Exact Non-Self-Similar Solutions of the Equation

In this paper, we obtain new exact non-self-similar solutions of the nonlinear diffusion equation
$$u_t=\Delta \ln u,\quad u\triangleq u(\mathbf x,t):\Omega\times\mathbb R^+\to\mathbb R, \quad\mathbf x\in\mathbb R^n,$$
where $\Omega\subset\mathbb R^n$ is the domain and $\mathbb R^+=\{t:0\le t<+\infty\}$, $u(\mathbf x,t)\ge0$ is the temperature of the medium.