Welington Vieira Assuncao, Jose Luiz Boldrini
We analyze a highly nonlinear system of partial differential equations that may be seen as a model for solidification or melting of certain viscoelastic materials subject to thermal effects; under the assumption that solid parts of the material may support damped vibrations. Phase change is controlled by a phase field equation with a potential including barriers at the pure solid and pure liquid states.
The present system is closely related to a model analyzed by Rocca and Rossi . They proved the existence of local in time solutions (global in the one dimensional case) assuming values just in the mushy zone, and thus such local solutions do not allow regions of pure solid or pure liquid states, except in the special one-dimensional case where pure liquid state is also allowed.
By including a suitable dissipation in the previous model and assuming constant latent heat, in this work we are able to prove global in time existence even for solutions that may touch the potential barriers; that is, they allow regions with pure solid or pure liquid.
Submitted February 8, 2013. Published September 11, 2013.
Math Subject Classifications: 76A10, 35A01, 35B45, 35B50, 35M33, 80A22.
Key Words: Nonlinear PDE system; degenerate PDE system; global solutions; uniqueness; phase transitions; thermoviscoelastic materials.
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| Welington Vieira Assunção |
Universidade Federal do ABC, Centro de Matemática
Computação e Cognição; Rua Santa Adélia,
Vila São Pedro 09210-170 Santo Andr, SP, Brazil
| José Luiz Boldrini |
Unicamp-IMECC; Rua Sérgio Buarque de Holanda
651; 13083-859 Campinas, SP, Brazil
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