Abstract
This paper is mainly about the infiltration equation
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.
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The author would like to express his sincere thanks to the anonymous reviewers for their truly helpful comments.
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The paper is supported by Natural Science Foundation of China (no: 11371297), Natural Science Foundation of Fujian province (no: 2015J01592), supported by Science Foundation of Xiamen University of Technology, China.
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Zhan, H. Infiltration Equation with Degeneracy on the Boundary. Acta Appl Math 153, 147–161 (2018). https://doi.org/10.1007/s10440-017-0124-3
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DOI: https://doi.org/10.1007/s10440-017-0124-3