Decay rate of the range component of solutions to some semilinear evolution equations (Q879915)

The author examines the rate of decay to zero, as \(t\to+\infty\), of the projection on the range of \(A\) of the solutions of an equation of the form \(u'+Au+| u| ^{p-1} = 0\) or \(u''+u'+Au+| u| ^{p-1} = 0\) in a bounded domain of \(\mathbb R^N\), where \(A=-\Delta\) with Neumann boundary conditions or \(A=-\Delta-\lambda_1I\) with Dirichlet boundary conditions. In the paper it is shown that this decay is much faster than the decay of the projection on the kernel; it is often exponential, but apparently not always.