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## Hadamard well-posedness for a structure acoustic model with a supercritical source and damping terms

 1 Department of Mathematics and Computer Science, Drake University, Des Moines, IA 50311-4505, USA 2 Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE 68588-0130, USA

* Corresponding author: Andrew R. Becklin

Received  February 2020 Revised  June 2020 Early access September 2020

This article is concerned with Hadamard's well posedness of a structural acoustic model consisting of a semilinear wave equation defined on a smooth bounded domain $\Omega\subset\mathbb{R}^3$ which is strongly coupled with a Berger plate equation acting only on a flat part of the boundary of $\Omega$. The system is influenced by several competing forces. In particular, the source term acting on the wave equation is allowed to have a supercritical exponent, in the sense that its associated Nemytskii operators is not locally Lipschitz from $H^1_{\Gamma_0}(\Omega)$ into $L^2(\Omega)$. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions. Moreover, we prove that such solutions depend continuously on the initial data, and uniqueness is obtained in two different scenarios.

Citation: Andrew R. Becklin, Mohammad A. Rammaha. Hadamard well-posedness for a structure acoustic model with a supercritical source and damping terms. Evolution Equations & Control Theory, doi: 10.3934/eect.2020093
##### References:
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Lasiecka, Exact controllability of finite energy states for an acoustic wave/plate interaction under the influence of boundary and localized controls, Adv. Differential Equations, 10 (2005), 901-930.   Google Scholar [6] A. V. Babin and M. I. Vishik, Attractors of evolution equations, in Studies in Mathematics and its Applications, Volume 25, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar [7] V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Mathematics in Science and Engineering, Volume 190, Academic Press Inc., Boston, MA, 1993.  Google Scholar [8] V. Barbu, Y. Guo, M. A. Rammaha and D. Toundykov, Convex integrals on Sobolev spaces, J. Convex Anal., 19 (2012), 837-852.   Google Scholar [9] J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 25 (1976), 895-917.  doi: 10.1512/iumj.1976.25.25071.  Google Scholar [10] L. Bociu, Local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping, Nonlinear Anal., 71 (2009), e560–e575. doi: 10.1016/j.na.2008.11.062.  Google Scholar [11] L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860.  doi: 10.3934/dcds.2008.22.835.  Google Scholar [12] L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683.  doi: 10.1016/j.jde.2010.03.009.  Google Scholar [13] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, Volume 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar [14] H. Brézis, Intégrales convexes dans les espaces de Sobolev, Israel J. Math., 13 (1973), 9-23.  doi: 10.1007/BF02760227.  Google Scholar [15] J. Cagnol, I. Lasiecka, C. Lebiedzik and J.-P. Zolésio, Uniform stability in structural acoustic models with flexible curved walls, J. Differential Equations, 186 (2002), 88-121.  doi: 10.1016/S0022-0396(02)00029-3.  Google Scholar [16] I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132.  Google Scholar [17] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512.  doi: 10.1007/s10884-004-4289-x.  Google Scholar [18] I. Chueshov and I. Lasiecka, Global attractors for Mindlin-Timoshenko plates and for their Kirchhoff limits, Milan J. Math., 74 (2006), 117-138.  doi: 10.1007/s00032-006-0050-8.  Google Scholar [19] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), 183 pp. doi: 10.1090/memo/0912.  Google Scholar [20] I. Chueshov, I. Lasiecka and and D. Toundykov, Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent, Discrete Contin. Dyn. Syst., 20 (2008), 459-509.  doi: 10.3934/dcds.2008.20.459.  Google Scholar [21] I. Chueshov, I. Lasiecka and and D. Toundykov, Global attractor for a wave equation with nonlinear localized boundary damping and a source term of critical exponent, J. Dynam. Differential Equations, 21 (2009), 269-314.  doi: 10.1007/s10884-009-9132-y.  Google Scholar [22] I. D. Chueshov, Introduction to the Theory of Infinite-dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, AKTA, Kharkiv, 1999, 436 pp.  Google Scholar [23] E. Feireisl, Attractors for wave equations with nonlinear dissipation and critical exponent, C. R. Acad. Sci. Paris Sér. I Math., 315 (1992), 551-555.   