Structure and bifurcation of pullback attractors in a non-autonomous Chafee-Infante equation (Q2845051)

This paper studies the non-autonomous version of the Chafee-Infante equation NEWLINE\[NEWLINEu_t=u_{xx}+\lambda u-\beta(t)u^3,\;\;0\leq x\leq \pi,\;\;t>\tauNEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0,t)=u(\pi,t)=0,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(x,\tau)=\phi(x).NEWLINE\]NEWLINE The autonomous (\(\beta\) constant) version of this problem is one of the few nonlinear partial differential equations whose global dynamics is completely understood. In this paper some of this understanding is extended to the non-autonomous case. Here the notion of global attractor used in the autonomous case is replace by that of the pullback attractor, and several results on this pullback attractor are proved under the sole assumption that \(\beta(t)\) is bounded from above and from below (by a positive constant), analogous to, though necessarily less precise than those available for the autonomous case. These include the existence of minimal and maximal bounded global solutions, upper and lower bounds on the dimension of the pullback attractor, and a description of the part of the pullback attractor in the positive cone. A notion of `non-autonomous equilibrium' is introduced, and a theorem ensuring existence of at least \(2N+1\) non-autonomous equilibria, when \(\lambda\in (\lambda_N,\lambda_{N+1})\) (\(\lambda_N=N^2\) are the eigenvalues of the linear part of the equation), is presented. Further results are obtained for the case of small non-autonomous perturbations, and for certain specific examples. The paper concludes with several open questions.