Journal of the
Korean Mathematical Society JKMS

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