Classical solutions to the one‐dimensional logarithmic diffusion equation with nonlinear Robin boundary conditions

Abstract

AbstractIn this paper, we investigate the behavior of classical solutions to the one‐dimensional (1D) logarithmic diffusion equation with nonlinear Robin boundary conditions, namely, where γ is a constant. Let u0 > 0 be a smooth function defined on [ − l, l], and which satisfies the compatibility condition We show that for γ > 0, classical solutions to the logarithmic diffusion equation above with initial data u0 are global and blow‐up in infinite time, and that for p > 2 there is finite time blow‐up. Also, we show that in the case of γ < 0, , 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 rates and blow‐down rates along sequences of times. Our approach is based on studying the Ricci flow on a cylinder endowed with a ‐symmetric metric, and some comparison arguments. 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.