The Cauchy Problem for Higher Order Abstract Differential Equations

✍ Scribed by Ti-Jun Xiao, Jin Liang (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 1998
• Tongue: English
• Leaves: 314
• Series: Lecture Notes in Mathematics 1701
• Edition: 1
• Category: Library

✦ Synopsis

The main purpose of this book is to present the basic theory and some recent de­ velopments concerning the Cauchy problem for higher order abstract differential equations u(n)(t) + ~ AiU(i)(t) = 0, t ~ 0, { U(k)(O) = Uk, 0 ~ k ~ n-l. where AQ, Ab . . . , A - are linear operators in a topological vector space E. n 1 Many problems in nature can be modeled as (ACP ). For example, many n initial value or initial-boundary value problems for partial differential equations, stemmed from mechanics, physics, engineering, control theory, etc. , can be trans­ lated into this form by regarding the partial differential operators in the space variables as operators Ai (0 ~ i ~ n - 1) in some function space E and letting the boundary conditions (if any) be absorbed into the definition of the space E or of the domain of Ai (this idea of treating initial value or initial-boundary value problems was discovered independently by E. Hille and K. Yosida in the forties). The theory of (ACP ) is closely connected with many other branches of n mathematics. Therefore, the study of (ACPn) is important for both theoretical investigations and practical applications. Over the past half a century, (ACP ) has been studied extensively.

✦ Table of Contents

Front Matter....Pages N2-XII
Laplace transforms and operator families in locally convex spaces....Pages 1-44
Wellposedness and solvability....Pages 45-83
Generalized wellposedness....Pages 85-140
Analyticity and parabolicity....Pages 141-176
Exponential growth bound and exponential stability....Pages 177-197
Differentiability and norm continuity....Pages 199-238
Almost periodicity....Pages 239-261
Back Matter....Pages 263-309

✦ Subjects

Ordinary Differential Equations