Weak stabilization for an evolution equation with nonlinear nonmonotone damping: The artificial problem method (Q2730788)

The following evolution equation is considered NEWLINE\[NEWLINE\frac{du}{dt}+Au+B(u)=0NEWLINE\]NEWLINE on a Hilbert space \(H\). Here, \(A\) is a linear, unbounded, conservative operator (i.e. \(\langle Au,u\rangle=0\) for all \(u\in D(A))\) and \(B:H\to H\) is an accretive operator, i.e., \(\langle B(u),u\rangle\geq 0\). Using the energy inequality (which is a consequence of the accretivity of \(B\)) for such general equations, the author proves stability results on weak solutions. The results are applied to some hybrid systems or to the wave equation with boundary or interior control.