Dynamics of solutions of a singular diffusion equation (Q5954505)

The aim of the article is to show that for any constant \(0 < c_1 < \infty\) and convex, smooth, bounded domain \(\Omega \subset \mathbb{R}^n\), and \(u_0\) satisfying \(c_1 \leq u_0 \in L^p(\Omega)\) for some \(p > \max(1,n/2)\) if \(u\) is the solution of NEWLINE\[NEWLINEu_t = \Delta (\log u), \quad \text{in } \Omega\times (0,\infty), \qquad u > 0 \quad \text{on } \overline{\Omega}\times (0,\infty),NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(x,t)= c_1 \quad \text{on } \partial\Omega\times (0,\infty), \qquad u(x,0) = u_0(x) \quad \text{in } x\in\Omega, NEWLINE\]NEWLINE and \(v = \log (u/c_1)\), \(w = e^{(\lambda_1/c_1)t}\), then the function \(w\) will converge uniformly to \(A\phi_1\) on \(\Omega\) for some constant \(A\) as \(t\to\infty\), where \(\lambda_1\) and \(\phi_1\) are the first positive eigenvalue and eigenfunction of the Laplace operator on \(\Omega\) with \(\|\phi_1\|_{L^2(\Omega)} = 1\), respectively.