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Keywords:
nonisothermal Ginzburg-Landau (Allen-Cahn) system; microforce balance; existence and uniqueness results; renormalized solutions; Moser iterations
nonisothermal Ginzburg-Landau (Allen-Cahn) system; microforce balance; existence and uniqueness results; renormalized solutions; Moser iterations
Summary:
The article deals with a nonlinear generalized Ginzburg-Landau (Allen-Cahn) system of PDEs accounting for nonisothermal phase transition phenomena which was recently derived by A. Miranville and G. Schimperna: Nonisothermal phase separation based on a microforce balance, Discrete Contin. Dyn. Syst., Ser. B, {\it 5} (2005), 753--768. The existence of solutions to a related Neumann-Robin problem is established in an $N \le 3$-dimensional space setting. A fixed point procedure guarantees the existence of solutions locally in time. Next, Sobolev embeddings, interpolation inequalities, Moser iterations estimates and results on renormalized solutions for a parabolic equation with $L^1$ data are used to handle a suitable a priori estimate which allows to extend our local solutions to the whole time interval. The uniqueness result is justified by proper contracting estimates.
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