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Singular solutions of parabolic $ p$-Laplacian with absorption


Authors: Xinfu Chen, Yuanwei Qi and Mingxin Wang
Journal: Trans. Amer. Math. Soc. 359 (2007), 5653-5668
MSC (2000): Primary 35K65, 35K15
DOI: https://doi.org/10.1090/S0002-9947-07-04336-X
Published electronically: May 8, 2007
MathSciNet review: 2327046
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider, for $ p\in(1,2)$ and $ q>1$, the $ p$-Laplacian evolution equation with absorption

$\displaystyle u_t = \hbox{div}\, ( \vert\nabla u\vert^{p-2} \nabla u) - u^q \quad \hbox{in } \mathbb{R}^n \times (0,\infty).$

We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in $ \mathbb{R}^n\times[0,\infty)\setminus\{(0,0)\}$, and satisfy $ u(x,0)=0$ for all $ x\not=0$. We prove the following:
(i)
When $ q\geq 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 $ \int_{\mathbb{R}^n}u(\cdot,t)\to c$ as $ t\searrow 0$.

Also, $ u_c\nearrow u_\infty$ as $ c\nearrow \infty$, where $ u_\infty$ is a singular solution that satisfies $ \int_{\mathbb{R}^n} u_\infty(\cdot,t) \to \infty$ as $ t\searrow 0$.

Furthermore, any singular solution is either $ u_\infty$ or $ u_c$ for some finite positive $ c$.


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Additional Information

Xinfu Chen
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: xinfu@pitt.edu

Yuanwei Qi
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: yqi@pegasus.cc.ucf.edu

Mingxin Wang
Affiliation: Department of Applied Mathematics, Southeast University, Nanjing 210018, People’s Republic of China
Email: mxwang@seu.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-07-04336-X
Keywords: $p$-Laplacian, fast diffusion, absorption, fundamental solution, very singular solution.
Received by editor(s): May 7, 2002
Received by editor(s) in revised form: May 15, 2006
Published electronically: May 8, 2007
Additional Notes: All the authors are grateful to the Hong Kong RGC Grant HKUST 630/95P given to the second author. The first author would like to thank the National Science Foundation for Grant DMS-9971043, USA. The third author thanks the PRC for NSF Grant NSFC-19831060.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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