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Hwang, Jin-soo

Weak and strong solutions for a strongly damped quasilinear membrane equation. (English) Zbl 1470.35240

Abstr. Appl. Anal. 2017, Article ID 4529847, 9 p. (2017).
Summary: We consider a strongly damped quasilinear membrane equation with Dirichlet boundary condition. The goal is to prove the well-posedness of the equation in weak and strong senses. By setting suitable function spaces and making use of the properties of the quasilinear term in the equation, we have proved the fundamental results on existence, uniqueness, and continuous dependence on data including bilinear term of weak and strong solutions.

MSC:

35L82 Pseudohyperbolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35D30 Weak solutions to PDEs
35D35 Strong solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
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\textit{J.-s. Hwang}, Abstr. Appl. Anal. 2017, Article ID 4529847, 9 p. (2017; Zbl 1470.35240)
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References:

[1] Hwang, J.-s.; Nakagiri, S.-i., Weak solutions of the equation of motion of membrane with strong viscosity, Journal of the Korean Mathematical Society, 44, 2, 443-453, (2007) · Zbl 1142.35515
[2] Banks, H. T.; Smith, R. C.; Wang, Y., Smart Material Structures, Modeling, Estimation and Control, (1996), John Wiley and Sons · Zbl 0882.93001
[3] Dautray, R.; Lions, J.-L., Mathematical Analysis and Numerical Methods for Science and Technology, Evolution Problems I, 5, (1992), Springer-Verlag
[4] Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, (1997), New York, NY, USA: Springer, New York, NY, USA · Zbl 0871.35001
[5] Kobayashi, T.; Pecher, H.; Shibata, Y., On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity, Mathematische Annalen, 296, 2, 215-234, (1993) · Zbl 0788.35001
[6] Hwang, J.-s.; Nakagiri, S.-i., Optimal control problems for the equation of motion of membrane with strong viscosity, Journal of Mathematical Analysis and Applications, 321, 1, 327-342, (2006) · Zbl 1101.49016
[7] Hwang, J.-s.; Nakagiri, S.-i., Parameter identification problem for the equation of motion of membrane with strong viscosity, Journal of Mathematical Analysis and Applications, 342, 1, 125-134, (2008) · Zbl 1143.35386
[8] Hwang, J.-s.; Nakagiri, S.-i.; Tanabe, H., Solutions of quasilinear wave equation with strong and nonlinear viscosity, Journal of the Korean Mathematical Society, 48, 4, 867-885, (2011) · Zbl 1222.35128
[9] Belmiloudi, A., Bilinear minimax control problems for a class of parabolic systems with applications to control of nuclear reactors, Journal of Mathematical Analysis and Applications, 327, 1, 620-642, (2007) · Zbl 1116.49014
[10] Bradley, M. E.; Lenhart, S., Bilinear spatial control of the velocity term in a Kirchhoff plate equation, Electronic Journal of Differential Equations, (2001) · Zbl 0974.49012
[11] Bradley, M. E.; Lenhart, S.; Yong, J., Bilinear optimal control of the velocity term in a Kirchhoff plate equation, Journal of Mathematical Analysis and Applications, 238, 2, 451-467, (1999) · Zbl 0936.49003
[12] Lenhart, S.; Liang, M., Bilinear optimal control for a wave equation with viscous damping, Houston Journal of Mathematics, 26, 3, 575-595, (2000) · Zbl 0976.49005
[13] Simon, J., Compact sets in the space Lp, Annali di Matematica Pura ed Applicata, 146, 4, 65-96, (1987) · Zbl 0629.46031
[14] Temam, R., Navier–Stokes Equations, (1984), Amsterdam, The Netherlands: North-Holland, Amsterdam, The Netherlands · Zbl 0572.35083
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.