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Multiplicity and stability of equilibrium solutions of a onedimensional fast diffusion problem. (English) Zbl 0557.35070
The author considers first the problem \((u^ m)_{xx}+\lambda u(1-u^ k)=0\) in (0,L), \(u(0)=u(L)=0\) where \(m\in (0,1)\), \(k\geq 1\) is an integer and \(\lambda >0\). He shows that there is some \(\lambda_ 0\) such that i) for \(\lambda <\lambda_ 0\), \(u=0\) is the only solution, ii) for \(\lambda =\lambda_ 0\) there is a unique positive solution, iii) for \(\lambda >\lambda_ 0\) there are two positive solutions \(\bar u_{\lambda}>\underline u_{\lambda}\). The author studies the stability of the equilibrium solutions with respect to the evolution problem \(u_ t=(u^ m)_{xx}+\lambda u(1-u^ k)\).
Reviewer: H.Brezis

35K55 Nonlinear parabolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B35 Stability in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)