Approximation of nonlinear wave equations with nonstandard anisotropic growth conditions

Haehnle, Jonas; Prohl, Andreas

doi:10.1090/S0025-5718-09-02231-5

Approximation of nonlinear wave equations with nonstandard anisotropic growth conditions
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by Jonas Haehnle and Andreas Prohl;

Math. Comp. 79 (2010), 189-208

DOI: https://doi.org/10.1090/S0025-5718-09-02231-5

Published electronically: July 1, 2009

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Abstract:

Weak solutions for nonlinear wave equations involving the $p(\mathbf {x})$-Laplacian, for $p: \Omega \rightarrow (1,\infty )$ are constructed as appropriate limits of solutions of an implicit finite element discretization of the problem. A simple fixed-point scheme with appropriate stopping criteria is proposed to conclude convergence for all discretization, regularization, perturbation, and stopping parameters tending to zero. Computational experiments are included to motivate interesting dynamics, such as blowup, and asymptotic decay behavior.

References

Stanislav N. Antontsev and Sergei I. Shmarev, Anisotropic parabolic equations with variable nonlinearity, Preprint, URL: http://cmaf.ptmat.fc.ul.pt/preprints/preprints.html, 2007.
S. Antontsev and S. Shmarev, Parabolic equations with anisotropic nonstandard growth conditions, Free boundary problems, Internat. Ser. Numer. Math., vol. 154, Birkhäuser, Basel, 2007, pp. 33–44. MR 2305342, DOI 10.1007/978-3-7643-7719-9_{4}
Hajer Bahouri and Jalal Shatah, Decay estimates for the critical semilinear wave equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 15 (1998), no. 6, 783–789 (English, with English and French summaries). MR 1650958, DOI 10.1016/S0294-1449(99)80005-5
J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser. (2) 28 (1977), no. 112, 473–486. MR 473484, DOI 10.1093/qmath/28.4.473
J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, 31–52. Partial differential equations and applications. MR 2026182, DOI 10.3934/dcds.2004.10.31
Abbès Benaissa and Soufiane Mokeddem, Decay estimates for the wave equation of $p$-Laplacian type with dissipation of $m$-Laplacian type, Math. Methods Appl. Sci. 30 (2007), no. 2, 237–247. MR 2285123, DOI 10.1002/mma.789
Piotr Bizoń, Threshold behavior for nonlinear wave equations, J. Nonlinear Math. Phys. 8 (2001), no. suppl., 35–41. Nonlinear evolution equations and dynamical systems (Kolimbary, 1999). MR 1821505, DOI 10.2991/jnmp.2001.8.s.7
Piotr Bizoń, Tadeusz Chmaj, and Zbisław Tabor, On blowup for semilinear wave equations with a focusing nonlinearity, Nonlinearity 17 (2004), no. 6, 2187–2201. MR 2097671, DOI 10.1088/0951-7715/17/6/009
M. Crouzeix and V. Thomée, The stability in $L_p$ and $W^1_p$ of the $L_2$-projection onto finite element function spaces, Math. Comp. 48 (1987), no. 178, 521–532. MR 878688, DOI 10.1090/S0025-5718-1987-0878688-2
Lars Diening, Lebesgue and Sobolev spaces with variable exponent, 2007, Habilitation Thesis.
Xian-Ling Fan and Qi-Hu Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), no. 8, 1843–1852. MR 1954585, DOI 10.1016/S0362-546X(02)00150-5
Xianling Fan and Dun Zhao, On the generalized Orlicz-Sobolev space $W^ {k,p(x)}(\Omega )$, J. Gansu Educ. College 12 (1998), no. 1, 1–6.
Xianling Fan and Dun Zhao, On the spaces $L^{p(x)}(\Omega )$ and $W^{m,p(x)}(\Omega )$, J. Math. Anal. Appl. 263 (2001), no. 2, 424–446. MR 1866056, DOI 10.1006/jmaa.2000.7617
