Welington Vieira Assuncao, Jose Luiz Boldrini
Abstract:
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 [23]. 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 email: welington.assuncao@ufabc.edu.br |
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José Luiz Boldrini Unicamp-IMECC; Rua Sérgio Buarque de Holanda 651; 13083-859 Campinas, SP, Brazil email: boldrini@ime.unicamp.br |
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