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Keywords:
nonlinear thermoelasticity; viscoelasticity; nonlinear wave equation; global solutions
Summary:
The global in time solvability of the one-dimensional nonlinear equations of thermoelasticity, equations of viscoelasticity and nonlinear wave equations in several space dimensions with some boundary dissipation is discussed. The blow up of the solutions which might be possible even for small data is excluded by allowing for a certain dissipative mechanism.
References:
[1] Agmon S.: On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Commun. Pure Appl. Math. 15 (1962), 119-147. MR 0147774 | Zbl 0109.32701
[2] Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Commun. Pure Appl. Math. 12 (1959), 623-727; II, ibid. 17 (1964), 35-92. MR 0125307 | Zbl 0093.10401
[3] Andrews G.: On the existence of solutions to the equation: $u_{tt} = u_{xxt} + \sigma(u_x)_x$. J. Diff. Eqns. 35 (1980), 200-231. MR 0561978 | Zbl 0415.35018
[4] Andrews G., Ball J.M.: Asymptotic behaviour and changes in phase in one-dimensional nonlinear viscoelasticity. J. Diff. Eqns. 44 (1982), 306-341. MR 0657784
[5] Ang D.D., Dinh A.P.N.: On the strongly damped wave equation: $u_{tt} - \Delta u - \Delta u_t + f(u) = 0$. SIAM J. Math. Anal. 19 (1988), 1409-1418. MR 0965260
[6] Aviles P., Sandefur J.: Nonlinear second order equations with applications to partial differential equations. J. Diff. Eqns. 58 (1985), 404-427. MR 0797319 | Zbl 0572.34004
[7] Bardos C., Lebeau G., Rauch J: Contrôle et stabilisation dans les problèmes hyperboliques. Appendix II in J.L. Lions Contrôlabilité exacte, perturbations et stabilisation de systémes distribués, I, Contrôlabilité exacte Masson, RMA 8, 1988. MR 0953547
[8] Bardos C., Lebeau G., Rauch J: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. submitted to SIAM. J. Cont. Optim. Zbl 0786.93009
[9] Chen G.: Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain. J. Math. pures et appl. 58 (1976), 249-273. MR 0544253
[10] Chrzȩszczyk A.: Some existence results in dynamical thermoelasticity. Part I. Nonlinear Case. Arch. Mech. 39 (1987), 605-617. MR 0976929
[11] Cleménts J.: Existence theorems for a quasilinear evolution equation. SIAM J. Appl. Math. 26 (1974), 745-752. MR 0372426
[12] Cleménts J.: On the existence and uniqueness of solutions of the equation $u_{tt} - (\partial/\partial x_i)\sigma_i(u_{x_i}) - \Delta_Nu_t = f$. Canad. Math. Bull. 18 (1975), 181-187. MR 0397200
[13] Dafermos C.M.: On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity. Arch. Rational Mech. Anal. 29 (1968), 241-271. MR 0233539 | Zbl 0183.37701
[14] Dafermos C.M.: The mixed initial-boundary value problem for the equations of non-linear one-dimensional visco-elasticity. J. Diff. Eqns. 6 (l969), 71-86. MR 0241831
[15] Dafermos C.M., Hsiao L.: Development of singularities in solutions of the equations of nonlinear thermoelasticity. Quart. Appl. Math. 44 (1986), 463-474. MR 0860899 | Zbl 0661.35009
[16] Dan W.: On a local in time solvability of the Neumann problem of quasilinear hyperbolic parabolic coupled systems. preprint, 1992. MR 1357364 | Zbl 0841.35003
[17] Dassios G., Grillakis M.: Dissipation rates and partition of energy in thermoelasticity. Arch. Rational Mech. Anal. 87 (1984), 49-91. MR 0760319 | Zbl 0563.73007
[18] Ebihara Y.: On some nonlinear evolution equations with the strong dissipation. J. Diff. Eqns. 30 (1978), 149-164 II ibid. 34 (1979), 339-352 III ibid. 45 (1982), 332-355. MR 0513267
[19] Ebihara Y.: Some evolution equations with the quasi-linear strong dissipation. J. Math. pures et appl. 58 (1987), 229-245. MR 0539221
[20] Engler H.: Strong solutions for strongly damped quasilinear wave equations. Contemporary Math. 64 (1987), 219-237. MR 0881465 | Zbl 0638.35054
[21] Feireisl E.: Forced vibrations in one-dimensional nonlinear thermoelasticity as a local coercive-like problem. Comment. Math. Univ. Carolinae 31 (1990), 243-255. MR 1077895 | Zbl 0718.73013
[22] Friedman A., Nečas J.: Systems of nonlinear wave equations with nonlinear viscosity. Pacific J. Math. 135 (1988), 29-55. MR 0965683
[23] Greenberg J.M.: On the existence, uniqueness, and stability of the equation $\rho_0X_{tt} = E(X_x)X_{xx} + X_{xxt}$. J. Math. Anal. Appl. 25 (1969), 575-591. MR 0240473
[24] Greenberg J.M., Li Ta-tsien: The effect of boundary damping for the quasilinear wave equation. J. Diff. Eqns. 52 (1984), 66-75. MR 0737964
[25] Greenberg J.M., MacCamy R.C., Mizel J.J.: On the existence, uniqueness, and stability of the equation $\sigma^{\prime} (u_x)u_{xx} - \lambda u_{xxt} = \rho_0u_{tt}$. J. Math. Mech. 17 (1968), 707-728.
