Interfaces in multidimensional diffusion equations with absorption terms
The author studies the properties of interfaces in solutions of the Cauchy problem for the nonlinear degenerate parabolic equation \[ {\partial u\over\partial t}=\Delta u^m- au^p\quad\text{in }\mathbb{R}^n\times (0,T],\quad u= 1,2,3 \] with the parameters \(m> 1\), \(p> 0\), \(a> 0\) satisfying the condition \(m+ p\geq 2\). The author shows that the velocity of the interface \(\Gamma(t)= \partial\overline{\{\text{supp }u(x,t)\}}\) is given by the formula \[ v= \Biggl[-{m\over m-1} \nabla u^{m-1}+ \nabla\Pi\Biggr]\Biggl|_{\Gamma(t)}, \] where \(\Pi\) is the solution of the degenerate elliptic equation \[ \text{div}(u\nabla\Pi)- au^P= 0,\quad \Pi= 0\quad\text{on }\Gamma(t). \] The author gives explicit formulas which represent the interface \(\Gamma(t)\) as a bijection from \(\Gamma(0)\). It is proved that the solution \(u\) and its interface \(\Gamma(t)\) are analytic functions of time \(t\).
scientific article; zbMATH DE number 1910238