首页|On the Well-Posedness Problem of the Anisotropic Porous Medium Equation with a Variable Diffusion Coefficient
On the Well-Posedness Problem of the Anisotropic Porous Medium Equation with a Variable Diffusion Coefficient
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The initial-boundary value problem of an anisotropic porous medium equa-tion ut=N∑i=1∂/∂(a(x,t)|u|αiuxi)+N∑i=1∂fi(u,x,t)/∂xi is studied.Compared with the usual porous medium equation,there are two differ-ent characteristics in this equation.One lies in its anisotropic property,another one is that there is a nonnegative variable diffusion coefficient a(x,t)additionally.Since a(x,t)may be degenerate on the parabolic boundary ∂Ω×(0,T),instead of the bound-edness of the gradient|▽u|for the usual porous medium,we can only show that▽u ∈ L∞(0,T;L2loc(Ω)).Based on this property,the partial boundary value conditions matching up with the anisotropic porous medium equation are discovered and two stability theorems of weak solutions can be proved naturally.
Anisotropic porous medium equationvariable diffusion coefficientstabilitypartial boundary condition
ZHAN Huashui
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School of Mathematics and Statistics,Xiamen University of Technology,Xiamen 361024,China
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