Solvability to some strongly degenerate parabolic problems

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:475

Article

Lavrentiev, Mikhail M.^1,2  Tani, Atusi³ 

[1] Russian Acad Sci, Siberian Div, Novosibirsk State Univ, Novosibirsk 630090, Russia

[2] Russian Acad Sci, Siberian Div, Inst Automat & Electrometry, Novosibirsk 630090, Russia

[3] Keio Univ, Dept Math, Yokohama, Kanagawa 2238522, Japan

关键词: Strongly degenerate parabolic equations;    Initial-boundary value problem;    Hyperbolic phenomena;    Global-in-time solutions;    Generalized distance;   

DOI  :  10.1016/j.jmaa.2019.02.056

来源: Elsevier

【 摘 要 】

Nonlinear parabolic equations of divergence form, u(t) = (phi(u) psi(u(x)))(x), are considered under the assumption that the material flux, phi(u) psi(v), is bounded for all values of arguments, u and v. In literature such equations have been referred to as strongly degenerate equations. This is due to the fact that the coefficient, phi(u) psi'(u(x)), of the second derivative, u(xx), can be arbitrarily small for large value of the gradient, u(x). The hyperbolic phenomena (unbounded growth of space derivatives within a finite time) have been established in literature for solutions to Cauchy problem for the above-mentioned equations. Accordingly one can expect a correct statement of the initial-boundary value problem for such equations only under additional assumptions on the problem data. In this paper we describe several restrictions, under which the initial-boundary value problems for strongly degenerate parabolic equations are well-posed. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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