[26] Godin P.: Private communication in 1992.
[27] Hrusa W.J., Messaoudi S.A.: On formation of singularities in one-dimensional nonlinear thermoelasticity. Arch. Rational Mech. Anal. 111 (1990), 135-151. MR 1057652 | Zbl 0712.73023
[28] Hrusa W.J., Tarabek M.A.: On smooth solutions of the Cauchy problem in one-dimensional nonlinear thermoelasticity. Quart. Appl. Math. 47 (1989), 631-644. MR 1031681 | Zbl 0692.73005
[29] Jiang S.: Global existence of smooth solutions in one- dimensional nonlinear thermoelasticity. Proc. Roy. Soc. Edinburgh 115A (1990), 257-274. MR 1069521 | Zbl 0723.35044
[30] Jiang S.: Far field behavior of solutions to the equations of nonlinear 1-d-thermoelasticity. Appl. Anal. 36 (1990), 25-35. MR 1040876 | Zbl 0672.35011
[31] Jiang S.: Rapidly decreasing behaviour of solutions in nonlinear 3-D-thermo-elasticity. Bull. Austral. Math. Soc. 43 (1991), 89-99. MR 1086721
[32] Jiang S.: Global solutions of the Dirichlet problem in one-dimensional nonlinear thermoelasticity. SFB 256 Preprint 138, Universität Bonn, 1990.
[33] Jiang S.: Global solutions of the Neumann problem in one-dimensional nonlinear thermoelasticity. to appear in Nonlinear TMA. MR 1174462 | Zbl 0786.73009
[34] Jiang S., Racke R.: On some quasilinear hyperbolic-parabolic initial boundary value problems. Math. Meth. Appl. Sci. 12 (1990), 315-339. MR 1048561 | Zbl 0706.35098
[35] Kawashima S.: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. Thesis, Kyoto University, 1983.
[36] Kawashima S., Okada M.: Smooth global solutions for the one-dimensional equations in magnetohydrodynamics. Proc. Japan Acad. Ser. A. 53 (1982), 384-387. MR 0694940 | Zbl 0522.76098
[37] Kawashima S., Shibata Y.: Global existence and exponential stability of small solutions to nonlinear viscoelasticity. to appear in Commun. Math. Phys. MR 1178142 | Zbl 0779.35066
[38] Kawashima S., Shibata Y.: On the Neumann problem of one-dimensional nonlinear thermoelasticity with time- independent external forces. preprint, 1992. MR 1314530
[39] Klainerman S., Majda A.: Formation of singularities for wave equations including the nonlinear vibrating string. Pure Appl. Math. 33 (1980), 241-263. MR 0562736 | Zbl 0443.35040
[40] Kobayashi T., Pecher H., Shibata Y.: On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity. preprint, 1992. MR 1219900 | Zbl 0788.35001
[41] Lagnese J.: Boundary stabilization of linear elastodynamic systems. SIAM J. Control Optim. 21 (1983), 968-984. MR 0719524 | Zbl 0531.93044
[42] MacCamy R.C., Mizel V.J.: Existence and nonexistence in the large of solutions of quasilinear wave equations. Arch. Rational Mech. Anal. 25 (1967), 299-320. MR 0216165 | Zbl 0146.33801
[43] Matsumura A.: Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with first order dissipation. Publ. RIMS Kyoto Univ. Ser. A 13 (1977), 349-379. MR 0470507
[44] Mizohata K., Ukai S.: The global existence of small amplitude solutions to the nonlinear acoustic wave equation. preprint, 1991, Dep. of Information Sci. Tokyo Inst. of Tech. MR 1231754 | Zbl 0794.35108
[45] Nagasawa T.: On the one-dimensional motion of the polytropic ideal gas non-fixed on the boundary. J. Diff. Eqns. 65 (1986), 49-67. MR 0859472 | Zbl 0598.34021
[46] Pecher H.: On global regular solutions of third order partial differential equations. J. Math. Anal. Appl. 73 (1980), 278-299. MR 0560948 | Zbl 0429.35057
[47] Ponce G.: Global existence of small solutions to a class of nonlinear evolution equation. Nonlinear Anal. TMA 9 (1985), 399-418. MR 0785713
[48] Ponce G., Racke R.: Global existence of small solutions to the initial value problem for nonlinear thermoelasticity. J. Diff. Eqns. 87 (1990), 70-83. MR 1070028 | Zbl 0725.35065
