Exact non-self-similar solutions of the equation \(u_t=\Delta\ln u\) (Q1810031)

This paper is devoted to the construction of exact, non-self-similar, anisotropic (in the sense of variables), explicit, nonnegative solutions of the nonlinear diffusion (heat) equation \[ u_t=\Delta\ln u\tag{1} \] where \(u= u(x,t): \Omega\times \mathbb{R}^+\to \mathbb{R}\), \(x\in\mathbb{R}^n\), \(\Omega\subset\mathbb{R}^n\) is the domain and \(\mathbb{R}^+= \{t\mid 0\leq t<\infty\}\), \(u(x,t)\geq 0\) is the temperature of the medium.