Abstract
We consider a class of non-autonomous, degenerate parabolic equations and we study the asymptotic behaviour of the solutions. Even if the equation depends explicitly upon the time, we prove that several asymptotic properties, valid for the autonomous case, are preserved in this more general situation. To our knowledge, it is the first time that the asymptotic behaviour of solutions to non-autonomous equations is studied.
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References
Aronson D.G., Peletier L.A.: Large time behaviour of solutions of the porous medium equation in bounded domains. J. Differ. Equ. 39(3), 378–412 (1981)
Aronson D., Crandall M.G., Peletier L.A.: Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Anal. 6(10), 1001–1022 (1982)
Bertsch M., Nanbu T., Peletier L.A.: Decay of solutions of a degenerate nonlinear diffusion equation. Nonlinear Anal. 6(6), 539–554 (1982)
Bertsch M., Peletier L.A.: The asymptotic profile of solutions of degenerate diffusion equations. Arch. Ration. Mech. Anal. 91(3), 207–229 (1985)
Di Benedetto E.: Degenerate Parabolic Equations. Springer, Berlin (1993)
Di Benedetto, E., Urbano, J.M., Vespri, V.: Current issues on singular and degenerate evolution equations, evolutionary equations. In: Handbook of Differential Equations, vol. 1, pp. 169–286. North-Holland, Amsterdam (2004)
Di Benedetto E., Gianazza U., Vespri V.: Local claustering on the non zero set of functions in W 1,1(E). Rend. Accad. Naz. Lincei Roma 17, 223–225 (2006)
Di Benedetto E., Gianazza U., Vespri V.: Harnack estimates for quasi-linear degenerate parabolic differential equations. Acta Math. 200(2), 181–209 (2008)
Berryman J.G., Holland C.J.: Stability of the separable solution for fast diffusion. Arch. Ration. Mech. Anal. 74(4), 379–388 (1980)
Fornaro S., Sosio M.: Intrinsic Harnack estimates for some doubly nonlinear degenerate parabolic equations. Adv. Differ. Equ. 13, 139–168 (2008)
Friedman A., Kamin S.: The asymptotic behavior of gas in an n-dimensional porous medium. Trans. Am. Math. Soc. 262(2), 551–556 (1980)
Gianazza U., Vespri V.: A Harnack inequality for a degenerate parabolic equation. J. Evol. Equ. 97, 3–22 (1998)
Kamin S., Peletier L.A.: Large time behaviour of solutions of the porous media equation with absorption. Israel J. Math. 55(2), 129–146 (1986)
Kamin S., Vázquez J.L.: Fundamental solutions and asymptotic behaviour for the p-Laplacian equation. Rev. Mat. Iberoamericana 4(2), 339–354 (1988)
Kamin S., Vázquez J.L.: Asymptotic behaviour of solutions of the porous medium equation with changing sign. SIAM J. Math. Anal. 22(1), 34–45 (1991)
Manfredi J.J., Vespri V.: Large time behaviour of solutions to a class of Doubly Nonlinear Parabolic equations. Electron. J. Differ. Equ. 2, 1–17 (1994)
Minty G.L.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)
Nanbu, T.: On some decay estimates of solutions for some nonlinear degenerate diffusion equations. Progress in Analysis, vols. I, II (Berlin, 2001), pp. 995–1003. World Sci. Publ., River Edge (2003)
Savaré G., Vespri V.: The asymptotic profile of solutions of a class of doubly nonlinear equations. Nonlinear Anal. 22(12), 1553–1565 (1994)
Vázquez J.L.: Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium. Trans. Am. Math. Soc. 277(2), 507–527 (1983)
Vázquez J.L.: Smoothing and decay estimates for nonlinear diffusion equations. Equations of porous medium type. In: Oxford Lecture Series in Mathematics and its Applications, vol. 33. Oxford University Press, Oxford (2006)
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Ragnedda, F., Vernier Piro, S. & Vespri, V. Large time behaviour of solutions to a class of non-autonomous, degenerate parabolic equations. Math. Ann. 348, 779–795 (2010). https://doi.org/10.1007/s00208-010-0496-4
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DOI: https://doi.org/10.1007/s00208-010-0496-4