In this paper we analyze how the dynamics of a class of superlinear indefinite reaction diffusion equations varies as the nodal behavior of a coefficient changes.To perform this analysis we use both theoretical and numerical tools. The analysis aids the numerical study, and the numerical study confirms and completes the analysis. The numerics in addition provides us with some further results for whichat-first glance analytical tools are not available yet. Our main analytical result shows that the problem possesses a unique positive solution which is linearly asymptotically stable if the trivial state is linearly unstable and the model admits some positive solution. This result is a relevant feature for superlinear indefinite problems, since our numerical computations show how these models can have an arbitrarily large number of positive solutions if the trivial state is unstable. 2000Academic Press

In this paper we analyze the uniqueness and the pointwise growth of the positive solutions of a nonlinear elliptic boundary-value problem of general sublinear type with a weight function multiplying the nonlinearity. When this function vanishes on some subdomain, the problem exhibits a bifurcation from infinity. In this case almost nothing is known about the pointwise growth of the positive solutions as the parameter approaches the critical value where the bifurcation from infinity occurs. In this work we show that the positive solutions grow to infinity in the region where the weight function vanishes and that on its support they stabilize to the minimal positive solution of the original equation subject to infinite Dirichlet boundary conditions. This behavior provides us with the uniqueness of the positive solution near the value of the parameter where the bifurcation from infinity occurs. Also, we solve the problem using spectral collocation methods coupled with path-following techniques to show how the main uniqueness result is optimal. Throughout the paper the mathematical analysis aids the numerical study, and the numerical study confirms and illuminates the analysis.

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