Asymptotic behaviour of solutions of \(u_t=\Delta\log u\) in a bounded domain
The author studies the large time behaviour of solutions of the equation \[ \frac{\partial u}{\partial t}=\Delta\log u,\quad u>0\tag{1} \] in a bounded cylindrical domain \(\Omega\times (0,\infty)\) with either finite or infinite lateral boundary values on \(\partial\Omega\times (0,\infty)\). The author shows that if \(u\) is the solution of (1), \(u=c_1\) on \(\partial\Omega\times (0,\infty)\), \(u(x,0)=u_0(x)\geq 0\) on \(\Omega\subset \mathbb{R}^n\), where \(\Omega\) is a smooth convex bounded domain, then for \(c_1=\infty\) the rescaled function \(w=\log(\frac ut)\) will converge uniformly on every compact subset of \(\Omega\) to the unique solution \(\psi\) of the equation \[ \Delta\psi-e^\psi=0,\quad \psi>0\text{ in }\Omega \] with \(\psi=\infty\) on \(\partial\Omega\) as \(t\to\infty\). Moreover he analyzes the case of finite \(c_1\).
finite or infinite lateral boundary values
scientific article; zbMATH DE number 1832813