Estabilidade, instabilidade e fenômenos de concentração em equações de reação e difusão: uma abordagem geométrica - Souza, Carolinne Stefane de (2022)
In this work, we adress the study of stability of a reaction-difusion equation in two domains: in a family of surfaces of revolution without boundary and in a bounded open interval whose diffusion function vanishes at a point inside this interval. In the first problem, in addition to proving the existence of stable non-constant stationary solutions for the Allen-Cahn equation in a family of surfaces of revolution without boundary, we analyze the asymptotic behavior and the instability of solutions when λ → ∞, where λ is a positive parameter whose inverse can be interpreted as a difusibility coefficient. While in the second problem, called degenerate Allen-Cahn problem, because we are dealing with a difusion function that vanishes at a point inside the interval, the main operator is not uniformly elliptical and for this reason, it is not possible to use results of the related literature nor the usual spaces. In this case, we prove that a specific function, namely a step function taking only two values – the stable zeros of the bistable function f –, is a stable non-constant stationary solution of degenerate problem.