The Cauchy Problem for Higher Order Abstract Differential Equations

Ti-Jun Xiao, Jin Liang (auth.)3540652388, 9783540652380

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.

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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