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Systems of nonlinear wave equations with damping and supercritical boundary and interior sources


Authors: Yanqiu Guo and Mohammad A. Rammaha
Journal: Trans. Amer. Math. Soc. 366 (2014), 2265-2325
MSC (2010): Primary 35L05, 35L20; Secondary 58J45
DOI: https://doi.org/10.1090/S0002-9947-2014-05772-3
Published electronically: January 7, 2014
MathSciNet review: 3165639
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Abstract: We consider the local and global well-posedness of the coupled nonlinear wave equations

$\displaystyle u_{tt}-\Delta u+g_1(u_t)=f_1(u,v),$    
$\displaystyle v_{tt}-\Delta v+g_2(v_t)=f_2(u,v)$    

in a bounded domain $ \Omega \subset \mathbb{R}^n$ with Robin and Dirichlét boundary conditions on $ u$ and $ v$ respectively. The nonlinearities $ f_1(u,v)$ and $ f_2(u,v)$ have supercritical exponents representing strong sources, while $ g_1(u_t)$ and $ g_2(v_t)$ act as damping. In addition, the boundary condition also contains a nonlinear source and a damping term. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data.

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  • [1] Keith Agre and M. A. Rammaha, Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations 19 (2006), no. 11, 1235-1270. MR 2278006 (2007i:35165)
  • [2] Claudianor O. Alves, Marcelo M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, and Daniel Toundykov, On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms, Discrete Contin. Dyn. Syst. Ser. S 2 (2009), no. 3, 583-608. MR 2525769 (2011a:35317), https://doi.org/10.3934/dcdss.2009.2.583
  • [3] ạng iñh Áng and A. Pham Ngoc Dinh, Mixed problem for some semilinear wave equation with a nonhomogeneous condition, Nonlinear Anal. 12 (1988), no. 6, 581-592. MR 945666 (89h:35207), https://doi.org/10.1016/0362-546X(88)90016-8
  • [4] Viorel Barbu, Yanqiu Guo, Mohammad A. Rammaha, and Daniel Toundykov, Convex integrals on Sobolev spaces, J. Convex Anal. 19 (2012), no. 3, 837-852. MR 3013761
  • [5] Viorel Barbu, Analysis and control of nonlinear infinite-dimensional systems, Mathematics in Science and Engineering, vol. 190, Academic Press Inc., Boston, MA, 1993. MR 1195128 (93j:49002)
  • [6] Lorena 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), no. 12, e560-e575. MR 2671860 (2011d:35320), https://doi.org/10.1016/j.na.2008.11.062
  • [7] Lorena Bociu and Irena Lasiecka, Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping, Appl. Math. (Warsaw) 35 (2008), no. 3, 281-304. MR 2453534 (2009h:35273), https://doi.org/10.4064/am35-3-3
  • [8] Lorena Bociu and Irena Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst. 22 (2008), no. 4, 835-860. MR 2434972 (2010b:35305), https://doi.org/10.3934/dcds.2008.22.835
  • [9] Lorena Bociu and Irena Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations 249 (2010), no. 3, 654-683. MR 2646044 (2011g:35251), https://doi.org/10.1016/j.jde.2010.03.009
  • [10] Haïm Brézis, Intégrales convexes dans les espaces de Sobolev, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 9-23 (1973) (French, with English summary). MR 0341077 (49 #5827)
  • [11] Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti, and Irena Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping--source interaction, J. Differential Equations 236 (2007), no. 2, 407-459. MR 2322019 (2008c:35189), https://doi.org/10.1016/j.jde.2007.02.004
  • [12] Igor Chueshov, Matthias Eller, and Irena Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations 27 (2002), no. 9-10, 1901-1951. MR 1941662 (2003m:35034), https://doi.org/10.1081/PDE-120016132
  • [13] Igor Chueshov and Irena Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc. 195 (2008), no. 912, viii+183. MR 2438025 (2009i:37200)
  • [14] Vladimir Georgiev and Grozdena Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations 109 (1994), no. 2, 295-308. MR 1273304 (95b:35141), https://doi.org/10.1006/jdeq.1994.1051
  • [15] Yanqiu Guo and Mohammad A. Rammaha, Blow-up of solutions to systems of nonlinear wave equations with supercritical sources, Applicable Analysis (2013), Vol. 92, No. 6, 1101-1115.
  • [16] Yanqiu Guo and Mohammad A. Rammaha, Global existence and decay of energy to systems of wave equations with damping and supercritical sources, Z. Angew. Math. Phys. 64 (2013), 621-658. MR 3068842
  • [17] Ryo Ikehata, Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal. 27 (1996), no. 10, 1165-1175. MR 1407454 (97i:35117), https://doi.org/10.1016/0362-546X(95)00119-G
  • [18] Konrad Jörgens, Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77 (1961), 295-308 (German). MR 0130462 (24 #A323)
  • [19] Herbert Koch and Irena Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, Evolution equations, semigroups and functional analysis (Milano, 2000), Progr. Nonlinear Differential Equations Appl., vol. 50, Birkhäuser, Basel, 2002, pp. 197-216. MR 1944164 (2003j:35199)
  • [20] Hideo Kubo and Masahito Ohta, Critical blowup for systems of semilinear wave equations in low space dimensions, J. Math. Anal. Appl. 240 (1999), no. 2, 340-360. MR 1731649 (2001f:35266), https://doi.org/10.1006/jmaa.1999.6585
  • [21] John E. Lagnese, Boundary stabilization of thin plates, SIAM Studies in Applied Mathematics, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1061153 (91k:73001)
