On bounded and unbounded global solutions of a supercritical semilinear heat equation

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Infinite-time blow-up for the 3-dimensional energy-critical heat equation (2017, background): “be bounded and go to zero, see [25, 26, 33]. They exist when p> n+2 n−2 but they have inﬁnite energy, see [17]. Inﬁnite time blow-up exists in that case but it has an entirely diﬀerent nature, see [28, 29]. The study of energy critical problems has attracted much attention in the last decade. For”
Locally uniform convergence to an equilibrium for nonlinear parabolic equations on $R^N$ (2015, background): “In [29], a positive, bounded, localized solution slowly oscillating up and down along a simply ordered family of equilibria was found (see also [31] for a related quasiconvergence result).”
Singularity and decay estimates in superlinear problems via liouville-type theorems. Part II: Parabolic equations (2007, background): “For n ≥ 11 and p larger than a certain exponent p∗(n) (> pS), it has even been shown [29] that unbounded global (classical) solutions of (1.”
Spreading Properties and Complex Dynamics for Monostable Reaction–Diffusion Equations (2012, background): “Related complex behaviors have also been exhibited for reaction-diffusion equations in bounded domains [ 19 ], or in ℝ N [ 20 ] with power-like nonlinearities, where the ω-limit set may be a continuum of equilibria.”