In this paper, we construct solutions u(t,x) of the heat equation on \({\mathbb{R}}^N\) such that \(t^{\frac {\mu } {2}} u(t, xt^\beta )\) has nontrivial limit points in \(C_0({\mathbb{R}}^N)\) as t → ∞ for certain values of μ > 0 and β > 1/2. We also show the existence of solutions of this type for nonlinear heat equations.
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Cazenave, T., Dickstein, F. & Weissler, F.B. Nonparabolic Asymptotic Limits of Solutions of the Heat Equation on \({\mathbb{R}}^N\) . J Dyn Diff Equat 19, 789–818 (2007). https://doi.org/10.1007/s10884-007-9076-z
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DOI: https://doi.org/10.1007/s10884-007-9076-z