Boundedness of global positive solutions of a porous medium equation with a moving localized source (Q2370751)

The authors study the boundedness of global positive solutions to the following initial boundary value problem \[ \begin{gathered} u_t= u^r(\Delta u+ af(u(u_0(t), t))),\quad x\in\Omega,\;t> 0,\\ u(x,t)= 0,\;x\in\partial\Omega,\;t> 0,\;u(x,0)= u_0(x),\;x\in\Omega,\end{gathered}\tag{1} \] where \(0< r< 1\) and \(a> 0\) are constants, \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with \(C^{3+\alpha}\) boundary \(\partial\Omega\) for some \(\alpha\in (0,1)\), \(x_0(t)\), \(t\geq 0\) is a moving point in \(\Omega\). Results regarding to global existence and finite time blow-up for (1) are presented.