The free boundary problem describing information diffusion in online social networks (Q1932437)

The authors consider the one-phase Stefan problem with unknown functions \( u(t,x)\), \(h(t)\): \[ \begin{aligned} u_t - d u_{xx} & = r(t)\,u\,(1 - u/K),\;t>0, \;0< x < h(t), \\ & u_x(t,0) = 0, \;t >0, \\ u(t,h(t)) & = 0, \;\;h'(t) = -\mu u_x(t,h(t)) = 0, \;t >0, \tag{1} \\ h(0) & = h_0, \;u(0,x) = u_0(x), \;0 \leq x \leq h_0,- \end{aligned} \] where \(d > 0\), \(0 < r_\infty := \lim_{t\to+\infty}r(t) \leq r(t) \leq r(0)\). The authors prove that the problem (1) has a unique solution, such that \((u,\, h) \in C^{\frac{1+\alpha}{2}, 1+\alpha}(D) \times C^{1+\alpha/2}([0,\infty))\), \( \alpha\in (0,1)\), \(0 < u(t,x) \leq M\), \(0 < h'(t) \leq C\) for all \(t>0, \;0< x < h(t)\). If \(h_\infty := \lim_{t\to+\infty}h(t) < \infty\), then \(\lim_{t\to+\infty} ||u(t,\cdot)||_{C([0,\,h(t)])}=0\); if \(h_\infty = \infty\), then \(\lim_{t\to+\infty} u(t,x) = K\) uniformly in any bounded set of \([0,\,\infty)\); here \(D := \{(t,x): t \in [0,\infty), \;x \in [0,\,h(t)]\}\), \(M, \,C\) are positive constants.