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Homogenization Of Nonlinear Dissipative Hyperbolic Problems Exhibiting Arbitrarily Many Spatial And Temporal Scales
Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.ORCID iD: 0000-0001-9984-2424
2016 (English)In: Networks and Heterogeneous Media, ISSN 1556-1801, E-ISSN 1556-181X, Vol. 11, no 4, 627-653 p.Article in journal (Refereed) Published
Abstract [en]

This paper concerns the homogenization of nonlinear dissipative hyperbolic problems partial derivative ttu(epsilon) (x, t) - del . (a(x/epsilon(q1),..., x/epsilon(qn), t/epsilon(r1),..., t/epsilon(rm)) del u(epsilon) (x, t)) +g (x/epsilon(r1),..., x/epsilon(rn), t/epsilon(r1), u(epsilon) (x, t), del u(epsilon) (x, t)) = f (x, t)

where both the elliptic coefficient a and the dissipative term a 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.

Place, publisher, year, edition, pages
2016. Vol. 11, no 4, 627-653 p.
Keyword [en]
Homogenization theory, nonlinear dissipative hyperbolic problems, multiscale convergence
National Category
Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-29510DOI: 10.3934/nhm.2016012ISI: 000387912500004Scopus ID: 2-s2.0-85002870806OAI: oai:DiVA.org:miun-29510DiVA: diva2:1055533
Available from: 2016-12-12 Created: 2016-12-12 Last updated: 2017-11-29Bibliographically approved

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Flodén, LiselottPersson, Jens

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