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| We present results on the properties of localization in time and space for solutions of doubly nonlinear parabolic equations with anisotropic and variable growth conditions. We derive the ranges of the variable exponents of nonlinearity where the solutions vanish or blowup in a finite time and study the asymptotic behavior of solutions for large time. Sufficient conditions of finite speed of propagation in space are established. We also study the effect of nonpropagation of disturbances from the data in certain directions due to the anisotropy of the diffusion operator. The results were obtained in collaboration with S.Antontsev. The presentation partially follows the papers
1. S.Antontsev, S.Shmarev Doubly degenerate parabolic equations with variable nonlinearity II: Blow-up and extinction in a finite time. Nonlinear Anal. 95 (2014), 483--498.
2. S.Antontsev, S.Shmarev. Localization of solutions of anisotropic parabolic equations. Nonlinear Anal. 71 (2009), no.12, e725--e737. |
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