# American Institute of Mathematical Sciences

December  2016, 11(4): 627-653. doi: 10.3934/nhm.2016012

## Homogenization of nonlinear dissipative hyperbolic problems exhibiting arbitrarily many spatial and temporal scales

 1 Department of Quality Technology and Management, Mechanical Engineering and Mathematics, Mid Sweden University, S-83125 Östersund, Sweden, Sweden

Received  May 2015 Revised  April 2016 Published  October 2016

This paper concerns the homogenization of nonlinear dissipative hyperbolic problems \begin{gather*} \partial _{tt}u^{\varepsilon }\left( x,t\right) -\nabla \cdot \left( a\left( \frac{x}{\varepsilon ^{q_{1}}},\ldots ,\frac{x}{\varepsilon ^{q_{n}}},\frac{t }{\varepsilon ^{r_{1}}},\ldots ,\frac{t}{\varepsilon ^{r_{m}}}\right) \nabla u^{\varepsilon }\left( x,t\right) \right) \\ +g\left( \frac{x}{\varepsilon ^{q_{1}}},\ldots ,\frac{x}{\varepsilon ^{q_{n}} },\frac{t}{\varepsilon ^{r_{1}}},\ldots ,\frac{t}{\varepsilon ^{r_{m}}} ,u^{\varepsilon }\left( x,t\right) ,\nabla u^{\varepsilon }\left( x,t\right) \right) =f(x,t) \end{gather*} where both the elliptic coefficient $a$ and the dissipative term $g$ are periodic in the $n+m$ first arguments where $n$ and $m$ may attain any non-negative integer value. The homogenization procedure is performed within the framework of evolution multiscale convergence which is a generalization of two-scale convergence to include several spatial and temporal scales. In order to derive the local problems, one for each spatial scale, the crucial concept of very weak evolution multiscale convergence is utilized since it allows less benign sequences to attain a limit. It turns out that the local problems do not involve the dissipative term $g$ even though the homogenized problem does and, due to the nonlinearity property, an important part of the work is to determine the effective dissipative term. A brief illustration of how to use the main homogenization result is provided by applying it to an example problem exhibiting six spatial and eight temporal scales in such a way that $a$ and $g$ have disparate oscillation patterns.
Citation: Liselott Flodén, Jens Persson. Homogenization of nonlinear dissipative hyperbolic problems exhibiting arbitrarily many spatial and temporal scales. Networks and Heterogeneous Media, 2016, 11 (4) : 627-653. doi: 10.3934/nhm.2016012
##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. [2] G. Allaire and M. Briane, Multiscale convergence and reiterated homogenization, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 297-342. doi: 10.1017/S0308210500022757. [3] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Existence and boundary stabilization of a nonlinear hyperbolic equation with time-dependent coefficients, Electron. J. Differential Equations, (1998), 21 pp. [4] M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano and J. S. Souza, Homogenization and uniform stabilization for a nonlinear hyperbolic equation in domains with holes of small capacity, Electron. J. Differential Equations, (2004), 19 pp. [5] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications, 17, The Clarendon Press, Oxford University Press, New York, 1999. [6] L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631. [7] L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245-265. doi: 10.1017/S0308210500032121. [8] L. Flodén, A. Holmbom, M. Olsson and J. Persson, Very weak multiscale convergence, Appl. Math. Lett., 23 (2010), 1170-1173. doi: 10.1016/j.aml.2010.05.005. [9] L. Flodén, A. Holmbom, M. Olsson Lindberg and J. Persson, Detection of scales of heterogeneity and parabolic homogenization applying very weak multiscale convergence, Ann. Funct. Anal., 2 (2011), 84-99. doi: 10.15352/afa/1399900264. [10] L. Flodén, A. Holmbom, M. Olsson Lindberg and J. Persson, Homogenization of parabolic equations with an arbitrary number of scales in both space and time, J. Appl. Math., 2014 (2014), Art. ID 101685, 16 pp. doi: 10.1155/2014/101685. [11] L. Flodén and M. Olsson, Reiterated homogenization of some linear