Victor A. Galaktionov and Stanislav I. Pohozaev, Blow-up, critical exponents and asymptotic spectra for nonlinear hyperbolic equations, Preprint, URL: ftp://ftp.maths.bath.ac.uk/pub/preprints/maths0010.ps.Z, 2000.
Hongjun Gao and To Fu Ma, Global solutions for a nonlinear wave equation with the $p$-Laplacian operator, Electron. J. Qual. Theory Differ. Equ. , posted on (1999), No. 11, 13. MR 1718343, DOI 10.14232/ejqtde.1999.1.11
Hongjun Gao and Hui Zhang, Global nonexistence of the solutions for a nonlinear wave equation with the $q$-Laplacian operator, J. Partial Differential Equations 20 (2007), no. 1, 71–79. MR 2317126
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, DOI 10.1006/jdeq.1994.1051
Ryo Ikehata, Tokio Matsuyama, and Mitsuhiro Nakao, Global solutions to the initial-boundary value problem for the quasilinear viscoelastic wave equation with a perturbation, Funkcial. Ekvac. 40 (1997), no. 2, 293–312. MR 1480280
Fritz John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28 (1979), no. 1-3, 235–268. MR 535704, DOI 10.1007/BF01647974
V. K. Kalantarov and O. A. Ladyženskaja, Formation of collapses in quasilinear equations of parabolic and hyperbolic types, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 69 (1977), 77–102, 274 (Russian, with English summary). MR 604036
Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation, Preprint, URL: http://arxiv.org/abs/math.AP/0610801, 2007.
Howard A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt}=-Au+{\cal F}(u)$, Trans. Amer. Math. Soc. 192 (1974), 1–21. MR 344697, DOI 10.1090/S0002-9947-1974-0344697-2
Howard A. Levine and Grozdena Todorova, Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy, Proc. Amer. Math. Soc. 129 (2001), no. 3, 793–805. MR 1792187, DOI 10.1090/S0002-9939-00-05743-9
Hans Lindblad, Counterexamples to local existence for nonlinear wave equations, Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1994) École Polytech., Palaiseau, 1994, pp. Exp. No. X, 5. MR 1298681
Salim A. Messaoudi and Belkacem Said Houari, Global non-existence of solutions of a class of wave equations with non-linear damping and source terms, Math. Methods Appl. Sci. 27 (2004), no. 14, 1687–1696. MR 2089156, DOI 10.1002/mma.522
Mitsuhiro Nakao, Energy decay for the quasilinear wave equation with viscosity, Math. Z. 219 (1995), no. 2, 289–299. MR 1337222, DOI 10.1007/BF02572366
Stanislav Pohozaev and Laurent Véron, Blow-up results for nonlinear hyperbolic inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 2, 393–420. MR 1784180
Andreas Prohl, Convergence of a finite element-based space-time discretization in elastodynamics, SIAM J. Numer. Anal. 46 (2008), no. 5, 2469–2483. MR 2421043, DOI 10.1137/070685166
Andreas Prohl and Isabelle Weindl, Convergence of an implicit finite element discretization for a class of parabolic equations with nonstandard anisotropic growth conditions, Preprint, URL: http://na.uni-tuebingen.de/preprints.shtml, 2007.
S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integral Transforms Spec. Funct. 16 (2005), no. 5-6, 461–482. MR 2138062, DOI 10.1080/10652460412331320322
Zhijian Yang, Blowup of solutions for a class of non-linear evolution equations with non-linear damping and source terms, Math. Methods Appl. Sci. 25 (2002), no. 10, 825–833. MR 1910734, DOI 10.1002/mma.312
Qihu Zhang, Existence of blow-up solutions to a class of $p(x)$-Laplacian problems, Int. J. Math. Anal. (Ruse) 1 (2007), no. 1-4, 79–88. MR 2340930
Dun Zhao, W. J. Quang, and Xian Ling Fan, On generalized Orlicz spaces $L^ {p(x)}(\Omega )$, J. Gansu Sci. 9 (1996), no. 2, 1–7.