Existence and decay of global solutions of some nonlinear degenerate parabolic equations
The paper is concerned with the initial-boundary value problem: \[ u_ t-\Delta (| u|^ mu)+\nabla \cdot (g_ 1(u),g_ 2(u),...,g_ n(u))+h(u)=0\quad on\quad \Omega \times R^+,\quad u(x,0)=u_ 0,\quad u|_{\partial \Omega}=0, \] where \(m\geq 0\), \(\Omega\) is a bounded domain of \(R^ n\) with smooth boundary \(\partial \Omega\). Its object is to show that a modified method of potential well can be applied, and a global solution exists if the initial value \(u_ 0\) is small in a certain sense; a decay estimate of such solutions as t tends to infinity is also derived. A uniqueness result is given for \(m=0\).
scientific article; zbMATH DE number 3935606
scientific article