The porous medium equation with nonlinear absorption and moving boundaries
Dirichlet and Neumann boundary value problem for the nonlinear parabolic equation \[ (1)\quad u_ t=[\phi (u)]_{xx}+f(u) \] are analysed, when \(\phi\) : \(R\to R\) is \(C^ 2\) and \(\phi (0)=0\) and \(\phi '(r)>0\) if \(r\neq 0\). The main examples are \(\phi (r)=dr^ m\), \(d>0\), \(m\geq 1\). The paper is divided into three sections. The first part deals with the existence of (generalized) solutions to the Dirichlet and Neumann problems with moving boundaries by using the theory of nonlinear semigroups. The solutions are constructed using approximations and the Crandall-Ligett theorem for the generation of semigroups in general Banach spaces. The second part deals with the asymptotic behavior of solutions, and the third part gives some differentiability properties of the generalized solutions. It is shown that the general types of behavior for the solution of equation (1) is similar to the semilinear case.
scientific article; zbMATH DE number 3987820
scientific article