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Critical exponent for the parabolic equation u t = Δu m + h (t)u p in a cone

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Abstract

In this paper, we study the initial-boundary value problem of porous medium equation u t = Δu m + h (t)u p in a cone D = (0,∞) × Ω, where h(t) ∼ t σ. Let ω 1 denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l 2+(n−2)l = ω 1. We prove that if \(m < p \leqslant m + \tfrac{{2(\sigma + 1)}} {{n + l}} + \sigma (m - 1) \) , then the problem has no global nonnegative solutions for any nonnegative u 0 unless u 0 = 0; if \(p > m + \tfrac{{2(\sigma + 1)}} {{n + l}} + \sigma (m - 1) \) , then the problem has global solutions for some u 0 ≥ 0.

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  1. Department of Mathematics, Jilin University, Changchun, 130012, P. R. China

    Chang Chun Liu & Song Zhe Lian

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  1. Chang Chun Liu
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  2. Song Zhe Lian
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Correspondence to Chang Chun Liu.

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Supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry

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Liu, C.C., Lian, S.Z. Critical exponent for the parabolic equation u t = Δu m + h (t)u p in a cone. Acta. Math. Sin.-English Ser. 28, 1623–1632 (2012). https://doi.org/10.1007/s10114-012-0026-2

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  • DOI: https://doi.org/10.1007/s10114-012-0026-2

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