Stability and localization of unbounded solutions of a nonlinear heat equation in a plane |
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Authors: | E. S. Kurkina I. M. Nikol’skii |
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Affiliation: | 1. Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia
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Abstract: | The article investigates unbounded solutions of the equation u t = div (u σgrad u) + u β in a plane. We numerically analyze the stability of two-dimensional self-similar solutions (structures) that increase with blowup. We confirm structural stability of the simple structure with a single maximum and metastability of complex structures. We prove structural stability of the radially symmetrical structure with a zero region at the center and investigate its attraction region. We study the effect of various perturbations of the initial function on the evolution of self-similar solutions. We further investigate how arbitrary compact-support initial distributions attain the self-similar mode, including distributions whose support is different from a disk. We show that the self-similar mode described by a simple radially symmetrical structure is achieved only in the central region, while the entire localization region does not have enough time to transform into a disk during blowup. We show for the first time that simple structures may merge into a complex structure, which evolves for a long time according to self-similar law. |
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