| Full text | |
| Author(s): |
Arrieta, Jose M.
;
Carvalho, Alexandre N.
;
Moreira, Estefani M.
;
Valero, Jose
Total Authors: 4
|
| Document type: | Journal article |
| Source: | Advances in Differential Equations; v. 29, n. 1-2, p. 26-pg., 2024-01-01. |
| Abstract | |
In this article, we study the scalar one-dimensional nonlocal quasilinear problem of the form ut = a(parallel to ux parallel to 2)uxx + nu f (u), with Dirichlet boundary conditions on the interval [0, pi], where a : R+- [m, M] subset of (0, +infinity) and f : R- R are continuous functions that satisfy suitable additional conditions. We give a complete characterization of the bifurcations and hyperbolicity for the corresponding equilibria. With respect to bifurcation, the existing result requires that the function a(center dot) be non-decreasing and shows that bifurcations are pitchfork supercritical bifurcations from zero. We extend these results to the case of a general smooth nonlocal diffusion function a(center dot) and show that bifurcations may be pitchfork or saddle-node, both subcritical or supercritical. Concern-ing hyperbolicity, we specifying necessary and sufficient conditions for its occurrence. We also explore some examples to exhibit the variety of possibilities, depending on the choice of the function a(center dot), that may occur as the parameter nu varies. (AU) | |
| FAPESP's process: | 23/01225-8 - Asymptotic and qualitative analysis of partial differential equations |
| Grantee: | Estefani Moraes Moreira |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |
| FAPESP's process: | 18/00065-9 - Gradient structure of skew product semiflows |
| Grantee: | Estefani Moraes Moreira |
| Support Opportunities: | Scholarships in Brazil - Doctorate |
| FAPESP's process: | 20/00104-4 - Comparison results between solutions of autonomous and non-autonomous problems: an investigation about existence of non-autonomos equilibria |
| Grantee: | Estefani Moraes Moreira |
| Support Opportunities: | Scholarships abroad - Research Internship - Doctorate |
| FAPESP's process: | 20/14075-6 - Dynamical systems and their attractors under perturbations |
| Grantee: | Alexandre Nolasco de Carvalho |
| Support Opportunities: | Research Projects - Thematic Grants |