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Well-posedness results for a model of damage in thermoviscoelastic materials. (English) Zbl 1152.35505
Summary: This paper deals with a phase transitions model describing the evolution of damage in thermoviscoelastic materials. The resulting system is highly non-linear, mainly due to the presence of quadratic dissipative terms and non-smooth constraints on the variables. Existence and uniqueness of a solution are proved, as well as regularity results, on a suitable finite time interval.

##### MSC:
 35Q72 Other PDE from mechanics (MSC2000) 74R20 Anelastic fracture and damage 35B65 Smoothness and regularity of solutions to PDEs
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##### References:
 [1] Baiocchi, C., Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert, Ann. mat. pura appl. (IV), 76, 233-304, (1967) · Zbl 0153.17202 [2] Barbu, V., Nonlinear semigroups and differential equations in Banach spaces, (1976), Noordhoff Leyden [3] Bonetti, E.; Bonfanti, G., Existence and uniqueness of the solution to a 3D thermoviscoelastic system, Electron. J. differential equations, 50, 1-15, (2003) · Zbl 1034.74022 [4] Bonetti, E.; Schimperna, G., Local existence to frémond’s model for damaging in elastic materials, Contin. mech. thermodyn., 16, 319-335, (2004) · Zbl 1066.74048 [5] Bonetti, E.; Schimperna, G.; Segatti, A., On a doubly nonlinear model for the evolution of damaging in viscoelastic materials, J. differential equations, 218, 91-116, (2005) · Zbl 1078.74048 [6] Bonfanti, G.; Frémond, M.; Luterotti, F., Global solution to a nonlinear system for irreversible phase changes, Adv. math. sci. appl., 10, 1-24, (2000) · Zbl 0956.35122 [7] Bonfanti, G.; Frémond, M.; Luterotti, F., Existence and uniqueness results to a phase transition model based on microscopic accelerations and movements, Nonlinear anal. real world appl., 5, 123-140, (2004) · Zbl 1092.80006 [8] Brézis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans LES espaces de Hilbert, North-holland math. studies, vol. 5, (1973), North-Holland Amsterdam · Zbl 0252.47055 [9] Dafermos, C.M., Global smooth solutions to the initial boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity, SIAM J. math. anal., 13, 397-408, (1982) · Zbl 0489.73124 [10] Francfort, G.A.; Suquet, P., Homogenization and mechanical dissipation in thermoviscoelasticity, Arch. ration. mech. anal., 96, 265-293, (1986) · Zbl 0621.73044 [11] Frémond, M., Non-smooth thermomechanics, (2001), Springer-Verlag Berlin · Zbl 0990.80001 [12] Frémond, M.; Kenmochi, N., Damage problems for viscous locking materials, Adv. math. sci. appl., 16, 697-716, (2006) · Zbl 1158.74310 [13] Frémond, M.; Kuttler, K.L.; Nedjar, B.; Shillor, M., One dimensional models of damage, Adv. math. sci. appl., 8, 541-570, (1998) · Zbl 0915.73041 [14] Lions, J.L., Quelques Méthodes de Résolution des problèmes aux limites non linéaires, (1969), Dunod, Gauthier-Villars Paris · Zbl 0189.40603 [15] Luterotti, F.; Schimperna, G.; Stefanelli, U., Global solution to a phase field model with irreversible and constrained phase evolution, Quart. appl. math., 60, 301-316, (2002) · Zbl 1032.35109 [16] Nirenberg, L., On elliptic partial differential equations, Ann. scuola norm. sup. Pisa (3), 13, 115-162, (1959) · Zbl 0088.07601 [17] Sassetti, M.; Tarsia, A., Su un’equazione non lineare Della corda vibrante, Ann. mat. pura appl., 161, 1-42, (1992) [18] Schimperna, G.; Stefanelli, U., Positivity of the temperature for phase transitions with micro-movements, Nonlinear anal. real world appl., 8, 257-266, (2007) · Zbl 1116.80015 [19] Simon, J., Compact sets in the space $$L^p(0, T; B)$$, Ann. mat. pura appl. (4), 146, 65-96, (1987) · Zbl 0629.46031
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