Measure-valued solutions and the phenomenon of blow-down in logarithmic diffusion
The author studies the existence, uniqueness and behavior of generalized solutions of an initial value problem for the equation \(u_t=\Delta\log(u+c)\), where \(x\in\mathbb R^2\), \(t>0\) and \(c\geq0\) is a constant. The initial value is a general nonnegative Radon measure of the form \(\mu(x)=\sum_{i=1}^\infty M_i\delta(x-x_i)+\mu_0(x)\), where \(\mu_0\) is non-atomic. In particular, the author shows that the solution becomes classical after a finite time and the singular part of the solution has the form \(u_{\text{sing}}(x,t)=\sum_i(M_i-4\pi t)_+\delta(x-x_i)\). The total mass of the solution is preserved if \(c>0\), while \(\int u(x,t)\,dx=(\mu(\mathbb R^2)-4\pi t)_+\) if \(c=0\).
scientific article; zbMATH DE number 5543869
scientific article