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

Jean C. Cortissoz and César Reyes

Mathematische Nachrichten, 2023, vol. 296, issue 9, 4086-4107

Abstract: 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

Date: 2023
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