Well-posedness of differential-operator problems. I: The Cauchy problem in spaces of distributions
Let \(A: D(A) \subset X \rightarrow X\) be a linear, closed, densely defined operator on a Banach space \(X\) and \[ X_A:=\{D(A), \;\| x\| _1:=\| x\| + \| Ax\| \}. \] Denote by \({\mathcal D}\) the space of Schwartz test functions, by \({\mathcal D}'(X)\) the space of abstract distributions and by \({\mathcal D}_0'(X)\) the subspace of distributions with support in \([0,\infty)\). Consider the Cauchy problem for first-order equations \[ u'(t) = Au(t),\quad t\geq 0,\quad u(0)=x. \tag{1} \] A distribution \(U\in {\mathcal D}_0'(X_A)\) is called a solution to the Cauchy problem (1) in the sense of distributions if \[ \forall \varphi \in {\mathcal D},\;\langle U,\varphi'\rangle + A\langle U,\varphi\rangle = -\langle\delta,\varphi\rangle x,\quad x\in X. \] The Cauchy problem is called well-posed in the sense of distributions if, for any \(x \in X\), there exists a unique solution \(U\in {\mathcal D}_0'(X_A)\) and, for any \(x_n \rightarrow 0\), the corresponding solution \(U_n \rightarrow 0\) in \({\mathcal D}_0' (X_A)\). The aim of this paper is to prove the equivalence of some statements on the well-posedness of the Cauchy problem (1). For example, problem (1) is well-posed in the sense of distributions iff \(A\) is the generator of a \(d\)-semigroup with dense range. An analog theorem in the space of exponential distributions is also given.
scientific article; zbMATH DE number 1292310
scientific article