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# Article

 Title: Large time behavior of solutions to a class of doubly nonlinear parabolic equations (English) Author: Zhan, Huashui Language: English Journal: Applications of Mathematics ISSN: 0862-7940 Volume: 53 Issue: 6 Year: 2008 Pages: 521-533 Summary lang: English . Category: math . Summary: We study the large time asymptotic behavior of solutions of the doubly degenerate parabolic equation $u_t=\mathop{{\rm div}} (u^{m-1}|Du|^{p-2}Du)-u^q$ with an initial condition $u(x,0)=u_0(x)$. Here the exponents $m$, $p$ and $q$ satisfy $m+p\geq 3$, $p>1$ and $q>m+p-2$. (English) Keyword: degenerate parabolic equation Keyword: large time asymptotic behavior MSC: 35B40 MSC: 35K15 MSC: 35K55 MSC: 35K65 idZBL: Zbl 1199.35188 idMR: MR2469063 DOI: 10.1007/s10492-008-0039-4 . Date available: 2010-07-20T12:37:32Z Last updated: 2015-05-17 Stable URL: http://hdl.handle.net/10338.dmlcz/140337 . Reference: [1] Benedetto, E. Di: Degenerate Parabolic Equations.Springer New York (1993). MR 1230384 Reference: [2] Ivanov, A. V. H.: Hölder estimates for quasilinear doubly degenerate parabolic equations.J. Sov. Math. 56 (1991), 2320-2347. Zbl 0729.35018, MR 1031986, 10.1007/BF01671935 Reference: [3] Kalashnikov, A. S.: Some problems of nonlinear parabolic equations of second order.Uspekhi Math. Nauk 42 (1987), 135-176. MR 0898624 Reference: [4] Kamin, S., Vazquez, J. L.: Fundamental solutions and asymptotic behaviour for the $p$-Laplacian equation.Rev. Mat. Iberoam. 4 (1988), 339-354. MR 1028745, 10.4171/RMI/77 Reference: [5] Ladyzhenskaya, O. A.: New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problem for them.Tr. Mat. Inst. Steklova 102 (1967), 95-118. Reference: [6] Ladyzhenskaya, O. A., Solonnikov, V. A., Ural'tseva, N. N.: Linear and Quasilinear Equation of Parabolic Type. Trans. Math. Monographs 23.American Mathematical Society (AMS) Providence (1968). Reference: [7] Manfredi, J., Vespri, V.: Large time behavior of solutions to a class of doubly nonlinear parabolic equations.Electron. J. Differ. Equ. 1994/02 (1994), 1-16. Zbl 0787.35047, MR 1262933 Reference: [8] Masayoshi, T.: On solutions of some doubly nonlinear degenerate parabolic equations with absorption.J. Math. Anal. Appl. 132 (1988), 187-212. MR 0942364, 10.1016/0022-247X(88)90053-4 Reference: [9] Winkler, M.: Large time behavior of solutions to degenerate parabolic equations with absorption.NoDEA, Nonlinear Differ. Equ. Appl. 8 (2001), 343-361. Zbl 0980.35077, MR 1841613, 10.1007/PL00001452 Reference: [10] Wu, Z., Zhao, J., Yin, J., Li, H.: Nonlinear Diffusion Equations.Word Scientific Singapore (2001). Zbl 0997.35001 Reference: [11] Yang, J., Zhao, J.: The asymptotic behavior of solutions of some doubly degenerate nonlinear parabolic equations.Northeast. Math. J. 11 (1995), 241-252. Zbl 0848.35067, MR 1350918 Reference: [12] Zhao, J., Yuan, H.: The Cauchy problem of some nonlinear doubly degenerate parabolic equations.Chin. Ann. Math., Ser. A 16 (1995), 181-196. Zbl 0828.35071, MR 1341930 .

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