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Solution to nonlinear parabolic equations related to P-Laplacian

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Abstract

Consider the following Cauchy problem:

$\begin{gathered} u_t = div(|\nabla u^m |^{p - 2} \nabla u^m ),(x,t) \in S_T = \mathbb{R}^N \times (0,T), \hfill \\ u(x,0) = \mu ,x \in \mathbb{R}^N \hfill \\ \end{gathered} $

where 1 < p < 2, 1 < m < \(\tfrac{1} {{p - 1}} \) , and µ is a σ-finite measure in ℝN. By the Moser’s iteration method, the existence of the weak solution is obtained, provided that \(\tfrac{{(m + 1)N}} {{mN + 1}} < p \) . In contrast, if \(\tfrac{{(m + 1)N}} {{mN + 1}} \geqslant p \) , there is no solution to the Cauchy problem with an initial value δ(x), where δ(x) is the classical Dirac function.

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Authors and Affiliations

  1. School of Sciences, Jimei University, Xiamen, 361021, Fujian, China

    Huashui Zhan

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  1. Huashui Zhan
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Correspondence to Huashui Zhan.

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Project supported by the Fujian Provincial Natural Science Foundation of China (No. 2012J01011) and Pan Jinglong’s Natural Science Foundation of Jimei University (No. ZC2010019).

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Zhan, H. Solution to nonlinear parabolic equations related to P-Laplacian. Chin. Ann. Math. Ser. B 33, 767–782 (2012). https://doi.org/10.1007/s11401-012-0729-9

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  • DOI: https://doi.org/10.1007/s11401-012-0729-9

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