  • [22] V. Lakshmikantham and S. Leela, Differential and integral inequalities: Theory and applications. Vol. I: Ordinary differential equations, Academic Press, New York, 1969, Mathematics in Science and Engineering, Vol. 55-I. MR 0379933 (52:837)
  • [23] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations 6 (1993), no. 3, 507-533. MR 1202555 (94c:35129)
  • [24] 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. MR 1108480 (92e:35102), https://doi.org/10.1007/BF01765322
  • [25] Irena Lasiecka, Mathematical control theory of coupled PDEs, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 75, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. MR 1879543 (2003a:93002)
  • [26] Irena Lasiecka and Daniel Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms, Nonlinear Anal. 64 (2006), no. 8, 1757-1797. MR 2197360 (2006k:35189), https://doi.org/10.1016/j.na.2005.07.024
  • [27] Howard A. Levine and James Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal. 137 (1997), no. 4, 341-361. MR 1463799 (99b:34110), https://doi.org/10.1007/s002050050032
  • [28] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), no. 3-4, 273-303. MR 0402291 (53 #6112)
  • [29] David R. Pitts and Mohammad A. Rammaha, Global existence and non-existence theorems for nonlinear wave equations, Indiana Univ. Math. J. 51 (2002), no. 6, 1479-1509. MR 1948457 (2003j:35219), https://doi.org/10.1512/iumj.2002.51.2215
  • [30] Petronela Radu, Weak solutions to the Cauchy problem of a semilinear wave equation with damping and source terms, Adv. Differential Equations 10 (2005), no. 11, 1261-1300. MR 2175336 (2007a:35110)
  • [31] Mohammad A. Rammaha, The influence of damping and source terms on solutions of nonlinear wave equations, Bol. Soc. Parana. Mat. (3) 25 (2007), no. 1-2, 77-90. MR 2379676 (2008k:35328), https://doi.org/10.5269/bspm.v25i1-2.7427
  • [32] Mohammad A. Rammaha and Sawanya Sakuntasathien, Critically and degenerately damped systems of nonlinear wave equations with source terms, Appl. Anal. 89 (2010), no. 8, 1201-1227. MR 2681440 (2011e:35198), https://doi.org/10.1080/00036811.2010.483423
  • [33] Mohammad A. Rammaha and Sawanya Sakuntasathien, Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms, Nonlinear Anal. 72 (2010), no. 5, 2658-2683. MR 2577827 (2011a:35320), https://doi.org/10.1016/j.na.2009.11.013
  • [34] Mohammad A. Rammaha and Theresa A. Strei, Global existence and nonexistence for nonlinear wave equations with damping and source terms, Trans. Amer. Math. Soc. 354 (2002), no. 9, 3621-3637 (electronic). MR 1911514 (2003f:35214), https://doi.org/10.1090/S0002-9947-02-03034-9
  • [35] Michael Reed, Abstract non-linear wave equations, Lecture Notes in Mathematics, Vol. 507, Springer-Verlag, Berlin, 1976. MR 0605679 (58 #29290)
  • [36] Sawanya Sakuntasathien, Global well-posedness for systems of nonlinear wave equations, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)-The University of Nebraska - Lincoln. MR 2711412
  • [37] R. Seeley, Interpolation in $ L^{p}$ with boundary conditions, Studia Math. 44 (1972), 47-60. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I. MR 0315432 (47 #3981)
  • [38] Irving Segal, Non-linear semi-groups, Ann. of Math. (2) 78 (1963), 339-364. MR 0152908 (27 #2879)
  • [39] R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs, vol. 49, American Mathematical Society, Providence, RI, 1997. MR 1422252 (98c:47076)
  • [40] Daniel Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), no. 1, 185-206. MR 1633000 (99e:35129)
  • [41] Roger Temam, Navier-Stokes equations, 3rd ed., Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. Theory and numerical analysis; With an appendix by F. Thomasset. MR 769654 (86m:76003)
  • [42] Daniel Toundykov, Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions, Nonlinear Anal. 67 (2007), no. 2, 512-544. MR 2317185 (2008f:35257), https://doi.org/10.1016/j.na.2006.06.007
  • [43] Enzo Vitillaro, Some new results on global nonexistence and blow-up for evolution problems with positive initial energy, Rend. Istit. Mat. Univ. Trieste 31 (2000), no. suppl. 2, 245-275. Workshop on Blow-up and Global Existence of Solutions for Parabolic and Hyperbolic Problems (Trieste, 1999). MR 1800451 (2001j:35210)
  • [44] Enzo Vitillaro, A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasg. Math. J. 44 (2002), no. 3, 375-395. MR 1956547 (2003k:35169), https://doi.org/10.1017/S0017089502030045

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Additional Information

Yanqiu Guo
Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588
Address at time of publication: Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
Email: s-yguo2@math.unl.edu, yanqiu.guo@weizmann.ac.il

Mohammad A. Rammaha
Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588
Email: mrammaha1@math.unl.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-05772-3
Keywords: Wave equations, damping and source terms, weak solutions, energy identity, nonlinear semigroups, monotone operators
Received by editor(s): July 4, 2011
Received by editor(s) in revised form: December 5, 2011
Published electronically: January 7, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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