Keywords
Iterative method
non-linear diffusion
solución débil
método iterativo
difusión no lineal
How to Cite
Abstract
We study the existence and uniqueness of the solution of a non-linear stratified diffusion problem. To this aim, we construct an alternative method based on successive substitutions of a linear approximation of the original problem. We use the theory of partial differential equations and mathematical induction to prove that each of the linear problems of the iteration has a unique weak solution. Finally, we prove that the sequence of weak solutions obtained is a Cauchy sequence that converges to the weak solution of the problem.
References
Bertsch, M.; Gurtin, M. E.; Hilhorst, D.; Peletier, L. (1984) “On interacting populations that disperse to avoid crowding: The effect of a sedentary colony”, Journal Mathematical Biology 19: 1–12.
Ciarlet, P.G. (2013) Linear and Nonlinear Functional Analysis with Applications Society for Industrial & Applied Mathematics, New York.
Deutray, R.; Lions. J-L. (2000) Mathematical Analysis and Numerical Methods for Science and Tecnology, Vol. 4-6. Springer-Verlag, Heidelberg.
DiBenedetto, E. (1993) Degenerate Parabolic Equations. Springer-Verlag, New York.
Evans, L.C. (2010) Partial Differential Equations. American Mathematical Society, Providence, Rhode Island.
Fisher, R.A. (1937) “The wave of advance of advantageous genes”, Annals of Eugenics 7: 353–369.
Gurtin, M.; MacCamy, M.(1977) “On the diffusion of biological populations”, Mathematical Biosciences 33: 35–49.
Murray, J.D. (2003) Mathematical Biology II. Spatial Models and Biomedical Applications. Springer-Verlag, New York.
Petrovskii, S.V.; Bai-Lian, L. (2006) Exactly Solvable Models of Biological Invasion. Chapman & Hall/CRC, Boca Raton FL.
Shigesada, N.; Kawasaki, K.; Takeda, Y. (1995) “Modeling stratified diffusion in biological invasions”, The American Naturalist 146(2): 229–251.
Shigesada, N.; Kawasaki, K. (1997) Biological Invasions: Theory and Practice. Oxford University Press, New York.
Showalter, R.E. (1997) Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs 49, American Mathematical Society, Providence RI.
Taira, K. (2016) Analytic Semigroups and Semilinear Initial Boundary Value Problems. Part of London Mathematical Society Lecture Note Series, 2nd Edition.
Tröltzsch, F. (2010) Optimal Control of Partial Differential Equations - Theory, Methods and Applications. Graduate Studies in Mathematics, Vol. 112. American Mathematical Society, Providence RI.
Vázquez, J.L. (2007) The Porous Medium Equation. Oxford University Press, Oxford.
Witelsky, T.P. (1995) “Merging traveling waves for the porous-Fisher’s equation", Applied Mathematics Letters 8(4): 57–62
Zeidler, E. (1990) Nonlinear Functional Analysis and its Applications II/B: Nonlinear Monotone Operators. Springer-Verlag, New York.