Measure-valued solutions of nonlinear parabolic equations with logarithmic diffusion
The authors study the existence and regularity of a Radon measure-valued solution for a class of Cauchy problems for nonlinear degenerate parabolic equations of the form \[ \begin{cases} u_t=\Delta (\psi(u)) & \text{in} \;\Omega\times(0,T),\\ u=0 & \text{on} \;\partial\Omega\times(0,T),\\ u=u_0 & \text{in} \;\Omega\times\{0\}, \end{cases} \] where \(\Omega\subset \mathbb{R}^N,\) \(N\geq2,\) is a bounded and smooth domain and the initial datum \(u_0\) is a bounded Radon measure. Regarding the function \(\psi\in C^\infty(\mathbb{R}),\) it is supposed that \(\psi(0)=0,\) \(\psi'(s)>0\) and there exist \(s_0>0\) and \(\beta>\alpha>0\) such that \[ \dfrac{\alpha}{1+|s|}\leq \psi'(s)\leq \dfrac{\beta}{1+|s|}\quad \text{for all } |s|\geq s_0. \] In particular, it is proved that a regularizing effect appears if the initial datum is diffused with respect to the ``\(C_2\)-capacity'', since in this case the solution becomes a summable function. Moreover, the uniqueness of measure-valued solutions is studied as well.
nonlinear degenerate parabolic equations
scientific article; zbMATH DE number 6496639
scientific article