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Behavior of blow-up solutions for quasilinear parabolic equations

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Abstract

We study the quasilinear parabolic equation (|u|q − 1u)t − Δpu = 0 in a multidimensional domain (0; T) × Ω under the condition u(t; x) = f(t; x) on (0; T) × 𝜕Ω, where the boundary function f blows-up at a finite time T, i.e., f(t; x) → ∞ as t → T. For p ⩾ q > 0 and the boundary function f with power-like behavior, the upper bounds of weak solutions of the problem are obtained. The behavior of solutions at the transition from the case where p > q to p = q is investigated. A general approach within the method of energy estimates to such problems is described.

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slov’yans’k, Ukraine

    Yevgeniia A. Yevgenieva

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  1. Yevgeniia A. Yevgenieva
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Correspondence to Yevgeniia A. Yevgenieva.

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 17, No. 2, pp. 278–295 April–June, 2020.

The study was financially supported by the National Academy of Sciences of Ukraine in the frame of projects 0120U100177 and 0119U1020890.

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Yevgenieva, Y.A. Behavior of blow-up solutions for quasilinear parabolic equations. J Math Sci 249, 804–816 (2020). https://doi.org/10.1007/s10958-020-04974-z

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