Geodesic
Existence of Global Weak Solutions to the Equations of One-Dimensional Nonlinear Thermoviscoelasticity with Discontinuous Data
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 11-19 
A. A. Amosov. Existence of Global Weak Solutions to the Equations of One-Dimensional Nonlinear Thermoviscoelasticity with Discontinuous Data. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 11-19. http://geodesic.mathdoc.fr/item/TM_2002_236_a1/
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     author = {A. A. Amosov},
     title = {Existence of {Global} {Weak} {Solutions} to the {Equations} of {One-Dimensional} {Nonlinear} {Thermoviscoelasticity} with {Discontinuous} {Data}},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The existence of global weak solutions to initial–boundary value problems for a system of quasilinear differential equations describing the dynamics of a one-dimensional Voigt-type thermoviscoelastic body is established. The initial and boundary data may be discontinuous functions. Only physically natural requirements are imposed on the data. In particular, it is required that the initial velocity and initial temperature should be such that the full energy is finite. The density of heat sources and a boundary heat flux may be functions from $L_1$. The functions defining the properties of the body may also be discontinuous in $x$.

[1] Dafermos C. M., “Global smooth solutions to the initial-boundary value problem of the equations of one-dimensional nonlinear thermoviscoelasticity”, SIAM J. Math. Anal., 13 (1982), 397–408 | DOI | MR | Zbl

[2] Dafermos C. M., Hsiao L., “Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity”, Nonlin. Anal. Th., Meth. and Appl., 6:5 (1982), 435–454 | DOI | MR | Zbl

[3] Kim J. U., “Global existence of solutions of the equations of one-dimensional thermoviscoelasticity with initial data in $BV$ and $L^1$”, Ann. Scuola Norm. Super. Piza, 10:3 (1983), 357–427 | MR | Zbl

[4] Zheng S., Shen W., “Global smooth solutions to the Caushy problem of equations of one-dimensional nonlinear thermoviscoelasticity”, J. Part. Diff. Equat., 2 (1989), 26–38 | MR | Zbl

[5] Jiang S., “Global large solutions to initial boundary value problems in one-dimensional nonlinear thermoviscoelasticity”, Quart. Appl. Math., 51:4 (1993), 731–744 | MR | Zbl

[6] Amosov A. A., “Suschestvovanie globalnykh obobschennykh reshenii uravnenii odnomernogo dvizheniya vyazkogo realnogo gaza i odnomernoi nelineinoi termovyazkouprugosti”, Dokl. RAN, 369:3 (1999), 295–298 | MR | Zbl

[7] Amosov A. A., Zlotnik A. A., “Poludiskretnyi metod resheniya uravnenii odnomernogo dvizheniya neodnorodnogo vyazkogo teploprovodnogo gaza s negladkimi dannymi”, Izv. vuzov. Matematika, 1997, no. 4, 3–19 | MR | Zbl

[8] Amosov A. A., “Suschestvovanie globalnykh obobschennykh reshenii uravnenii odnomernogo dvizheniya vyazkogo realnogo gaza s razryvnymi dannymi”, Dif. uravneniya, 36:4 (2000), 486–499 | MR | Zbl

[9] Andrews G., “On the existence of solutions to the equation $u_{tt}=u_{xxt}+\sigma(u_x)_x$”, J. Diff. Equat., 35 (1980), 200–231 | DOI | MR