Stability for a class of semilinear parabolic equations (Q1891561)

A class of semilinear initial value problems in a general Banach space \(X\), \(u'(t)= Au(t)+ f(u(t))\), \(t> 0\), \(u(0)= u_ 0\), is considered, where \(A: D(A)\subset X\mapsto X\) is a sectorial operator and \(f\) is a sufficiently regular nonlinear function defined in an intermediate space \(X_ \alpha\) between \(X\) and \(D(A)\), with \(f(0)= 0\). It is supposed that \(\| e^{tA}\|_{L(X)}\leq M_ 0\), \(\| te^{tA}\|_{L(X)}\leq M_ 1\), \(t> 0\). Under suitable growth assumptions on \(f\) is a global existence and asymptotic decay theorem proved in the case where the initial datum is small in \(X_ \alpha\). The used methods are: method of contraction and method of a priori estimates. The abstract results are applied to some nonlinear evolution problems of the second and fourth order.