*-solutions of evolution equations in Hilbert space
The paper gives sufficient conditions under which ''variation-of- constants'' solutions of semilinear evolution equations in a Hilbert space \(u'(t)=A(t)u(t)+g(u,t)\) are ''*-solutions'', i.e. solutions of \[ (u(t)- u(s),y)=\int^{t}_{s}((u(\sigma),A^*(\sigma)y)+(g(u,\sigma),y))d\sigma \] for all \(0\leq s<t\leq T\) and all \(y\in D=\cap_{0\leq t\leq T}D(A(t))\), assuming that this set is dense, and under which these two notions of solutions are equivalent. The case where D is not dense is also treated. As a motivation, the authors mention a class of stochastic evolution equations.
scientific article; zbMATH DE number 3991967
scientific article