Linearized stability for nonlinear evolution equations (Q1397627)

Let \(A\) be a multivalued operator from a Banach space \(X\) into itself. Consider the Cauchy problem \[ u'(t)+Au(t)\ni 0,\quad t\geq 0, \qquad u(0)=u_0.\tag{1} \] In the first part of this interesting article, the author proves a general principle of linearized stability at an equilibrium point of (1). More precisely, under the assumptions that \(A\) is \(\omega\)-accretive and has a resolvent differential \(\widehat{A}\) at the equilibrium point \(x_e\), (1) is proved to be locally exponentially stable at \(x_e\), provided \(\widehat{A}- \widehat{\omega}I\) is accretive for some \(\widehat{\omega}>0\). The above principle is then used to obtain linearized stability results when \(A=B+G\), with minimal differentiability conditions on \(B\) and \(G\) and for partial differential delay equations.