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Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions

  • * Corresponding author: Jesús Ildefonso Díaz

    * Corresponding author: Jesús Ildefonso Díaz
The research of J.I. Díaz was partially supported by the project ref. MTM2014-57113 of the DGISPI (Spain) and the Research Group MOMAT (Ref. 910480) of the UCM. The results of this paper were started during the visit of the first author to the UCM, on October 2015, under the support of the University Aboubekr Belkaid of Tlemcen (Algeria). This author wants to thank this support as well as the received hospitality from the Instituto de Matemática Interdisciplar of the UCM.
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  • We study stability of the nonnegative solutions of a discontinuous elliptic eigenvalue problem relevant in several applications as for instance in climate modeling. After giving the explicit expresion of the S-shaped bifurcation diagram $\left( \lambda ,{{\left\| {{\mu }_{\lambda }} \right\|}_{\infty }} \right)$ we show the instability of the decreasing part of the bifurcation curve and the stability of the increasing part. This extends to the case of non-smooth nonlinear terms the well known 1971 result by M.G. Crandall and P.H. Rabinowitz concerning differentiable nonlinear terms. We point out that, in general, there is a lacking of uniquenees of solutions for the associated parabolic problem. Nevertheless, for nondegenerate solutions (crossing the discontinuity value of u in a transversal way) the comparison principle and the uniqueness of solutions hold. The instability is obtained trough a linearization process leading to an eigenvalue problem in which a Dirac delta distribution appears as a coefficient of the differential operator. The stability proof uses a suitable change of variables, the continuuity of the bifurcation branch and the comparison principle for nondegenerate solutions of the parabolic problem.

    Mathematics Subject Classification: 35B35, 35J61, 35P30, 35K58, 86A10.


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  • Figure 1.  A qualitative description of the bifurcation curve

    Figure 2.  Security neighborhood

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