Classification of singular solutions of porous media equations with absorption

    Authors

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

    Comments

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    Abbreviated Journal Title

    Proc. R. Soc. Edinb. Sect. A-Math.

    Keywords

    HEAT-EQUATION; PARABOLIC EQUATIONS; Mathematics, Applied; Mathematics

    Abstract

    We consider, for m is an element of (0, 1) and q > 1, the porous media equation with absorption u(t) = Delta u(m) - 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 results. When q > = m + 2/n, there does not exist any such singular solution. When q < m + 2/n, there exists, for every c > 0, a unique singular solution u = u((c)), called the fundamental solution with initial mass c, which satisfies f(Rn) u(center dot, t) - > c as t SE arrow 0. Also, there exists a unique singular solution u = u(infinity) called the very singular solution, which satisfies f(Rn) u(infinity) (center dot, t) - > infinity as t SE arrow 0. In addition, any singular solution is either u(infinity) or u((c)) for some finite positive c, u((c1)) < u((c2)) when c(1) < c(2), and u(,) NE arrow u(infinity) as c NE arrow infinity. Furthermore, u(infinity) is self-similar in the sense that u(infinity)(x, t) = t(-alpha) w(vertical bar x vertical bar t(-alpha beta)) for alpha = 1/(q - 1),beta = 1/2(q - m), and some smooth function w defined on [0, infinity), so that integral(Rn) u(infinity)(n beta) (center dot, t) is a finite positive constant independent of t > 0.

    Journal Title

    Proceedings of the Royal Society of Edinburgh Section a-Mathematics

    Volume

    135

    Publication Date

    1-1-2005

    Document Type

    Article

    Language

    English

    First Page

    563

    Last Page

    584

    WOS Identifier

    WOS:000230595200006

    ISSN

    0308-2105

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