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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

##### MSC:
 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)
##### Keywords:
existence; uniqueness; multiplicity; stability; equilibrium solutions