Acta Applicandae Mathematicae

, Volume 153, Issue 1, pp 147–161 | Cite as

Infiltration Equation with Degeneracy on the Boundary

Article
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Abstract

This paper is mainly about the infiltration equation
$$ {u_{t}}= \operatorname{div} \bigl(a(x)|u|^{\alpha }{ \vert { \nabla u} \vert ^{p-2}}\nabla u\bigr),\quad (x,t) \in \Omega \times (0,T), $$
where \(p>1\), \(\alpha >0\), \(a(x)\in C^{1}(\overline{\Omega })\), \(a(x)\geq 0\) with \(a(x)|_{x\in \partial \Omega }=0\). If there is a constant \(\beta \) such that \(\int_{\Omega }a^{-\beta }(x)dx\leq c\), \(p>1+\frac{1}{\beta }\), then the weak solution is smooth enough to define the trace on the boundary, the stability of the weak solutions can be proved as usual. Meanwhile, if for any \(\beta >\frac{1}{p-1}\), \(\int_{\Omega }a^{-\beta }(x)dxdt=\infty \), then the weak solution lacks the regularity to define the trace on the boundary. The main innovation of this paper is to introduce a new kind of the weak solutions. By these new definitions of the weak solutions, one can study the stability of the weak solutions without any boundary value condition.

Keywords

Infiltration equation Weak solution Boundary degeneracy Stability 

Mathematics Subject Classification

35K65 35K92 35K85 35R35 

Notes

Acknowledgements

The author would like to express his sincere thanks to the anonymous reviewers for their truly helpful comments.

Competing Interests

The author declares that he has no competing interests.

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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.School of Applied MathematicsXiamen University of TechnologyXiamenChina

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