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December  2021, 10(4): 797-836. doi: 10.3934/eect.2020093

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 Published  December 2021 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 and Control Theory, 2021, 10 (4) : 797-836. doi: 10.3934/eect.2020093
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. 

[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.

[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. 

[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.

[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. 

[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.

[7]

V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Mathematics in Science and Engineering, Volume 190, Academic Press Inc., Boston, MA, 1993.

[8]

V. BarbuY. GuoM. A. Rammaha and D. Toundykov, Convex integrals on Sobolev spaces, J. Convex Anal., 19 (2012), 837-852. 

[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.

[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.

[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.

[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.

[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.

[14]

H. Brézis, Intégrales convexes dans les espaces de Sobolev, Israel J. Math., 13 (1973), 9-23.  doi: 10.1007/BF02760227.

[15]

J. CagnolI. LasieckaC. 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.

[16]

I. ChueshovM. 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.

[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.

[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.

[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.

[20]

I. ChueshovI. 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.

[21]

I. ChueshovI. 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.

[22]

I. D. Chueshov, Introduction to the Theory of Infinite-dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, AKTA, Kharkiv, 1999, 436 pp.

[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. 

[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.

[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.

[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.

[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.

[28]

Y. GuoM. A. RammahaS. SakuntasathienE. 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.

[29] M. S. Howe, Acoustics of Fluid-Structure Interactions, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9780511662898.
[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[39]

P. PeiM. 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.

[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.

[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.

[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.

[43]

S. Sakuntasathien, Global well-posedness for systems of nonlinear wave equations, Ph.D. Thesis, The University of Nebraska, Lincoln, 2008, 124 pp.

[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.

[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.

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. 

[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.

[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. 

[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.

[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. 

[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.

[7]

V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Mathematics in Science and Engineering, Volume 190, Academic Press Inc., Boston, MA, 1993.

[8]

V. BarbuY. GuoM. A. Rammaha and D. Toundykov, Convex integrals on Sobolev spaces, J. Convex Anal., 19 (2012), 837-852. 

[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.

[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.

[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.

[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.

[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.

[14]

H. Brézis, Intégrales convexes dans les espaces de Sobolev, Israel J. Math., 13 (1973), 9-23.  doi: 10.1007/BF02760227.

[15]

J. CagnolI. LasieckaC. 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.

[16]

I. ChueshovM. 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.

[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.

[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.

[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.

[20]

I. ChueshovI. 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.

[21]

I. ChueshovI. 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.

[22]

I. D. Chueshov, Introduction to the Theory of Infinite-dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, AKTA, Kharkiv, 1999, 436 pp.

[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. 

[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.

[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.

[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.

[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.

[28]

Y. GuoM. A. RammahaS. SakuntasathienE. 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.

[29] M. S. Howe, Acoustics of Fluid-Structure Interactions, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9780511662898.
[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[39]

P. PeiM. 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.

[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.

[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.

[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.

[43]

S. Sakuntasathien, Global well-posedness for systems of nonlinear wave equations, Ph.D. Thesis, The University of Nebraska, Lincoln, 2008, 124 pp.

[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.

[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.

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