Boundary value problems for pseudo-differential equations with singular symbols

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

$$\displaystyle\begin{array}{rcl} L[u]& \equiv & \frac{\partial ^{m}u} {\partial t^{m}} +\sum _{ k=0}^{m-1}A_{ k}(t)\frac{\partial ^{k}u} {\partial t^{k}} = f(t,x),\quad t \in (T_{1},T_{2}),\ x \in \varOmega,{}\end{array}$$

(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)

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 $G \subset R^{n}$.