Google Scholar [24] E. Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent, J. Differential Equations, 116 (1995), 431-447.  doi: 10.1006/jdeq.1995.1042.  Google Scholar [25] M. Grobbelaar-Van Dalsen, On a structural acoustic model with interface a Reissner-Mindlin plate or a Timoshenko beam, J. Math. Anal. Appl., 320 (2006), 121-144.  doi: 10.1016/j.jmaa.2005.06.034.  Google Scholar [26] M. Grobbelaar-Van Dalsen, On a structural acoustic model which incorporates shear and thermal effects in the structural component, J. Math. Anal. Appl., 341 (2008), 1253-1270.  doi: 10.1016/j.jmaa.2007.10.073.  Google Scholar [27] Y. Guo and M. A. Rammaha, Systems of nonlinear wave equations with damping and supercritical boundary and interior sources, Trans. Amer. Math. Soc., 366 (2014), 2265-2325.  doi: 10.1090/S0002-9947-2014-05772-3.  Google Scholar [28] Y. Guo, M. A. Rammaha, S. Sakuntasathien, E. S. Titi and D. Toundykov, Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differential Equations, 257 (2014), 3778-3812.  doi: 10.1016/j.jde.2014.07.009.  Google Scholar [29] M. S. Howe, Acoustics of Fluid-Structure Interactions, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9780511662898.  Google Scholar [30] N. J. Kass, Damped Wave Equations of the p-Laplacian Type with Supercritical Sources, Ph.D. Thesis, The University of Nebraska, Lincoln, 2018, 116 pp.  Google Scholar [31] N. J. Kass and M. A. Rammaha, Local and global existence of solutions to a strongly damped wave equation of the $p$-Laplacian type, Commun. Pure Appl. Anal., 17 (2018), 1449-1478.  doi: 10.3934/cpaa.2018070.  Google Scholar [32] N. J. Kass and M. A. Rammaha, On wave equations of the $p$-Laplacian type with supercritical nonlinearities, Nonlinear Anal., 183 (2019), 70-101.  doi: 10.1016/j.na.2019.01.005.  Google Scholar [33] H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, In Evolution equations, semigroups and functional analysis (Milano, 2000), Progr. Nonlinear Differential Equations Appl., Volume 50, Birkhäuser, Basel, 2002, 197–216.  Google Scholar [34] I. Lasiecka, Boundary stabilization of a 3-dimensional structural acoustic model, J. Math. Pures Appl. (9), 78 (1999), 203-232.  doi: 10.1016/S0021-7824(01)80009-X.  Google Scholar [35] I. Lasiecka, Mathematical control theory of coupled PDEs, in CBMS-NSF Regional Conference Series in Applied Mathematics, Volume 75, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898717099.  Google Scholar [36] I. Lasiecka and A. A. Ruzmaikina, Finite dimensionality and regularity of attractors for a 2-D semilinear wave equation with nonlinear dissipation, J. Math. Anal. Appl., 270 (2002), 16-50.  doi: 10.1016/S0022-247X(02)00006-9.  Google Scholar [37] I. Lasiecka and R. Triggiani, Sharp regularity theory for second order hyperbolic equations of Neumann type. Ⅰ. ${L_2}$ nonhomogeneous data, Ann. Mat. Pura Appl. (4), 157 (1990), 285-367.  doi: 10.1007/BF01765322.  Google Scholar [38] I. Lasiecka and R. Triggiani, Regularity theory of hyperbolic equations with nonhomogeneous Neumann boundary conditions. Ⅱ. General boundary data, J. Differential Equations, 94 (1991), 112-164.  doi: 10.1016/0022-0396(91)90106-J.  Google Scholar [39] P. Pei, M. A. Rammaha and and D. Toundykov, Local and global well-posedness of semilinear Reissner–Mindlin–Timoshenko plate equations, Nonlinear Anal., 105 (2014), 62-85.  doi: 10.1016/j.na.2014.03.024.  Google Scholar [40] P. Pei, M. A. Rammaha, and D. Toundykov, Weak solutions and blow-up for wave equations of $p$-Laplacian type with supercritical sources, J. Math. Phys., 56 (2015), 081503, 30 pp. doi: 10.1063/1.4927688.  Google Scholar [41] D. Pražák, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping, J. Dynam. Differential Equations, 14 (2002), 763-776.  doi: 10.1023/A:1020756426088.  Google Scholar [42] M. A. Rammaha and S. Sakuntasathien, Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms, Nonlinear Anal., 72 (2010), 2658-2683.  doi: 10.1016/j.na.2009.11.013.  Google Scholar [43] S. Sakuntasathien, Global well-posedness for systems of nonlinear wave equations, Ph.D. Thesis, The University of Nebraska, Lincoln, 2008, 124 pp.  Google Scholar [44] R. E. Showalter, Monotone Operators in {B}anach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, Volume 49, American Mathematical Society, Providence, RI, 1997.  Google Scholar [45] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Volume 68, Applied Mathematical Sciences, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

