Large time behaviour of solutions to a class of non-autonomous degenerate parabolic equations

In this paper we considered a class of non-autonomous, degenerate parabolic equations and we studied the asymptotic behavior of the solutions. Even if the equation depends explicitly upon the time, we proved that several asymptotic properties, valid for the autonomous case, are preserved in this more general situation. More precisely, let $\Omega$ be a bounded domain in $R^N$ with $C^1$ boundary. We considered in $\Omega \times (t > 0)$ the following initial boundary value problem $$u_t = div A(x, t, u,\nabla u), (x, t) \in \Omega \times (0, \infty)$$ $$u(x, t) = 0, (x, t) \in \partial \Omega \times (0, \infty)$$ $$u(x, 0) = u_0(x) \geq 0, x ∈ \Omega $$ where $u_0 \in L^1(\Omega)$ and $ \int_\Omega u_0(x)dx > 0$. The functions $A := (A_1, . . . , A_N )$ are assumed to be only measurable and to satisfy the following structure conditions: $$A(x, t, u,\nablau)\nablau \geq c_0|\nablau|^p, $$ and $$|A(x, t, u, \nablau)| ≤ c1|\nablau |^{p−1}$$ with $p > 2$ and $ c_0, c_1$ given positive constants. Our aim was to study the asymptotic behavior of the weak solutions. In the last few years, several papers were devoted to the study of the asymptotic behavior of solutions to the porous media and the p-Laplace equations. Among them we quote [ACP], [AP], [BN], [BP], [FK], [KP], [KV1], [KV2], [N] and [V]. We refer the reader to the recent monograph by [VM]. To our knowledge, in all these references the authors use elliptic results to study the asymptotic behavior of the solutions. If, from one side, this makes the proof simple and very elegant, on the other hand it looks like this method cannot be applied in the case of time-dependent coefficients. Studying the asymptotic behavior of singular porous medium equations, in [BH] Berryman and Holland introduced a different approach, so to say, more parabolic; namely, relying on the properties of the evolution equations, they were able to study the asymptotic behavior of the solutions and derive the elliptic properties of the asymptotic limit as a by-product. This approach was extended in [MV] and [SV] to the case of degenerate parabolic equations. In both these papers the asymptotic behavior is studied in the case of the prototype equations. In this paper we exploited the techniques introduced in [MV] and [SV] to study the asymptotic behavior of an initial boundary value problem with coefficients only measurable and time depending. We remark that to our knowledge it was the first time that the large time behavior of solutions of non-autonomous equations is studied. This generalization to the non-autonomous case was based on recent results about Harnack estimates for quasilinear parabolic equations proved in [DGV], that al-lowed us to avoid the use of the maximum principle and of the Rayleigh quotients. This paper was recently generalized by the Authors to the case of more general structure conditions (in [RVV1]) add to the case of singular quasilinear parabolic equations (in [RVV2]) From Google Scholar database it results that this paper was quoted 3 times. REFERENCES [ACP]. Aronson, D., Crandall, M.G., Peletier, L.A.: Stabilization of solutions of a degenerate non-linear diffusion problem. Nonlinear Anal. 6(10), 1001–1022 (1982) [AP] Aronson, D.G., Peletier, L.A.: Large time behaviour of solutions of the porous medium equa-tion in bounded domains. J. Differ. Equ. 39(3), 378–412 (1981) [BH] Berryman, J.G.,Holland, C.J.: Stability of the separable solution for fast diffusion. Arch. Ra-tion. Mech. Anal. 74(4), 379–388 (1980) [BNP]. Bertsch, M., Nanbu, T., Peletier, L.A.: Decay of solutions of a degenerate nonlinear diffu-sion equation. Nonlinear Anal. 6(6), 539–554 (1982) [BP]. Bertsch, M., Peletier, L.A.: The asymptotic profile of solutions of degenerate diffusion equa-tions.Arch. Ration. Mech. Anal. 91(3), 207–229 (1985) [DGV]. Di Benedetto, E., Gianazza, U., Vespri, V.: Harnack estimates for quasi-linear degenerate parabolic differential equations. Acta Math. 200(2), 181–209 (2008) [FK] Friedman, A., Kamin, S.: The asymptotic behavior of gas in an n-dimensional porous me-dium. Trans. Am. Math. Soc. 262(2), 551–556 (1980) [KP]. 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) [KV1] Kamin, S., Vázquez, J.L.: Fundamental solutions and asymptotic behaviour for the p-Laplacian equation. Rev. Mat. Iberoamericana 4(2), 339–354 (1988) [KV2] 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) [MV] 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) [N] 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) [RVV1] Ragnedda, F., Vernier-Piro S., Vespri V.; Asymptotic time behaviour for non-autonomous degenerate parabolic equations with forcing terms. NonLinear Anal. 71 (2009) 2316-2321. [RVV2] Ragnedda, F., Vernier-Piro S., Vespri V.; Pointwise estimates for solutions of singular par-abolic problems in $R^N \times [0;+\infty)$. Submitted. [SV] Savare’, G., Vespri, V.: The asymptotic profile of solutions of a class of doubly nonlinear equations.Nonlinear Anal. 22(12), 1553–1565 (1994) [V]. Vazquez, 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) [VM]. Vazquez, 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. Ox-ford University Press, Oxford (2006)