Volterra and Functional Differential Equations

DIFFERENTIABILITY PROPERTIES OF PSEUDOPARABOLIC POINT CONTROL PROBLEMS

L. W. White

Department of Mathematics and Energy Resources Center

The University of Oklahoms

Norman, Oklahoma

DOI: 10.1201/9781003420026-25

1. INTRODUCTION

In this paper, we study the following problem: Let Ω be a nonempty bounded open subset of ℝ^P,P=2 or 3, with a smooth boundary Γ, and let Q = Ω × (0,T), Σ = Γ × (0,T), and a ∈ Ω. Consider the pseudoparabolic problem

$$\{\begin{array}{l}\text{M}{y}_t+\text{Ly}=\text{v}\left(\text{t}\right)\varphi \left(\text{x}-\text{a}\right)\text{ in Q}\hfill \\ \text{y}\left(\text{x},0;\text{v}\right)=0\text{ in }\Omega \hfill \\ \text{y}\left(\text{x},\text{t};\text{v}\right)=0\text{ on Σ}\hfill \end{array}$$ (1)

Where M=N[x] and L=L(x) are second order symeetrie uniformly elliptic operators. The function may ɸ be an “approximate identity” with the properties:

$$\{\begin{array}{}\end{array}$$