show all references

##### References:
 [1] K. Agre and M. A. Rammaha, Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations, 19 (2006), 1235-1270.   Google Scholar [2] G. Avalos, Wellposedness of a structural acoustics model with point control, In Differential Geometric Methods in the Control of Partial Differential Equations, Contemp. Math., Volume 268, Amer. Math. Soc., Providence, RI, 2000, 1–22. doi: 10.1090/conm/268/04305.  Google Scholar [3] G. Avalos and I. Lasiecka, Uniform decay rates for solutions to a structural acoustics model with nonlinear dissipation, Appl. Math. Comput. Sci., 8 (1998), 287-312.   Google Scholar [4] G. Avalos and I. Lasiecka, Exact controllability of structural acoustic interactions, J. Math. Pures Appl. (9), 82 (2003), 1047-1073.  doi: 10.1016/S0021-7824(03)00016-3.  Google Scholar [5] G. Avalos and I. Lasiecka, Exact controllability of finite energy states for an acoustic wave/plate interaction under the influence of boundary and localized controls, Adv. Differential Equations, 10 (2005), 901-930.   Google Scholar [6] A. V. Babin and M. I. Vishik, Attractors of evolution equations, in Studies in Mathematics and its Applications, Volume 25, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar [7] V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Mathematics in Science and Engineering, Volume 190, Academic Press Inc., Boston, MA, 1993.  Google Scholar [8] V. Barbu, Y. Guo, M. A. Rammaha and D. Toundykov, Convex integrals on Sobolev spaces, J. Convex Anal., 19 (2012), 837-852.   Google Scholar [9] J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 25 (1976), 895-917.  doi: 10.1512/iumj.1976.25.25071.  Google Scholar [10] L. Bociu, Local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping, Nonlinear Anal., 71 (2009), e560–e575. doi: 10.1016/j.na.2008.11.062.  Google Scholar [11] L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860.  doi: 10.3934/dcds.2008.22.835.  Google Scholar [12] L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683.  doi: 10.1016/j.jde.2010.03.009.  Google Scholar [13] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, Volume 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar [14] H. Brézis, Intégrales convexes dans les espaces de Sobolev, Israel J. Math., 13 (1973), 9-23.  doi: 10.1007/BF02760227.  Google Scholar [15] J. Cagnol, I. Lasiecka, C. Lebiedzik and J.-P. Zolésio, Uniform stability in structural acoustic models with flexible curved walls, J. Differential Equations, 186 (2002), 88-121.  doi: 10.1016/S0022-0396(02)00029-3.  Google Scholar [16] I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132.  Google Scholar [17] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512.  doi: 10.1007/s10884-004-4289-x.  Google Scholar [18] I. Chueshov and I. Lasiecka, Global attractors for Mindlin-Timoshenko plates and for their Kirchhoff limits, Milan J. Math., 74 (2006), 117-138.  doi: 10.1007/s00032-006-0050-8.  Google Scholar [19] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), 183 pp. doi: 10.1090/memo/0912.  Google Scholar [20] I. Chueshov, I. Lasiecka and and D. Toundykov, Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent, Discrete Contin. Dyn. Syst., 20 (2008), 459-509.  doi: 10.3934/dcds.2008.20.459.  Google Scholar [21] I. Chueshov, I. Lasiecka and and D. Toundykov, Global attractor for a wave equation with nonlinear localized boundary damping and a source term of critical exponent, J. Dynam. Differential Equations, 21 (2009), 269-314.  doi: 10.1007/s10884-009-9132-y.  Google Scholar [22] I. D. Chueshov, Introduction to the Theory of Infinite-dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, AKTA, Kharkiv, 1999, 436 pp.  Google Scholar [23] E. Feireisl, Attractors for wave equations with nonlinear dissipation and critical exponent, C. R. Acad. Sci. Paris Sér. I Math., 315 (1992), 551-555.   Google Scholar [24] E. Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent, J. Differential Equations, 116 (1995), 431-447.  doi: 10.1006/jdeq.1995.1042.  Google Scholar [25] M. Grobbelaar-Van Dalsen, On a structural acoustic model with interface a Reissner-Mindlin plate or a Timoshenko beam, J. Math. Anal. Appl., 320 (2006), 121-144.  