A general result on measure solutions for semilinear evolution equations (Q1587212)

Generalized solutions are considered to semilinear evolution equations of the type \[ u'(t)= Au(t)+ f(t,u),\quad t>0, \] where \(A\) is the infinitesimal generator of a \(C_0\)-semigroup in a Banach space \(X\ni u\) and \(f= f(t,u)\) is measurable with respect to \(t\), continuous with respect to \(u\), and fulfills some linear growth condition. Extending the solution concept to measures allows to prove existence results under rather weak assumptions. The author firstly proves existence if \(f= f(t,u)\) is continuous and bounded on \(X\ni u\). Afterwards, the assumptions are weakened, and the main result is the existence also for continuous \(f\) that is bounded on bounded subsets of \(X\). Finally, the author deals with the autonomous case \(f= f(u)\), where \(A\) is the generator of an analytic semigroup. The paper ends with some remarks on quasilinear equations with \(A= A(u)\).