Singular solutions of parabolic $p$-Laplacian with absorption
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- by Xinfu Chen, Yuanwei Qi and Mingxin Wang PDF
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Abstract:
We consider, for $p\in (1,2)$ and $q>1$, the $p$-Laplacian evolution equation with absorption \[ u_t = \operatorname {div} ( |\nabla u|^{p-2} \nabla u) - u^q \quad \mathrm {in}\ \mathbb {R}^n \times (0,\infty ).\] 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:
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[(i)] When $q\geq p-1+p/n$, there does not exist any such singular solution.
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[(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
- MR Author ID: 261335
- 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
- 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.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2327046