Abstract
We consider a quasilinear parabolic Cauchy problem with spatial anisotropy of orthotropic type and study the spatial localization of solutions. Assuming that the initial datum is localized with respect to a coordinate having slow diffusion rate, we bound the corresponding directional velocity of the support along the flow. The expansion rate is shown to be optimal for large times.
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Acknowledgements
We thank J. L. Vázquez (U. Autónoma de Madrid) for a discussion on non-uniqueness phenomena for the Cauchy problem. S. Mosconi and V. Vespri are members of GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica). F. G. Düzgün is partially funded by Hacettepe University BAP through project FBI-2017-16260; S. Mosconi is partially funded by the grant PdR 2016-2018 - linea di intervento 2: “Metodi Variazionali ed Equazioni Differenziali” of the University of Catania.
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Düzgün, F.G., Mosconi, S. & Vespri, V. Anisotropic Sobolev embeddings and the speed of propagation for parabolic equations. J. Evol. Equ. 19, 845–882 (2019). https://doi.org/10.1007/s00028-019-00493-w
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DOI: https://doi.org/10.1007/s00028-019-00493-w