Volume 3, Issue 3
Periodic Solutions to Porous Media Equations of Parabolic-elliptic Type

N.Kenmochi , D.Kröner & M.Kubo

DOI:

J. Part. Diff. Eq., 3 (1990), pp. 63-77.

Published online: 1990-03

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  • Abstract

This paper is concerned with a equation, which is a model of filtration in partially saturated porous media, with mixed boundary condition of Dirichlet-Neumann type {∂_tb(u) - ∇ • a [∇u + k(b(u))] = f \qquad in \quad (0, ∞) × Ω u = h(t, x) \qquad on \quad (0, ∞) × Γ_0 v • a [∇u + k(b(u))] = g(t, x) \qquad on \quad (0, ∞) × Γ_1 We have proved that there exists one and only one periodic solution of the problem under the data f, g and h with same period. Moreover, we have proved that the unique periodic solution ω is asymptotically statble in the sense that for any solution u of the problem b(u(t)) - b(ω(t)) → 0\qquad in L²(Ω) as t → ∞.

  • Keywords

Filtration equation periodic solutions asymptotic stability

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@Article{JPDE-3-63, author = {}, title = {Periodic Solutions to Porous Media Equations of Parabolic-elliptic Type}, journal = {Journal of Partial Differential Equations}, year = {1990}, volume = {3}, number = {3}, pages = {63--77}, abstract = { This paper is concerned with a equation, which is a model of filtration in partially saturated porous media, with mixed boundary condition of Dirichlet-Neumann type {∂_tb(u) - ∇ • a [∇u + k(b(u))] = f \qquad in \quad (0, ∞) × Ω u = h(t, x) \qquad on \quad (0, ∞) × Γ_0 v • a [∇u + k(b(u))] = g(t, x) \qquad on \quad (0, ∞) × Γ_1 We have proved that there exists one and only one periodic solution of the problem under the data f, g and h with same period. Moreover, we have proved that the unique periodic solution ω is asymptotically statble in the sense that for any solution u of the problem b(u(t)) - b(ω(t)) → 0\qquad in L²(Ω) as t → ∞.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5807.html} }
TY - JOUR T1 - Periodic Solutions to Porous Media Equations of Parabolic-elliptic Type JO - Journal of Partial Differential Equations VL - 3 SP - 63 EP - 77 PY - 1990 DA - 1990/03 SN - 3 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5807.html KW - Filtration equation KW - periodic solutions KW - asymptotic stability AB - This paper is concerned with a equation, which is a model of filtration in partially saturated porous media, with mixed boundary condition of Dirichlet-Neumann type {∂_tb(u) - ∇ • a [∇u + k(b(u))] = f \qquad in \quad (0, ∞) × Ω u = h(t, x) \qquad on \quad (0, ∞) × Γ_0 v • a [∇u + k(b(u))] = g(t, x) \qquad on \quad (0, ∞) × Γ_1 We have proved that there exists one and only one periodic solution of the problem under the data f, g and h with same period. Moreover, we have proved that the unique periodic solution ω is asymptotically statble in the sense that for any solution u of the problem b(u(t)) - b(ω(t)) → 0\qquad in L²(Ω) as t → ∞.
N.Kenmochi , D.Kröner & M.Kubo . (2019). Periodic Solutions to Porous Media Equations of Parabolic-elliptic Type. Journal of Partial Differential Equations. 3 (3). 63-77. doi:
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