Analysis of parabolic problems on partitioned domains with nonlinear conditions at the interface: application to mass transfer through semi-permeable membranes. (English) Zbl 1101.35044
Summary: We address a problem governed by linear parabolic partial differential equations set in two adjoining domains, coupled by nonlinear interface conditions of Neumann type. In particular, we address the existence and uniqueness of strong solutions by applying the strong maximum principle, the Schauder fixed point theorem and the fundamental solutions of linear parabolic partial differential equations.
In the first part of this work, we consider the properties of a linear parabolic partial differential equation set on a single domain with a nonlinear boundary condition. After having addressed the well-posedness and some comparison results for the problem on one domain, in the second part of this work we address the case of coupled problems on adjoining domains. In both cases, we complete the understanding of the behavior of the solution of the problems at hand by means of numerical simulations.
The theoretical results obtained here are applied to study the behavior of a biological model for the transfer of chemicals through thin biological membranes. This model represents the dynamics of the concentration \(u\) of a chemical solution separated from the exterior by a semi-permeable membrane.
The analysis of the two-domain problem that we carry out could also be used to investigate the convergence property of iterative substructuring methods applied to the approximation of multidomain problems with nonlinear coupling of Neumann type.
In the first part of this work, we consider the properties of a linear parabolic partial differential equation set on a single domain with a nonlinear boundary condition. After having addressed the well-posedness and some comparison results for the problem on one domain, in the second part of this work we address the case of coupled problems on adjoining domains. In both cases, we complete the understanding of the behavior of the solution of the problems at hand by means of numerical simulations.
The theoretical results obtained here are applied to study the behavior of a biological model for the transfer of chemicals through thin biological membranes. This model represents the dynamics of the concentration \(u\) of a chemical solution separated from the exterior by a semi-permeable membrane.
The analysis of the two-domain problem that we carry out could also be used to investigate the convergence property of iterative substructuring methods applied to the approximation of multidomain problems with nonlinear coupling of Neumann type.
MSC:
| 35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 74G25 | Global existence of solutions for equilibrium problems in solid mechanics (MSC2010) |
| 92C40 | Biochemistry, molecular biology |
Keywords:
existence; uniqueness; strong maximum principle; Schauder fixed point theorem; nonlinear boundary conditionReferences:
| [1] | Amman, H., J. Diff. Eqns.72, 201 (1988), DOI: 10.1016/0022-0396(88)90156-8. |
| [2] | Arrieta, J. M.Carvalho, A. N.Rodríguez-Bernal, A., J. Diff. Eqns.156, 376 (1999), DOI: 10.1006/jdeq.1998.3612. |
| [3] | Brézis, H., Nonlinear Functional Analysis, ed. Zarantonello, E. (Academic Press, 1971) pp. 101-156. |
| [4] | Chipot, M.Fila, M.Quittner, P., Acta Math. Univ. Comenian. (N.S.)60, 35 (1991). |
| [5] | Duvaut, G.; Lions, J. L., Inequalities in Mechanics and Physiscs, 1976, Springer-Verlag · Zbl 0331.35002 |
| [6] | Evans, L., Nonlinear Anal.1, 593 (1977), DOI: 10.1016/0362-546X(77)90020-7. |
| [7] | Friedman, A., J. Math. Mech.8, 161 (1959). · Zbl 0101.31102 |
| [8] | Friedman, A., Pac. J. Math.8, 201 (1959). |
| [9] | Friedman, A., Partial Differential Equations of Parabolic Type, 1964, Prentice-Hall · Zbl 0144.34903 |
| [10] | Friedman, M. H., Principles and Models of Biological Transport, 1986, Springer-Verlag |
| [11] | Galaktionov, V. A.Vazquez, J. L., Discr. Cont. Dyn. Sys.8, 399 (2002). · Zbl 1010.35057 |
| [12] | Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, 2001, Springer-Verlag · Zbl 1042.35002 |
| [13] | Hopf, E., Proc. Amer. Math. Soc.3, 791 (1952), DOI: 10.2307/2032182. |
| [14] | Kedem, O.Katchalsky, A., Biochim. Biophys. Acta229 (1958), DOI: 10.1016/0006-3002(58)90330-5. |
| [15] | Ladyzenskaya, O. A.; Solonnikov, V. A.; Ueal’ceva, N. N., Linear and Quasilinear Equations of Parabolic Type, 1968, Amer. Math. Soc. · Zbl 0174.15403 |
| [16] | Levine, H., SIAM Rev.32, 262 (1990), DOI: 10.1137/1032046. |
| [17] | Nirenberg, L., Comm. Pure Appl. Math.6, 167 (1953). · Zbl 0050.09601 |
| [18] | Quarteroni, A.; Valli, A., Domain Decomposition Methods for Partial Differential Equations, 1999, Oxford Univ. Press · Zbl 0931.65118 |
| [19] | Walter, W., SIAM J. Math. Anal.6, 85 (1975), DOI: 10.1137/0506008. |
| [20] | Walter, W., Nonlinear Anal.30, 4695 (1997), DOI: 10.1016/S0362-546X(96)00259-3. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
