Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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THE STABILITY OF WEAK SOLUTIONS TO AN ANISOTROPIC POLYTROPIC INFILTRATION EQUATION

  • Zhan, Huashui (School of Applied Mathematics Xiamen University of Technology)
  • Received : 2020.06.30
  • Accepted : 2020.12.23
  • Published : 2021.09.01
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Abstract

This paper considers an anisotropic polytropic infiltration equation with a source term $$u_t={\sum\limits_{i=1}^{N}}{\frac{{\partial}}{{\partial}x_i}}\(a_1(x){\mid}u{\mid}^{{\alpha}_i}{\mid}u_{x_i}{\mid}^{p_i-2}u_{x_i}\)+f(x,t,u)$$, where pi > 1, αi > 0, ai(x) ≥ 0. The existence of weak solution is proved by parabolically regularized method. Based on local integrability $u_{x_i}{\in}W_{loc}^{1,p_i}(\Omega)$, the stability of weak solutions is proved without boundary value condition by the weak characteristic function method. One of the essential characteristics of an anisotropic equation different from an isotropic equation is found originally.

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Acknowledgement

The author would like to thank the reviewers for all useful and helpful comments on our manuscript.

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