Smooth solution for the porous medium equation in a bounded domain
There is considered the problem with unknown function (1) \(u(x,t): u_t=\Delta u^m\) in \(\Omega\subset \mathbb{R}^n\), \(u|_{t=0}=u_0\) in \(\Omega\), \(u_{\partial \Omega}=0\), where \(m>1\), \(u_0\) is nonnegative and compactly supported in \(\Omega\) function. After the substitution \(f=u^m\) the problem (1) is reduced to the problem, for which the short time existence of the solution \(f\) in the weighted Hölder space is established. To prove this the authors obtain solvability and coercive estimate of the solution of a corresponding model linearized problem in \(\mathbb{R}^n_+\) for the equation \(f_t-x_n^\alpha \Delta f=g\), \(\alpha\in(0,1)\).
scientific article; zbMATH DE number 5577109