Abstract
This work is concerned with the fast diffusion equation
with prescribed positive data on a smoothly bounded domain \({\Omega \subset \mathbb{R}^n}\), n ≥ 3 and positive m < 1. We consider solutions with boundary data u = a > 0 and initial data u 0(x) ≥ a that are continuous for \({x\ne 0\in \Omega}\) and have a singularity at x = 0. By skilfully choosing the behaviour of u 0 near 0 and under the further condition m < (n − 2)/n, we construct global in-time solutions u(x, t) that oscillate as t → ∞ between divergence to infinity at times t 2i → ∞ and convergence to a at times t 2i–1 → ∞. This happens locally uniformly in x.
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Vázquez, J.L., Winkler, M. Highly time-oscillating solutions for very fast diffusion equations. J. Evol. Equ. 11, 725–742 (2011). https://doi.org/10.1007/s00028-011-0107-1
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DOI: https://doi.org/10.1007/s00028-011-0107-1