zbMATH — the first resource for mathematics

Linear distribution differential equations. (English) Zbl 0552.34004
Let \(c\in {\mathbb{R}}\), let \({\mathcal P}^ 0\) be the set of Borel measures with support in \(x\leq c\). Let \({\mathcal B}^ 0\) be the set of all locally bounded Borel measurable functions with support in \(x\geq c\). Let \(\mu\in {\mathcal D}'({\mathbb{R}})\). Let \(j\geq 0\) be an integer. If for some \(\eta\in {\mathcal P}^ 0\) \(D^ j\mu =\eta\) then \(\mu\) is said to be in \({\mathcal P}^ j\). If \(D^ j\eta =\mu\) then \(\mu\) is said to be in \({\mathcal P}^{-j}\). \({\mathcal B}^ j\) and \({\mathcal B}^{-j}\) are defined in the same way with \({\mathcal B}^ 0\) as space of departure. For \(a\in {\mathcal P}^{-k}\) and \(f\in {\mathcal B}^ k\) af defines a distribution in a natural way. Let \(n=2\ell-1\) with \(\ell \geq 1\), \(\ell\) an integer. The present paper treats the Cauchy problem with zero initial data for (*) \(u^{(n)}+a_{n-1}u^{(n-1)}+... +a_ 0u=f\). Here \(a_ i\in {\mathcal P}^{i+1-\ell}\), \(0\leq i<n\), and \(f\in {\mathcal P}^{\ell-1}\). Then there is a unique \(u\in {\mathcal B}^{\ell -1}\) solving (*). A corresponding theorem is proved for \(n=2\ell\). If \(\ell =1\) one also requires (**) \(a_{n-1}(\{x\})\neq -1.\) Further information on equations with measures as coefficients is found in the author’s earlier articles [Matematiche 36, 151-171 (1981), Ann. Mat. Pura Appl., IV. Ser. 132, 177-187 (1982; Zbl 0522.34004), Rend. Semin. Mat., Torino, Fascicolo speciale 1983, 207-219].

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
46F99 Distributions, generalized functions, distribution spaces