A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation

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Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems (2007, methodology): “- they can be proved directly by monotonicity techniques based on the maximum principle (such as moving planes or spheres [17, 9, 10, 6, 25, 7], or intersectioncomparison [ 23 ] for certain parabolic problems).”
Locally uniform convergence to an equilibrium for nonlinear parabolic equations on $R^N$ (2015, background): “By [28], no such homoclinic solution exists if 1 < p < pS, and in particular, for any p > 1, if N = 1 or N = 2.”
Infinite-time blow-up for the 3-dimensional energy-critical heat equation (2017, unknown): “Positive radial global solutions must be bounded and go to zero, see [26, 28, 32].”
Optimal Liouville theorems for superlinear parabolic problems (2020, background): “The result in [22] can also be formulated as follows: If p < pS and we consider radial functions only, then any positive classical bounded solution of (1) has to be stationary.”
Threshold Solutions and Sharp Transitions for Nonautonomous Parabolic Equations on $${\mathbb{R}^N}$$ (2011, background): “It is known that in the class of radially symmetric solutions the Liouville theorem holds in the Sobolevsubcritical range 1 < p < pS = (N + 2)/(N − 2)+ (see [30, 31, 3]).”