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We give an astonishingly simple proof of a Liouville type theorem of Fila and Yanagida which asserts that there do not exist nontrivial ancient solutions to the nonlinear heat equation $u_t=\Delta u+|u|^{p-1}u$ that are smaller in absolute value than the radial singular steady state, provided that the exponent $p$ is larger or equal to the critical Sobolev exponent and smaller than that of Joseph and Lundgren. Our proof is based on a sweeping argument and does not use similarity variables. Our proof is based on a sweeping argument and does not use similarity variables. Using this approach, we can also prove an analogous Liouville type theorem for the steady state problem in higher dimensions.

A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart