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

In this paper, we investigate the behavior of classical solutions to the one‐dimensional (1D) logarithmic diffusion equation with nonlinear Robin boundary conditions, namely, ∂tu=∂xxloguin−l,l×0,∞∂xu±l,t=±2γup±l,t,$$\begin{equation*} {\left\lbrace \def\eqcellsep{&}\begin{array}{l}\partial _t u=\partial _{xx} \log u\quad \mbox{in}\quad {\left[-l,l\right]}\times {\left(0, \infty \right)}\\[3pt] \displaystyle \partial _x u{\left(\pm l, t\right)}=\pm 2\gamma u^{p}{\left(\pm l, t\right)}, \end{array} \right.} \end{equation*}$$where γ is a constant. Let u0 > 0 be a smooth function defined on [ − l, l], and which satisfies the compatibility condition ∂xlogu0±l=±2γu0p−1±l.$$\begin{equation*} \partial _x \log u_0{\left(\pm l\right)}= \pm 2\gamma u_0^{p-1}{\left(\pm l\right)}. \end{equation*}$$We show that for γ > 0, p≤32$p\le \frac{3}{2}$ 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