Singular solutions of parabolic p-Laplacian with absorption

    Authors

    X. F. Chen; Y. W. Qi;M. X. Wang

    Comments

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    Abbreviated Journal Title

    Trans. Am. Math. Soc.

    Keywords

    p-Laplacian; fast diffusion; absorption; fundamental solution; very; singular solution; POROUS-MEDIA EQUATION; LARGE TIME BEHAVIOR; HEAT-EQUATION; UNIQUENESS; Mathematics

    Abstract

    We consider, for p is an element of (1, 2) and q > 1, the p-Laplacian evolution equation with absorption u(t) = div(|del(u)|(p-2)del u)-u(q) in R-n x (0,infinity). We are interested in those solutions, which we call singular solutions, that are non- negative, non- trivial, continuous in R-n x [0, infinity) \ {(0, 0)}, and satisfy u(x, 0) = 0 for all x not equal 0. We prove the following: (i) When q > = p - 1 + p/n, there does not exist any such singular solution. (ii) When q < p - 1 + p/n, there exists, for every c > 0, a unique singular solution u = u(c) that satisfies integral R-n u(., t). c as t SE arrow 0. Also, uc NE arrow u(infinity) as c NE arrow 8, where u(infinity) is a singular solution that satisfies integral R-n u(infinity)(., t) -> infinity as t SE arrow 0. Furthermore, any singular solution is either u(infinity) or u(c) for some finite positive c.

    Journal Title

    Transactions of the American Mathematical Society

    Volume

    359

    Issue/Number

    11

    Publication Date

    1-1-2007

    Document Type

    Article

    Language

    English

    First Page

    5653

    Last Page

    5668

    WOS Identifier

    WOS:000249224700023

    ISSN

    0002-9947

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