Self-similar solutions in a sector for a quasilinear parabolic equation

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December 2012, 7(4): 857-879. doi: 10.3934/nhm.2012.7.857

Self-similar solutions in a sector for a quasilinear parabolic equation

Bendong Lou ^1,

1.	Department of Mathematics, Tongji University, Shanghai 200092

Received January 2012 Revised October 2012 Published December 2012

Abstract
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We study a two-point free boundary problem in a sector for a quasilinear parabolic equation. The boundary conditions are assumed to be spatially and temporally "self-similar" in a special way. We prove the existence, uniqueness and asymptotic stability of an expanding solution which is self-similar at discrete times. We also study the existence and uniqueness of a shrinking solution which is self-similar at discrete times.

Keywords: Discrete self-similar solution, quasilinear parabolic equation., spatially and temporally inhomogeneous boundary condition.

Mathematics Subject Classification: Primary: 35C06, 35C07; Secondary: 35K59, 35B4.

Citation: Bendong Lou. Self-similar solutions in a sector for a quasilinear parabolic equation. Networks and Heterogeneous Media, 2012, 7 (4) : 857-879. doi: 10.3934/nhm.2012.7.857

References:

[1]	P. Brunovský, P. Poláčik and B. Sandstede, Convergence in general periodic parabolic equation in one space dimension, Nonlinear Anal., 18 (1992), 209-215. doi: 10.1016/0362-546X(92)90059-N.
[2]	Y.-L. Chang, J.-S. Guo and Y. Kohsaka, On a two-point free boundary problem for a quasilinear parabolic equation, Asymptotic Anal., 34 (2003), 333-358.
[3]	X. Chen and J.-S. Guo, Motion by curvature of planar curves with end points moving freely on a line, Math. Ann., 350 (2011), 277-311. doi: 10.1007/s00208-010-0558-7.
[4]	H.-H. Chern, J.-S. Guo and C.-P. Lo, The self-similar expanding curve for the curvature flow equation, Proc. Amer. Math. Soc., 131 (2003), 3191-3201. doi: 10.1090/S0002-9939-03-07055-2.
[5]	G. Dong, Initial and nonlinear oblique boundary value problems for fully nonlinear parabolic equations, J. Partial Differential Equations, 1 (1988), 12-42.
[6]	A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964.
[7]	M.-H. Giga, Y. Giga and H. Hontani, Selfsimilar expanding solutions in a sector for a crystalline flow, SIAM J. Math. Anal., 37 (2005), 1207-1226. doi: 10.1137/040614372.
[8]	J.-S. Guo and B. Hu, A shrinking two-point free boundary problem for a quasilinear parabolic equation, Quart. Appl. Math., 64 (2006), 413-431.
[9]	J.-S. Guo and Y. Kohsaka, Self-similar solutions of two-point free boundary problem for heat equation, in "Nonlinear Diffusion Systems and Related Topics" RIMS Kokyuroku 1258, Kyoto University, (2002), 94-107.
[10]	D. Hilhorst, R. van der Hout, M. Mimura and I. Ohnishi, A mathematical study of the one-dimensional Keller and Rubinow model for Liesegang bands, J. Stat. Phys., 135 (2009), 107-132. doi: 10.1007/s10955-009-9701-9.
[11]	J. B. Keller and S. I. Rubinow, Recurrent precipitation and Liesegang rings, J. Chem. Phys., 74 (1981), 5000-5007. doi: 10.1063/1.441752.
[12]	Y. Kohsaka, Free boundary problem for quasilinear parabolic equation with fixed angle of contact to a boundary, Nonlinear Anal., 45 (2001), 865-894. doi: 10.1016/S0362-546X(99)00422-8.
[13]	G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., NJ, 1996.
[14]	O. A. Ladyzhenskia, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasi-linear Equations of Parabolic Type," Amer. Math. Soc., Providence, Rhode Island, 1968.
[15]	B. Lou, H. Matano and K. I. Nakamura, Recurrent traveling waves in a two-dimensional saw-toothed cylinder and their average speed, preprint.
[16]	H. Matano, K. I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit, Netw. Heterog. Media, 1 (2006), 537-568. doi: 10.3934/nhm.2006.1.537.
[17]	D. A. V. Stow, "Sedimentary Rocks in the Field: A Color Guide," Academic Press, 2005.
[18]	K. H. W. J. Ten Tusscher and A. V. Panfilov, Wave propagation in excitable media with randomly distributed obstacles, Multiscale Model. Simul., 3 (2005), 265-282. doi: 10.1137/030602654.

