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Erschienen in:
2015 | OriginalPaper | Buchkapitel
4. Boundary value problems for pseudo-differential equations with singular symbols
verfasst von : Sabir Umarov
Erschienen in: Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols
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Abstract
Let \(\varOmega \subset \mathbb{R}^{n}\) be a bounded domain with a smooth boundary or \(\varOmega = \mathbb{R}^{n}.\) This chapter discusses well-posedness problems of general boundary value problems for pseudo-differential and differential-operator equations of the form where f(t, x) is defined on \((T_{1},T_{2})\times \varOmega,\) \(-\infty <T_{1} <T_{2} \leq \infty,\) and \(\varphi _{k}(x),\,x \in \varOmega,\,k = 0,\ldots,m - 1,\) are given functions; A
k
(t) and b
kj
, \(k = 0,\ldots,m - 1,\,j = 0,\ldots,m - 1,\) are operators acting on some spaces (specified below) of functions defined on Ω; and \(t_{jk} \in [T_{1},T_{2}],\,j,k = 0,\ldots,m - 1.\) For example, when \(\varOmega = \mathbb{R}^{n},\) the latter operators may act as ΨDOSS defined on the space of distributions \(\varPsi _{-G,p^{'}}^{'}(\mathbb{R}^{n})\) with an appropriate \(G \subset R^{n}\).
$$\displaystyle\begin{array}{rcl} L[u]& \equiv & \frac{\partial ^{m}u} {\partial t^{m}} +\sum _{ k=0}^{m-1}A_{ k}(t)\frac{\partial ^{k}u} {\partial t^{k}} = f(t,x),\quad t \in (T_{1},T_{2}),\ x \in \varOmega,{}\end{array}$$
(4.1)
$$\displaystyle\begin{array}{rcl} B_{k}[u]& \equiv & \sum _{j=0}^{m-1}b_{ kj}\frac{\partial ^{j}u(t_{kj},x)} {\partial t^{j}} =\varphi _{k}(x),\quad x \in \varOmega,\,k = 0,\mathop{\ldots },m - 1,{}\end{array}$$
(4.2)