Global Existence and Uniqueness of Weak and Regular Solutions of Shallow Shells with Thermal Effects
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We study a dynamical thin shallow shell whose elastic deformations are described by a nonlinear system of Marguerre–Vlasov’s type under the presence of thermal effects. Our main result is the proof of a global existence and uniqueness of a weak solution in the case of clamped boundary conditions. Standard techniques for uniqueness do not work directly in this case. We overcame this difficulty using recent work due to Lasiecka (Appl Anal 4:1376–1422, 1998).
KeywordsExistence Uniqueness Global weak solutions
Mathematics Subject Classification58G20 58Z05 35Q72
We would like to express our sincere thanks to the Referee of this Journal for his (or hers) suggestions which helped us to present this final version in much better form than our first version. The first author (GPM) would to acknowledge the partial support he obtained by the Brazilian Research Council (CNPq) though Project 3036/2013-0. He is also grateful to PRONEX / FAPERJ (E 26/110-560/2010) from the Brazilian Government. The second author (FTC) would to acknowledge the partial support from FAPERGS-Brazil Grant 1947-2551/13-3 and from CAPES-Brazil though project BEX 12220/13-2. She also would like to thanks the Department of Mathematics of the University of Memphis by the atmosphere, seminars and hospitality that she received when she was doing her post-doc research there during 2014. Her special gratitude to Prof. Jerry Goldstein, Prof. Gisele Goldstein and Prof. Irena Lasiecka.
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