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Zbl 0552.34004
Linear distribution differential equations.
(English)
[J] Comment. Math. Univ. St. Pauli 33, 119-126 (1984). ISSN 0010-258X

Let $c\in {\bbfR}$, let ${\cal P}\sp 0$ be the set of Borel measures with support in $x\le c$. Let ${\cal B}\sp 0$ be the set of all locally bounded Borel measurable functions with support in $x\ge c$. Let $\mu\in {\cal D}'({\bbfR})$. Let $j\ge 0$ be an integer. If for some $\eta\in {\cal P}\sp 0$ $D\sp j\mu =\eta$ then $\mu$ is said to be in ${\cal P}\sp j$. If $D\sp j\eta =\mu$ then $\mu$ is said to be in ${\cal P}\sp{-j}$. ${\cal B}\sp j$ and ${\cal B}\sp{-j}$ are defined in the same way with ${\cal B}\sp 0$ as space of departure. For $a\in {\cal P}\sp{-k}$ and $f\in {\cal B}\sp k$ af defines a distribution in a natural way. Let $n=2\ell-1$ with $\ell \ge 1$, $\ell$ an integer. The present paper treats the Cauchy problem with zero initial data for (*) $u\sp{(n)}+a\sb{n-1}u\sp{(n-1)}+... +a\sb 0u=f$. Here $a\sb i\in {\cal P}\sp{i+1-\ell}$, $0\le i<n$, and $f\in {\cal P}\sp{\ell-1}$. Then there is a unique $u\in {\cal B}\sp{\ell -1}$ solving (*). A corresponding theorem is proved for $n=2\ell$. If $\ell =1$ one also requires (**) $a\sb{n-1}(\{x\})\ne -1.$ 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 2000:
*34A12 Initial value problems for ODE
46F99 Generalized functions, etc.

Keywords: distributions as coefficients; distribution test functions; Cauchy problem; measures as coefficients

Citations: Zbl 0522.34004

Cited in: Zbl 0688.34003 Zbl 0643.34004

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