## 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].

### MSC:

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

Zbl 0522.34004