show all references

References:

[1]	P. Brunovský, P. Poláčik and B. Sandstede, Convergence in general periodic parabolic equation in one space dimension, Nonlinear Anal., 18 (1992), 209-215. doi: 10.1016/0362-546X(92)90059-N.
[2]	Y.-L. Chang, J.-S. Guo and Y. Kohsaka, On a two-point free boundary problem for a quasilinear parabolic equation, Asymptotic Anal., 34 (2003), 333-358.
[3]	X. Chen and J.-S. Guo, Motion by curvature of planar curves with end points moving freely on a line, Math. Ann., 350 (2011), 277-311. doi: 10.1007/s00208-010-0558-7.
[4]	H.-H. Chern, J.-S. Guo and C.-P. Lo, The self-similar expanding curve for the curvature flow equation, Proc. Amer. Math. Soc., 131 (2003), 3191-3201. doi: 10.1090/S0002-9939-03-07055-2.
[5]	G. Dong, Initial and nonlinear oblique boundary value problems for fully nonlinear parabolic equations, J. Partial Differential Equations, 1 (1988), 12-42.
[6]	A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964.
[7]	M.-H. Giga, Y. Giga and H. Hontani, Selfsimilar expanding solutions in a sector for a crystalline flow, SIAM J. Math. Anal., 37 (2005), 1207-1226. doi: 10.1137/040614372.
[8]	J.-S. Guo and B. Hu, A shrinking two-point free boundary problem for a quasilinear parabolic equation, Quart. Appl. Math., 64 (2006), 413-431.
[9]	J.-S. Guo and Y. Kohsaka, Self-similar solutions of two-point free boundary problem for heat equation, in "Nonlinear Diffusion Systems and Related Topics" RIMS Kokyuroku 1258, Kyoto University, (2002), 94-107.
[10]	D. Hilhorst, R. van der Hout, M. Mimura and I. Ohnishi, A mathematical study of the one-dimensional Keller and Rubinow model for Liesegang bands, J. Stat. Phys., 135 (2009), 107-132. doi: 10.1007/s10955-009-9701-9.
[11]	J. B. Keller and S. I. Rubinow, Recurrent precipitation and Liesegang rings, J. Chem. Phys., 74 (1981), 5000-5007. doi: 10.1063/1.441752.
[12]	Y. Kohsaka, Free boundary problem for quasilinear parabolic equation with fixed angle of contact to a boundary, Nonlinear Anal., 45 (2001), 865-894. doi: 10.1016/S0362-546X(99)00422-8.
[13]	G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., NJ, 1996.
[14]	O. A. Ladyzhenskia, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasi-linear Equations of Parabolic Type," Amer. Math. Soc., Providence, Rhode Island, 1968.
[15]	B. Lou, H. Matano and K. I. Nakamura, Recurrent traveling waves in a two-dimensional saw-toothed cylinder and their average speed, preprint.
[16]	H. Matano, K. I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit, Netw. Heterog. Media, 1 (2006), 537-568. doi: 10.3934/nhm.2006.1.537.
[17]	D. A. V. Stow, "Sedimentary Rocks in the Field: A Color Guide," Academic Press, 2005.
[18]	K. H. W. J. Ten Tusscher and A. V. Panfilov, Wave propagation in excitable media with randomly distributed obstacles, Multiscale Model. Simul., 3 (2005), 265-282. doi: 10.1137/030602654.

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