The Effect of Varying Coefficients on the Dynamics of a Class of Superlinear Indefinite Reaction–Diffusion Equations

Abstract

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

High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems

“…On the other hand, if b ≥ b * , then (1) admits a positive solution if, and only if, λ ≤ π 2 . These results are direct consequences of the general theory developed in J. López-Gómez [14], H. Amann and J. López-Gómez [2], and R. Gómez-Reñasco and J. López-Gómez [11,12], where some pioneering findings by H. Berestycki et al [3,4], and S. Alama and G. Tarantello [1] were substantially sharpened.…”

“…According to (11), t max (m) is decreasing with respect to m. Moreover, by continuous dependence, lim…”

“…So, the technical details of the proof in this special case are omitted here in. According to (12), the unique solution of (9) blows up in a time t max (m) ≤ α if m ≥ m * , while it is globally defined in the time interval [0, α] if m < m * .…”

Multiple positive solutions for a superlinear problem: A topological approach

“…Generally speaking, this fact suggests the existence of 2 n − 1 positive solutions (for µ large) when the weight function presents n positive humps separated by n − 1 negative ones. It is interesting to observe that already in [22], for the one dimensional case and f (x, s) = λs + a(x)u p , p > 1, Gómez-Reñasco and López-Gómez conjectured this result (for λ < 0 sufficiently small) based on some numerical evidence (see also [1]). The same multiplicity result holds for the indefinite sublinear case, that is for equation u + a(x)u p = 0, 0 < p < 1, (cf.…”

Nonnegative solutions for the degenerate logistic indefinite sublinear equation

“…In the last years the case m = 1 (q = 1 and p = 2) has attracted much attention, see [2], [3], [9], [10], [17], [22], [26] and references therein.…”