{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T21:44:15Z","timestamp":1766180655748,"version":"3.37.3"},"reference-count":19,"publisher":"Wiley","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:p>The main contribution of this paper is the homogenization of the linear parabolic equation<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:msub><mml:mrow><mml:mo>\u2202<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:msub><mml:msup><mml:mrow><mml:mi>u<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u03b5<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mo>-<\/mml:mo><mml:mo>\u2207<\/mml:mo><mml:mo>\u00b7<\/mml:mo><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mo>\/<\/mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>\u03b5<\/mml:mi><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:mrow><mml:mo>,<\/mml:mo><mml:mo>.<\/mml:mo><mml:mo>.<\/mml:mo><mml:mo>.<\/mml:mo><mml:mo>,<\/mml:mo><mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mo>\/<\/mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>\u03b5<\/mml:mi><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:mrow><mml:mo>,<\/mml:mo><mml:mrow><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mo>\/<\/mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>\u03b5<\/mml:mi><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>r<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:mrow><mml:mo>,<\/mml:mo><mml:mo>.<\/mml:mo><mml:mo>.<\/mml:mo><mml:mo>.<\/mml:mo><mml:mo>,<\/mml:mo><mml:mrow><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mo>\/<\/mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>\u03b5<\/mml:mi><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>r<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:mrow><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mo>\u2207<\/mml:mo><mml:msup><mml:mrow><mml:mi>u<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u03b5<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mo>=<\/mml:mo><mml:mi>f<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>exhibiting an arbitrary finite number of both spatial and temporal scales. We briefly recall some fundamentals of multiscale convergence and provide a characterization of multiscale limits for gradients, in an evolution setting adapted to a quite general class of well-separated scales, which we name by jointly well-separated scales (see appendix for the proof). We proceed with a weaker version of this concept called very weak multiscale convergence. We prove a compactness result with respect to this latter type for jointly well-separated scales. This is a key result for performing the homogenization of parabolic problems combining rapid spatial and temporal oscillations such as the problem above. Applying this compactness result together with a characterization of multiscale limits of sequences of gradients we carry out the homogenization procedure, where we together with the homogenized problem obtain<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math>local problems, that is, one for each spatial microscale. To illustrate the use of the obtained result, we apply it to a case with three spatial and three temporal scales with<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:msub><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>=<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:math>,<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:msub><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>=<\/mml:mo><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:math>, and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mn mathvariant=\"normal\">0<\/mml:mn><mml:mo>&lt;<\/mml:mo><mml:msub><mml:mrow><mml:mi>r<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>&lt;<\/mml:mo><mml:msub><mml:mrow><mml:mi>r<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:math>.<\/jats:p>","DOI":"10.1155\/2014\/101685","type":"journal-article","created":{"date-parts":[[2014,2,24]],"date-time":"2014-02-24T16:01:24Z","timestamp":1393257684000},"page":"1-16","source":"Crossref","is-referenced-by-count":9,"title":["Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time"],"prefix":"10.1155","volume":"2014","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5702-7552","authenticated-orcid":true,"given":"Liselott","family":"Flod\u00e9n","sequence":"first","affiliation":[{"name":"Department of Quality Technology and Management, Mechanical Engineering and Mathematics, Mid Sweden University, S-83125 \u00d6stersund, Sweden"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6742-5781","authenticated-orcid":true,"given":"Anders","family":"Holmbom","sequence":"additional","affiliation":[{"name":"Department of Quality Technology and Management, Mechanical Engineering and Mathematics, Mid Sweden University, S-83125 \u00d6stersund, Sweden"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2942-6841","authenticated-orcid":true,"given":"Marianne","family":"Olsson Lindberg","sequence":"additional","affiliation":[{"name":"Department of Quality Technology and Management, Mechanical Engineering and Mathematics, Mid Sweden University, S-83125 \u00d6stersund, Sweden"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9984-2424","authenticated-orcid":true,"given":"Jens","family":"Persson","sequence":"additional","affiliation":[{"name":"Department of Quality Technology and Management, Mechanical Engineering and Mathematics, Mid Sweden University, S-83125 \u00d6stersund, 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