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Existence and decay of global solutions of some nonlinear degenerate parabolic equations. (English) Zbl 0584.35062
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 \(m=0\).
Reviewer: C.-Y.Chan

MSC:
35K65 Degenerate parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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