Abstract
In this note we consider the problem 237–1 where D is a smooth function such that D > 0 and D′ ≤ 0 on [0,1]. We denote the unique solution of Problem P ° by u. We furthermore consider for i = 0,1 the problems 237–2 Inspired by a conjecture of Hagan and Brenner [2] in a higher dimensional context we show that and The proof of (1.1) is based on a simple comparison argument. This inequality is in fact a special case of results due to Benilan and Diaz [1].
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References
Benilan Ph. and J.I. Diaz, Comparison of solutions of nonlinear evolution problems with different nonlinear terms, Israel J. of Math. 42, (1982) 241–257.
Hagan P.S. and D. Brenner, Sorption limits with increasing diffusion coefficients, SIAM J. Applied Math. 48, (1988), 917–920.
Ladyzenskaja O.A., V.A. Solonnikov and N.N. Ural’Ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, Amer. Math. Soc, Providence R.I., 1968.
Protter M.H. and H.F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, N.J., 1967.
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© 1992 Springer Science+Business Media New York
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Hilhorst, D., Hilhorst, H.J. (1992). On a conjecture by Hagan and Brenner. In: Lloyd, N.G., Ni, W.M., Peletier, L.A., Serrin, J. (eds) Nonlinear Diffusion Equations and Their Equilibrium States, 3. Progress in Nonlinear Differential Equations and Their Applications, vol 7. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0393-3_17
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DOI: https://doi.org/10.1007/978-1-4612-0393-3_17
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6741-6
Online ISBN: 978-1-4612-0393-3
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