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A non-autonomous bifurcation problem for a non-local scalar one-dimensional parabolic equation

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    * Corresponding author
The first author was supported by NSFC Grant 11671367. The second author was supported by Grants FAPESP 2018/10997-6 and CNPq 306213/2019-2. The third author was supported by FAPESP Grant 2019/20341-3. The fourth author was supported by Grants FAPESP 2018/00065-9 and CAPES-Scholarship 7547361/D
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  • In this paper we study the asymptotic behaviour of solutions for a non-local non-autonomous scalar quasilinear parabolic problem in one space dimension. Our aim is to give a fairly complete description of the forward asymptotic behaviour of solutions for models with Kirchhoff type diffusion. In the autonomous case we use the gradient structure, symmetry properties and comparison results to obtain a sequence of bifurcations of equilibria, analogous to what is seen in the local diffusivity case. We provide conditions so that the autonomous problem admits at most one positive equilibrium and analyse the existence of sign changing equilibria. Also using symmetry and the comparison results (developed here) we construct what is called non-autonomous equilibria to describe part of the asymptotics of the associated non-autonomous non-local parabolic problem.

    Mathematics Subject Classification: Primary: 35B09, 35B51, 35B32; Secondary: 35B41, 35B06, 35B40.


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  • Figure 1.  Region bounded by the positive equilibria $\phi^+_{1, b_1}$ and $\phi^+_{1, b_2}$

    Figure 2.  The set $X^+_2, $ the functions that lie between $\phi^+_{2, b_1}$ and $\phi^+_{2, b_2}$

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