[49] Potier-Ferry M.: On the mathematical foundation of elastic stability, I. Arch. Rational Mech. Anal. 78 (1982), 55-72. MR 0654552
[50] Qin T.: The global smooth solutions of second order quasilinear hyperbolic equations with dissipation boundary condition. Chinese Anals Math. 9B (1988), 251-269. MR 0968461
[51] Quinn J.P., Russell D.L.: Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping. Proc. Roy. Soc. Edinburgh 77A (1977), 97-127. MR 0473539 | Zbl 0357.35006
[52] Rabinowitz P.: Periodic solutions of nonlinear partial differential equations. Commun. Pure Appl. Math. 20 (1967), 145-205 II ibid. 22 (1969), 15-39. MR 0206507
[53] Racke R.: On the Cauchy problem in nonlinear 3-d-thermoelasticity. Math. Z. 203 (1990), 649-682. MR 1044071 | Zbl 0701.73002
[54] Racke R.: Blow-up in nonlinear three-dimensional thermoelasticity. Math. Meth. Appl. Sci. 12 (1990), 267-273. MR 1043758 | Zbl 0705.35081
[55] Racke R.: Mathematical aspects in nonlinear thermoelasticity. SFB 256 Lecture Note Series { 25}, 1992.
[56] Racke R.: Lectures on nonlinear evolution equation. Initial value problems. Ser. Aspects of Mathematics'', Fridr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1992. MR 1158463
[57] Racke R., Shibata Y.: Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity. Arch. Rational Mech. Anal. 116 (1991), 1-34. MR 1130241 | Zbl 0756.73012
[58] Racke R., Shibata Y., Zheng S.: Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity. to appear in Quart. Appl. Math. MR 1247439 | Zbl 0804.35132
[59] Rybka P.: Dynamical modelling of phase transitions by means of viscoelasticity in many dimensions. to appear in Proc. Roy. Soc. Edinburgh 121A (1992). MR 1169897 | Zbl 0758.73004
[60] Shibata Y.: Neumann problem for one-dimensional nonlinear thermoelasticity. to appear in Banach Center Publication. MR 1205848
[61] Shibata Y, Nakamura G.: On a local existence theorem of Neumann problem for some quasilinear hyperbolic systems of 2nd order. Math. Z. 202 (1989), 1-64. MR 1007739
[62] Shibata Y., Kikuchi M.: On the mixed problem for some quasilinear hyperbolic system with fully nonlinear boundary condition. J. Diff. Eqns. 80 (1989), 154-197. MR 1003254 | Zbl 0689.35055
[63] Shibata Y., Zheng S.: On some nonlinear hyperbolic systems with damping boundary conditions. Nonlinear Anal. TMA 17 (1991), 233-266. MR 1120976 | Zbl 0772.35031
[64] Slemrod M.: Global existence, uniqueness, and asymptotic stability of classical smooth solutions in the one-dimensional non-linear thermoelasticity. Arch. Rational Mech. Anal. 76 (1981), 97-133. MR 0629700
[65] Tanabe H.: Equations of evolution. Monographs and Studies in Mathematics, Pitman, London, San Francisco, Melbourne, l979. Zbl 0417.35003
[66] Webb G.F.: Existence and asymptotic behavior for a strongly damped nonlinear wave equation. Canada J. Math. 32 (1980), 631-643. MR 0586981 | Zbl 0414.35046
[67] Yamada Y.: Some remarks on the equation $u_{tt} - \sigma(y_x)y_{xx} -y_{xtx} = f$. Osaka J. Math. 17 (1980), 303-323. MR 0587752
[68] Zheng S.: Global solutions and applications to a class of quasilinear hyperbolic-parabolic coupled systems. Sci. Sinica Ser. A 27 (1984), 1274-1286. MR 0794293 | Zbl 0581.35056
[69] Zheng S., Shen W.: Global solutions to the Cauchy problem of quasilinear hyperbolic parabolic coupled systems. Sci. Sinica Ser. A 3 (1987), 1133-1149. MR 0942420 | Zbl 0649.35013
[70] Zuazua E.: Stability and decay for a class of nonlinear hyperbolic problems. Asymptotic Anal. 1 (1988), 161-185. MR 0950012 | Zbl 0677.35069

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