The stability of weak solutions to an anisotropic polytropic infiltration equation
J. Korean Math. Soc. 2021 Vol. 58, No. 5, 1109-1129
https://doi.org/10.4134/JKMS.j200369
Published online July 21, 2021
Printed September 1, 2021
Huashui Zhan
Xiamen University of Technology
Abstract : This paper considers an anisotropic polytropic infiltration equation with a source term $$ {u_t}= \sum_{i=1}^N\frac{\partial }{\partial x_i}\left(a_i(x)|u|^{\alpha_i}{\left| {u_{x_i}} \right|^{p_i-2}}u_{x_i}\right)+f(x,t,u), $$ where $p_i>1$, $\alpha_i >0$, $a_i(x)\geq 0$. The existence of weak solution is proved by parabolically regularized method. Based on local integrability $u_{x_i}\in W^{1,p_i}_{loc}(\Omega)$, 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.
Keywords : The anisotropic polytropic infiltration equation, the weak characteristic function method, stability, boundary value condition
MSC numbers : 35K55, 35B35, 35L70
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