and nonlinear monotone parabolic operators, Can. Appl. Math. Q., 14 (2006), 149-183. [12] L. Flodén and M. Olsson, Homogenization of some parabolic operators with several time scales, Appl. Math., 52 (2007), 431-446. doi: 10.1007/s10492-007-0025-2. [13] M. Hairer, E. Pardoux and A. Piatnitski, Random homogenisation of a highly oscillatory singular potential, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 571-605. doi: 10.1007/s40072-013-0018-y. [14] A. Holmbom, Homogenization of parabolic equations. An alternative approach and some corrector-type results, Appl. Math., 42 (1997), 321-343. doi: 10.1023/A:1023049608047. [15] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969. [16] D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence, Int. J. Pure Appl. Math., 2 (2002), 35-86. [17] A. K. Nandakumaran and M. Rajesh, Homogenization of a nonlinear degenerate parabolic differential equation, Electron. J. Differential Equations, (2001), 19 pp. [18] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043. [19] G. Nguetseng, Deterministic homogenization of a semilinear elliptic partial differential equation of order $2m$, Math. Rep. (Bucur.), 8 (2006), 167-195. [20] G. Nguetseng, H. Nnang and N. Svanstedt, $G$-convergence and homogenization of monotone damped hyperbolic equations, Banach J. Math. Anal., 4 (2010), 100-115. doi: 10.15352/bjma/1272374674. [21] G. Nguetseng, H. Nnang and N. Svanstedt, Asymptotic analysis for a weakly damped wave equation with application to a problem arising in elasticity, J. Funct. Spaces Appl., 8 (2010), 17-54. doi: 10.1155/2010/291670. [22] G. Nguetseng, H. Nnang and N. Svanstedt, Deterministic homogenization of quasilinear damped hyperbolic equations, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 1823-1850. doi: 10.1016/S0252-9602(11)60364-0. [23] G. Nguetseng and J. L. Woukeng, Deterministic homogenization of parabolic monotone operators with time dependent coefficients, Electron. J. Differential Equations, (2004), 23 pp. [24] G. Nguetseng and J. L. Woukeng, $\Sigma$-convergence of nonlinear parabolic operators, Nonlinear Anal., 66 (2007), 968-1004. doi: 10.1016/j.na.2005.12.035. [25] H. Nnang, Deterministic homogenization of weakly damped nonlinear hyperbolic-parabolic equations, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 539-574. doi: 10.1007/s00030-011-0142-1. [26] L. S. Pankratov and I. D. Chueshov, Averaging of attractors of nonlinear hyperbolic equations with asymptotically degenerate coefficients, Mat. Sb., 190 (1999), 99-126. doi: 10.1070/SM1999v190n09ABEH000427. [27] E. Pardoux and A. Piatnitski, Homogenization of a singular random one-dimensional PDE with time-varying coefficients, Ann. Probab., 40 (2012), 1316-1356. doi: 10.1214/11-AOP650. [28] J. Persson, Selected Topics in Homogenization, Mid Sweden University Doctoral Thesis 127, 2012. (URL: http://www.diva-portal.org/smash/get/diva2:527223/FULLTEXT01.pdf.) [29] J. Persson, Homogenization of monotone parabolic problems with several temporal scales, Appl. Math., 57 (2012), 191-214. doi: 10.1007/s10492-012-0013-z. [30] N. Svanstedt, Convergence of quasi-linear hyperbolic equations, J. Hyperbolic Differ. Equ., 4 (2007), 655-677. doi: 10.1142/S0219891607001306. [31] N. Svanstedt and J. L. Woukeng, Periodic homogenization of strongly nonlinear reaction-diffusion equations with large reaction terms, Appl. Anal., 92 (2013), 1357-1378. doi: 10.1080/00036811.2012.678334. [32] M. I. Vishik and B. Fidler, Quantative averaging of global attractors of hyperbolic wave equations with rapidly oscillating coefficients, Uspekhi Mat. Nauk., 57 (2002), 75-94. doi: 10.1070/RM2002v057n04ABEH000534. [33] J. L. Woukeng and D. Dongo, Multiscale homogenization of nonlinear hyperbolic equations with several time scales, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 843-856. doi: 10.1016/S0252-9602(11)60281-6. [34] E. Zeidler, Nonlinear Functional Analysis and its Applications IIA. Linear Monotone Operators, Springer Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

