On the one dimensional Logarithmic diffusion equation with nonlinear Robin boundary conditions

Sat, 27 Feb 2021 00:00:00 GMT

Jean Cortissoz, César Reyes — In this paper we investigate the one dimensional (1D) logarithmic diffusion equation with nonlinear Robin boundary conditions, namely, \{ beginarrayl ∂ₜ u=∂ₓₓ log u in [−l,l]× (0, ∞)\ ∂ₓ u(± l, t)=± 2γu^(p)(± l, t), endarray where γ is a constant. Let u₀>0 be a smooth function defined on [−l,l], and which satisfies the compatibility condition ∂ₓ log u₀(± l)= ± 2γu₀^(p-1)(± l). We show that for γ> 0, p≤ 3/2 solutions to the logarithmic diffusion equation above with initial data u₀ are global and blow-up in infinite time, and for p>2 there is finite time blow-up. Also, we show that in the case of γ<0, p≥ 3/2, solutions to the logarithmic diffusion equation with initial data u₀ are global and blow-down in infinite time, but if p≤ 1 there is finite time blow-down. For some of the cases mentioned above, and some particular families of examples, we provide blow-up and blow-down rates. Our approach is partly based on studying the Ricci flow on a cylinder endowed with a S¹-symmetric metric. Then, we bring our ideas full circle by proving a new long time existence result for the Ricci flow on a cylinder without any symmetry assumption. Finally, we show a blow-down result for the logarithmic diffusion equation on a disc.

César Reyes — Neckpinch

On the one dimensional Logarithmic diffusion equation with nonlinear Robin boundary conditions