doi: 10.1016/j.jmaa.2005.06.034.  Google Scholar [26] M. Grobbelaar-Van Dalsen, On a structural acoustic model which incorporates shear and thermal effects in the structural component, J. Math. Anal. Appl., 341 (2008), 1253-1270.  doi: 10.1016/j.jmaa.2007.10.073.  Google Scholar [27] Y. Guo and M. A. Rammaha, Systems of nonlinear wave equations with damping and supercritical boundary and interior sources, Trans. Amer. Math. Soc., 366 (2014), 2265-2325.  doi: 10.1090/S0002-9947-2014-05772-3.  Google Scholar [28] Y. Guo, M. A. Rammaha, S. Sakuntasathien, E. S. Titi and D. Toundykov, Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differential Equations, 257 (2014), 3778-3812.  doi: 10.1016/j.jde.2014.07.009.  Google Scholar [29] M. S. Howe, Acoustics of Fluid-Structure Interactions, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9780511662898.  Google Scholar [30] N. J. Kass, Damped Wave Equations of the p-Laplacian Type with Supercritical Sources, Ph.D. Thesis, The University of Nebraska, Lincoln, 2018, 116 pp.  Google Scholar [31] N. J. Kass and M. A. Rammaha, Local and global existence of solutions to a strongly damped wave equation of the $p$-Laplacian type, Commun. Pure Appl. Anal., 17 (2018), 1449-1478.  doi: 10.3934/cpaa.2018070.  Google Scholar [32] N. J. Kass and M. A. Rammaha, On wave equations of the $p$-Laplacian type with supercritical nonlinearities, Nonlinear Anal., 183 (2019), 70-101.  doi: 10.1016/j.na.2019.01.005.  Google Scholar [33] H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, In Evolution equations, semigroups and functional analysis (Milano, 2000), Progr. Nonlinear Differential Equations Appl., Volume 50, Birkhäuser, Basel, 2002, 197–216.  Google Scholar [34] I. Lasiecka, Boundary stabilization of a 3-dimensional structural acoustic model, J. Math. Pures Appl. (9), 78 (1999), 203-232.  doi: 10.1016/S0021-7824(01)80009-X.  Google Scholar [35] I. Lasiecka, Mathematical control theory of coupled PDEs, in CBMS-NSF Regional Conference Series in Applied Mathematics, Volume 75, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898717099.  Google Scholar [36] I. Lasiecka and A. A. Ruzmaikina, Finite dimensionality and regularity of attractors for a 2-D semilinear wave equation with nonlinear dissipation, J. Math. Anal. Appl., 270 (2002), 16-50.  doi: 10.1016/S0022-247X(02)00006-9.  Google Scholar [37] I. Lasiecka and R. Triggiani, Sharp regularity theory for second order hyperbolic equations of Neumann type. Ⅰ. ${L_2}$ nonhomogeneous data, Ann. Mat. Pura Appl. (4), 157 (1990), 285-367.  doi: 10.1007/BF01765322.  Google Scholar [38] I. Lasiecka and R. Triggiani, Regularity theory of hyperbolic equations with nonhomogeneous Neumann boundary conditions. Ⅱ. General boundary data, J. Differential Equations, 94 (1991), 112-164.  doi: 10.1016/0022-0396(91)90106-J.  Google Scholar [39] P. Pei, M. A. Rammaha and and D. Toundykov, Local and global well-posedness of semilinear Reissner–Mindlin–Timoshenko plate equations, Nonlinear Anal., 105 (2014), 62-85.  doi: 10.1016/j.na.2014.03.024.  Google Scholar [40] P. Pei, M. A. Rammaha, and D. Toundykov, Weak solutions and blow-up for wave equations of $p$-Laplacian type with supercritical sources, J. Math. Phys., 56 (2015), 081503, 30 pp. doi: 10.1063/1.4927688.  Google Scholar [41] D. Pražák, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping, J. Dynam. Differential Equations, 14 (2002), 763-776.  doi: 10.1023/A:1020756426088.  Google Scholar [42] M. A. Rammaha and S. Sakuntasathien, Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms, Nonlinear Anal., 72 (2010), 2658-2683.  doi: 10.1016/j.na.2009.11.013.  Google Scholar [43] S. Sakuntasathien, Global well-posedness for systems of nonlinear wave equations, Ph.D. Thesis, The University of Nebraska, Lincoln, 2008, 124 pp.  Google Scholar [44] R. E. Showalter, Monotone Operators in {B}anach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, Volume 49, American Mathematical Society, Providence, RI, 1997.  Google Scholar [45] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Volume 68, Applied Mathematical Sciences, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar
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