10.1007/BF01215257

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