Properties of Lions's \(d\)-semigroups and generalized well-posedness of the Cauchy problem (Q1128365)

Let \(X\) be a Banach space and let \(A\): \(D(A) \subset X\to X\) be a closed linear densely defined operator. Consider the Cauchy problem \[ u'(t)= Au(t),\quad t\geq 0,\quad u(0)=x.\tag{1} \] Denote by \(S\) the Schwartz space of infinitely differentiable functions \(\varphi: \mathbb{R}\to\mathbb{R}\) with compact supports. Let \(S'(X)\) be the space of abstract distributions and \(S_0'(X)\) the subspace of distributions with supports in \(\mathbb{R}_+\). A distribution \(u\in S_0'(X)\) is called a solution to (1) in the sense of distributions if for any \(\varphi \in S\) the equality holds \[ u(\varphi') +Au(\varphi)= -\delta (\varphi)x \tag{2} \] \((x\in X)\). The problem (1) is said to be well-posed in the sense of distributions if for any \(x\in X\) there exists a unique solution to (2) and for any sequence \(x_n\to 0\) the corresponding sequence \(\{u_n\}\) tends to zero in \(S_0'(X)\). The author gives conditions equivalent to the well-posedness of the Cauchy problem (1) in the sense of distributions. Those conditions are formulated mainly in terms of distribution semigroups or integrated semigroups. Few examples are presented illustrating the obtained results.