Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains

By Peter Poláčik, F. Simondon. 2002

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A Review on the Cahn-Hilliard Equation: Classical Results and Recent Advances in Dynamic Boundary Conditions (2021, background): “…examples have been given in the literature such that even for certain simple semilinear second order parabolic equation with a specific smooth nonlinear term, it admits a globally bounded solution whose ω -limit set is di ff eomorphic to the unit circle S 1 , namely, a continuum (see e.g., [131]).”
Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level (2005, background): “In general, it cannot be improved with the statement that the whole trajectory converges, see [32] and references therein.”
Global existence versus blow-up results for a fourth order parabolic PDE involving the Hessian (2015, background): “10 cannot be improved with the statement that the whole trajectory converges, see [31, 32] and references therein for second order equations.”
Locally uniform convergence to an equilibrium for nonlinear parabolic equations on $R^N$ (2015, background): “For a general f = f(u), in dimension N ≥ 2, the convergence issue remains unresolved; it is known, however, that nonconvergent bounded solutions do exist if f is allowed to depend on x: f = f(x, u), see [23, 24].”