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Diffusivity determination in nonlinear diffusion

Published online by Cambridge University Press:  16 July 2009

Carmen Cortázar ,
Manuel Elgueta  and
Juan Luis Vázquez
Carmen Cortázar
Affiliation:
Departamento de Matemática, Pontificia Universidad Católica, Casilla 6177, Santiago de Chile, Chile
Manuel Elgueta
Affiliation:
Departamento de Matemática, Pontificia Universidad Católica, Casilla 6177, Santiago de Chile, Chile
Juan Luis Vázquez
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain
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Abstract

We are concerned with the problem of determining the diffusivity D of a diffusion process governed by the equation ut = (Dux)x', under the assumption that D depends on u. The main point consists in the observation that there exist solutions of travelling-wave type and that the dependence D = D(u) can be explicitly found in terms of the profile of such solutions. The property of finite propagation speed is required for this method to work. We propose two concrete implementations of the inverse problem, and give a rigorous mathematical proof of our statements. We also describe the application of the travelling-wave method to another interesting class of nonlinear parabolic equations.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 2 , Issue 2 , June 1991 , pp. 159 - 169
Copyright
Copyright © Cambridge University Press 1991

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References

Aronson, D. G. 1986 The porous medium equation, in: Fasano, A. & Primicerio, M., eds., Nonlinear Diffusion Problems. Lecture Notes in Mathematics, Vol. 1224, Springer-Verlag.Google Scholar
Buckmaster, J. 1977 Viscous sheets advancing over dry beds. J. Fluid Mech. 81, 735–756.CrossRefGoogle Scholar
Bardos, C., Golse, F., Perthame, B. & Sentis, R. 1988 The nonaccretive radiative transfer equation: existence of solutions and Rosseland approximation. J. Fund. Anal. 77, 434–460.Google Scholar
Cortazar, C. & Elgueta, M. 1989 Localization and boundedness of the solutions of the Neumann problem for a filtration equation. J. Nonlinear Analysis. 3, 33–41CrossRefGoogle Scholar
Chandrasekhas, S. 1960 Radiative Transfer Equation. Dover.Google Scholar
Crank, J. 1975 The Mathematics of Dffusion. Clarendon Press.Google Scholar
Esteban, J. R. & Vazquen, J. L. 1988 Homogeneous diffusion in R with power-like non-linear diffusivity. Arch. Rat. Mech. Anal. 103, 39–80.Google Scholar
Gilding, B. H. & Peletier, L. A. 1976 On a class of similarity solutions of the porous media equation. J. Math. Anal. Appl. 55, 351–364.Google Scholar
Gilding, B. H. & Peletier, L. A. 1977 On a class of similarity solutions of the porous media equation II. J. Math. Anal. Appl. 57, 552–538.CrossRefGoogle Scholar
Gurtin, M. & MacCamy, R. C. 1977 On the diffusion of biological populations. Math. Biosciences 33, 35–49.CrossRefGoogle Scholar
Kalashnikov, A. S. 1978 On a nonlinear filtration equation appearing in the theory of non- stationary filtration. Trud. Sem. 137–146 (in Russian).Google Scholar
King, J. R. & Please, C. P. 1991 Diffusion of dopant in crystalline silicon – an asymptotic analysis. IMA J. Appl. Math. (to appear).Google Scholar
Lorenzi, A. 1985 An inverse problem for a quasilinear parabolic equation. Ann. Mat. Pura. Appl. 142, 145–169.CrossRefGoogle Scholar
Larsen, E. W. & Pomraning, G. C. 1980 Asymptotic analysis of nonlinear Marshak waves. SIAM J. Appl. Math. 39, 201–212.Google Scholar
Martinson, L. K. & Pavlov, K. B. 1971 Unsteady shear flows of a conducting fluid with a rheological power law. Magnit. Gidrodimamika 2, 50–58.Google Scholar
Mercier, B. 1987 Application of accetive operators to the radiative transfer equation. SIAM J. Math. Anal. 18, 393–408.Google Scholar
Muskat, M. 1937 The Flow of Fluids Through Porous Media. McGraw-Hill.CrossRefGoogle Scholar
Muzylev, M. 1985 Uniqueness of the solution of an inverse problem of nonlinear heat conduction. Vych. Mat. i Mat. Fiz. 25, 1346–1352 (in Russian).Google Scholar
Oleinik, O. A., Kalashnikov, A. S. & Czhou, Y. L. 1958 The Cauchy problem and boundary problems for equations of the type of nonstationary filtration. Izv. Akad. Nauk SSSR, Ser. Mat. 22, 667–704 (in Russian).Google Scholar
Peletier, L. A. 1974 A necessary and sufficient condition for the existence of an interface in flows through porous media. Arch. Rat. Mech. Anal. 56, 183–190.Google Scholar
Pilant, M. & Rundell, W. 1988 Fixed point methods for a nonlinear parabolic inverse coefficient problem. Comm. P. D. E. 113, 469–493.CrossRefGoogle Scholar
Rosenau, P. & Hyman, J. M. 1986 Plasma diffusion across a magnetic field. Physica 20D, 444–446.Google Scholar
Tayler, A. B. & King, J. R. 1987 Free boundaries in semi-conductor fabrication. Proceedings International Conference on Free Boundaries and their Applications, Irsee, Bavaria, Germany.Google Scholar
Zel'dovich, Y. B. & Raizer, Y. P. 1966 Physics of shock waves and high-temperature hydrodynamic phenomena. Academic Press.Google Scholar