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##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. [2] G. Allaire and M. Briane, Multiscale convergence and reiterated homogenization, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 297-342. doi: 10.1017/S0308210500022757. [3] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Existence and boundary stabilization of a nonlinear hyperbolic equation with time-dependent coefficients, Electron. J. Differential Equations, (1998), 21 pp. [4] M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano and J. S. Souza, Homogenization and uniform stabilization for a nonlinear hyperbolic equation in domains with holes of small capacity, Electron. J. Differential Equations, (2004), 19 pp. [5] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications, 17, The Clarendon Press, Oxford University Press, New York, 1999. [6] L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631. [7] L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245-265. doi: 10.1017/S0308210500032121. [8] L. Flodén, A. Holmbom, M. Olsson and J. Persson, Very weak multiscale convergence, Appl. Math. Lett., 23 (2010), 1170-1173. doi: 10.1016/j.aml.2010.05.005. [9] L. Flodén, A. Holmbom, M. Olsson Lindberg and J. Persson, Detection of scales of heterogeneity and parabolic homogenization applying very weak multiscale convergence, Ann. Funct. Anal., 2 (2011), 84-99. doi: 10.15352/afa/1399900264. [10] L. Flodén, A. Holmbom, M. Olsson Lindberg and J. Persson, Homogenization of parabolic equations with an arbitrary number of scales in both space and time, J. Appl. Math., 2014 (2014), Art. ID 101685, 16 pp. doi: 10.1155/2014/101685. [11] L. Flodén and M. Olsson, Reiterated homogenization of some linear and nonlinear monotone parabolic operators, Can. Appl. Math. Q., 14 (2006), 149-183. [12] L. Flodén and M. Olsson, Homogenization of some parabolic operators with several time scales, Appl. Math., 52 (2007), 431-446. doi: 10.1007/s10492-007-0025-2. [13] M. Hairer, E. Pardoux and A. Piatnitski, Random homogenisation of a highly oscillatory singular potential, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 571-605. doi: 10.1007/s40072-013-0018-y. [14] A. Holmbom, Homogenization of parabolic equations. An alternative approach and some corrector-type results, Appl. Math., 42 (1997), 321-343. doi: 10.1023/A:1023049608047. [15] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969. [16] D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence, Int. J. Pure Appl. Math., 2 (2002), 35-86. [17] A. K. Nandakumaran and M. Rajesh, Homogenization of a nonlinear degenerate parabolic differential equation, Electron. J. Differential Equations, (2001), 19 pp. [18] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043. [19] G. Nguetseng, Deterministic homogenization of a semilinear elliptic partial differential equation of order $2m$, Math. Rep. (Bucur.), 8 (2006), 167-195. [20] G. Nguetseng, H. Nnang and N. Svanstedt, $G$-convergence and homogenization of monotone damped hyperbolic equations, Banach J. Math. Anal., 4 (2010), 100-115. doi: 10.15352/bjma/1272374674. [21] G. Nguetseng, H. Nnang and N. Svanstedt, Asymptotic analysis for a weakly damped wave equation with application to a problem arising in elasticity, J. Funct. Spaces Appl., 8 (2010), 17-54. doi: 10.1155/2010/291670. [22] G. Nguetseng, H. Nnang and N. Svanstedt, Deterministic homogenization of quasilinear damped hyperbolic equations, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 1823-1850. doi: 10.1016/S0252-9602(11)60364-0. [23] G. Nguetseng and J. L. Woukeng, Deterministic homogenization of parabolic monotone operators with time dependent coefficients, Electron. J. Differential Equations, (2004), 23 pp. [24] G. Nguetseng and J. L. Woukeng, $\Sigma$-convergence of nonlinear parabolic operators, Nonlinear Anal., 66 (2007), 968-1004. doi: 10.1016/j.na.2005.12.035. [25] H. Nnang, Deterministic homogenization of weakly damped nonlinear hyperbolic-parabolic equations, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 539-574. doi: 10.1007/s00030-011-0142-1. [26] L. S. Pankratov and I. D. Chueshov, Averaging of attractors of nonlinear hyperbolic equations with asymptotically degenerate coefficients, Mat. Sb., 190 (1999), 99-126. doi: 10.1070/SM1999v190n09ABEH000427. [27] E. Pardoux and A. Piatnitski, Homogenization of a singular random one-dimensional PDE with time-varying coefficients, Ann. Probab., 40 (2012), 1316-1356. doi: 10.1214/11-AOP650. [28] J. Persson, Selected Topics in Homogenization, Mid Sweden University Doctoral Thesis 127, 2012. (URL: http://www.diva-portal.org/smash/get/diva2:527223/FULLTEXT01.pdf.) [29] J. Persson, Homogenization of monotone parabolic problems with several temporal scales, Appl. Math., 57 (2012), 191-214. doi: 10.1007/s10492-012-0013-z. [30] N. Svanstedt, Convergence of quasi-linear hyperbolic equations, J. Hyperbolic Differ. Equ., 4 (2007), 655-677. doi: 10.1142/S0219891607001306. [31] N. Svanstedt and J. L. Woukeng, Periodic homogenization of strongly nonlinear reaction-diffusion equations with large reaction terms, Appl. Anal., 92 (2013), 1357-1378. doi: 10.1080/00036811.2012.678334. [32] M. I. Vishik and B. Fidler, Quantative averaging of global attractors of hyperbolic wave equations with rapidly oscillating coefficients, Uspekhi Mat. Nauk., 57 (2002), 75-94. doi: 10.1070/RM2002v057n04ABEH000534. [33] J. L. Woukeng and D. Dongo, Multiscale homogenization of nonlinear hyperbolic equations with several time scales, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 843-856. doi: 10.1016/S0252-9602(11)60281-6. [34] E. Zeidler, Nonlinear Functional Analysis and its Applications IIA. Linear Monotone Operators, Springer Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.
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