Analytic semigroups, degenerate elliptic operators and applications to nonlinear Cauchy problems
The nonlinear problem \[ u_ t=\phi (\Theta (u))\Delta (\chi (u)),\quad x\text{ in } {\bar \Omega},\quad t\geq 0;\quad u(x,0)=u_ 0(x) \] is considered, where \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\) with \({\mathbb{C}}^{\infty}\) boundary. \(\phi\), \(\Theta\), \(\chi\) are smooth functions and \(u_ 0\) is positive and sufficiently regular. A large class of problems in applications is of this form. He studies the problem by analyzing the associated degenerate elliptic problem \((\lambda -E)u=f\), \(\lambda\in {\mathbb{C}}.\) He proves \(L^ p(\Omega)\) and \(C^{\alpha}({\bar \Omega})\)-norm estimates for the solution of the elliptic problem.
scientific article; zbMATH DE number 4165485
scientific article