Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity. (English) Zbl 0697.73033
The following boundary and initial data problem
\[
\partial^ 2_ tu(x,t)-\partial_ t\Delta u(x,t)-div_ x(g(| \nabla u(x,t)|^ 2)\nabla u(x,t)=f(x,t),\quad x\in \Omega,\quad 0<t<T,
\]
\[ u(x,t)=0,\quad x\in \partial \Omega,\quad 0<t<T, \]
\[ u(x,0)=u_ 0(x),\quad \partial_ tu(x,0)=u_ 1(x),\quad x\in \Omega, \] is considered. The equation of the problem is describing the antiplane shear motion of certain viscoelastic solids. For the above problem the author obtains results regarding the existence of unique global smooth solutions for large data in general domains in two dimensions, by imposing some natural conditions on the function g. The mechanical interpretations of these results are pointed out. Some other boundary conditions are considered.
\[ u(x,t)=0,\quad x\in \partial \Omega,\quad 0<t<T, \]
\[ u(x,0)=u_ 0(x),\quad \partial_ tu(x,0)=u_ 1(x),\quad x\in \Omega, \] is considered. The equation of the problem is describing the antiplane shear motion of certain viscoelastic solids. For the above problem the author obtains results regarding the existence of unique global smooth solutions for large data in general domains in two dimensions, by imposing some natural conditions on the function g. The mechanical interpretations of these results are pointed out. Some other boundary conditions are considered.
Reviewer: G.Ciobanu
MSC:
| 74D10 | Nonlinear constitutive equations for materials with memory |
| 35A25 | Other special methods applied to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |
| 35D10 | Regularity of generalized solutions of PDE (MSC2000) |
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
Keywords:
third order partial differential equation; boundary and initial data problem; existence of unique global smooth solutions; large data; general domains in two dimensionsReferences:
| [1] | Andrews, G.: On the existence of solutions to the equationu tt =u xxt +?(u x ) x . J. Differ. Equations35, 200-231 (1980) · Zbl 0415.35018 · doi:10.1016/0022-0396(80)90040-6 |
| [2] | Andrews, G., Ball, J.M.: Asymptotic behavior and changes in phase in one-dimensional non-linear viscoelasticity. J. Differ. Equations44, 306-341 (1982) · Zbl 0501.35011 · doi:10.1016/0022-0396(82)90019-5 |
| [3] | Brezis, H., Wainger, S.: A note on limiting cases of Sobolev imbeddings. Commun. Partial Differ. Equations5, 773-789 (1980) · Zbl 0437.35071 · doi:10.1080/03605308008820154 |
| [4] | Clements, J.: Existence theorems for a quasilinear evolution equation. SIAM J. Appl. Math.26, 745-752 (1974) · doi:10.1137/0126066 |
| [5] | Ebihara, Y.: On some nonlinear evolution equations with strong dissipation. J. Differ. Equations45, 332-355 (1982) · doi:10.1016/0022-0396(82)90032-8 |
| [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 |
| [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 · Zbl 0614.34055 |
| [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) |
| [9] | Engler, H.: An alternative proof of the Brezis-Wainger inequality, Commun. Partial Differ. Equations.14, 541-544 (1989) · Zbl 0688.46016 |
| [10] | Friedman, A., Ne?as, J.: Systems of nonlinear wave equations with nonlinear viscosity. Pac. J. Math.135, 27-56 (1988) |
| [11] | Goldstein, J.A.: Semigroups of linear operators and applications. New York: Oxford University Press 1985 · Zbl 0592.47034 |
| [12] | Greenberg, J.M., MacCamy, R.C., Mizel, V.J.: On the existence, uniqueness, and stability of the equation ??(u x )u xx +?u xxt =?0 u tt . J. Math. Mech.17, 707-728 (1968) · Zbl 0157.41003 |
| [13] | Greenberg, J.M.. On the existence, uniqueness, and stability of the equation ?0 X tt =E(X x )X xx +X xxt . J. Math. Anal. Appl.25, 575-591 (1969) · Zbl 0192.44803 · doi:10.1016/0022-247X(69)90257-1 |
| [14] | Grisvard, P.: Elliptic problems in nonsmooth domains. Boston: Pitman 1985 · Zbl 0695.35060 |
| [15] | Knowles, J.K.: On finite antiplane shear for incompressible elastic materials. J. Austral. Math. Soc. Ser. B,19, 400-415 (1975/76) · Zbl 0363.73045 · doi:10.1017/S0334270000001272 |
| [16] | Lady?enskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and quasilinear equations of parabolic type. AMS, Providence, R.I., 1968 |
| [17] | 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 |
| [18] | Pego, R.L.: Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability. Arch. Rational. Mech. Anal.97, 353-394 (1987) · Zbl 0648.73017 · doi:10.1007/BF00280411 |
| [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 · Zbl 0719.73013 |
| [21] | Truesdell, C., Noll, W.: The nonlinear field theories of mechanics, Handbuch der Physik, vol. III/1. Berlin Heidelberg New York: Springer 1965 · Zbl 0779.73004 |
| [22] | Yamada, N.: Note on certain nonlinear evolution equations of second order. Proc. Japan Acad., Ser. A55, 167-171 (1979) · Zbl 0436.47054 · doi:10.3792/pjaa.55.167 |
| [23] | Yamada, Y.: Some remarks on the equationy tt ??(y x )y xx ?y xtx =f. Osaka J. Math.17, 303-323 (1980) · Zbl 0446.35071 |
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