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Volume 37, Issue 2
On the Well-Posedness Problem of the Anisotropic Porous Medium Equation with a Variable Diffusion Coefficient

Huashui Zhan

DOI: 10.4208/jpde.v37.n2.2

J. Part. Diff. Eq., 37 (2024), pp. 135-149.

Published online: 2024-06

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  • Abstract

The initial-boundary value problem of an anisotropic porous medium equation $$u_t=\sum^N_{i=1}\frac{\partial}{\partial x_i}(a(x,t)|u|^{\alpha_i}u_{x_i})+\sum^N_{i=1}\frac{\partial f_i(u,x,t)}{\partial x_i}$$ is studied. Compared with the usual porous medium equation, there are two different 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 boundedness of the gradient $|∇u|$ for the usual porous medium, we can only show that $∇u∈ L^∞(0,T;L^2_{ {\rm loc}}(Ω)).$ 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.

  • Keywords

Anisotropic porous medium equation, variable diffusion coefficient, stability, partial boundary condition.

  • AMS Subject Headings

35L65, 35K55, 35B05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-37-135, author = {Zhan , Huashui}, title = {On the Well-Posedness Problem of the Anisotropic Porous Medium Equation with a Variable Diffusion Coefficient}, journal = {Journal of Partial Differential Equations}, year = {2024}, volume = {37}, number = {2}, pages = {135--149}, abstract = {

The initial-boundary value problem of an anisotropic porous medium equation $$u_t=\sum^N_{i=1}\frac{\partial}{\partial x_i}(a(x,t)|u|^{\alpha_i}u_{x_i})+\sum^N_{i=1}\frac{\partial f_i(u,x,t)}{\partial x_i}$$ is studied. Compared with the usual porous medium equation, there are two different 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 boundedness of the gradient $|∇u|$ for the usual porous medium, we can only show that $∇u∈ L^∞(0,T;L^2_{ {\rm loc}}(Ω)).$ 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.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v37.n2.2}, url = {http://global-sci.org/intro/article_detail/jpde/23205.html} }
TY - JOUR T1 - On the Well-Posedness Problem of the Anisotropic Porous Medium Equation with a Variable Diffusion Coefficient AU - Zhan , Huashui JO - Journal of Partial Differential Equations VL - 2 SP - 135 EP - 149 PY - 2024 DA - 2024/06 SN - 37 DO - http://doi.org/10.4208/jpde.v37.n2.2 UR - https://global-sci.org/intro/article_detail/jpde/23205.html KW - Anisotropic porous medium equation, variable diffusion coefficient, stability, partial boundary condition. AB -

The initial-boundary value problem of an anisotropic porous medium equation $$u_t=\sum^N_{i=1}\frac{\partial}{\partial x_i}(a(x,t)|u|^{\alpha_i}u_{x_i})+\sum^N_{i=1}\frac{\partial f_i(u,x,t)}{\partial x_i}$$ is studied. Compared with the usual porous medium equation, there are two different 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 boundedness of the gradient $|∇u|$ for the usual porous medium, we can only show that $∇u∈ L^∞(0,T;L^2_{ {\rm loc}}(Ω)).$ 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.

Huashui Zhan. (2024). On the Well-Posedness Problem of the Anisotropic Porous Medium Equation with a Variable Diffusion Coefficient. Journal of Partial Differential Equations. 37 (2). 135-149. doi:10.4208/jpde.v37.n2.2
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