zbMATH — the first resource for mathematics

Global attractors of one-dimensional parabolic equations: Sixteen examples. (English) Zbl 0814.35056
The author considers global attractors of semiflows generated by dissipative one dimensional parabolic equations \[ u_ t = u_{xx} + f(x,u,u_ x), \quad - 1 < x < 1, \tag{1} \] under Neumann boundary conditions. He lists (describes in terms of number of equilibria, their Morse indices and existing heteroclinic connections) sixteen examples of such attractors that arise if the equation has no more than 9 hyperbolic equilibria. A discussion of general properties of the attractor of (1) is also included.

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
34D45 Attractors of solutions to ordinary differential equations
35K57 Reaction-diffusion equations
35B32 Bifurcations in context of PDEs
Full Text: EuDML