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

Bendong Lou; Bendong Lou

doi:10.3934/nhm.2012.7.857

Networks and Heterogeneous Media

2012, Volume 7, Issue 4: 857-879. doi: 10.3934/nhm.2012.7.857

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

Bendong Lou ^{1
,}

1.
Department of Mathematics, Tongji University, Shanghai 200092

Received: 01 January 2012 Revised: 01 October 2012
Primary: 35C06, 35C07; Secondary: 35K59, 35B40.

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

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Abstract

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.

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