The well-posedness problem of an anisotropic porous medium equation with a convection term

The initial boundary value problem of an anisotropic porous medium equation is considered in this paper. The existence of a weak solution is proved by the monotone convergent method. By showing that $\nabla u\in L^{\infty}(0,T; L^{2}_{\mathrm{loc}}(\Omega ))$ ∇ u ∈ L ∞ ( 0 , T ; L loc 2 ( Ω ) ) , according to different boundary value conditions, some stability theorems of weak solutions are obtained. The unusual thing is that the partial boundary value condition is based on a submanifold Σ of $\partial \Omega \times (0,T)$ ∂ Ω × ( 0 , T ) and, in some special cases, $\Sigma = \{(x,t)\in \partial \Omega \times (0,T): \prod a_{i}(x,t)>0 \}$ Σ = { ( x , t ) ∈ ∂ Ω × ( 0 , T ) : ∏ a i ( x , t ) > 0 } .

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