1st Edition

Delay Differential Evolutions Subjected to Nonlocal Initial Conditions

    388 Pages
    by Chapman & Hall

    Filling a gap in the literature, Delay Differential Evolutions Subjected to Nonlocal Initial Conditions reveals important results on ordinary differential equations (ODEs) and partial differential equations (PDEs). It presents very recent results relating to the existence, boundedness, regularity, and asymptotic behavior of global solutions for differential equations and inclusions, with or without delay, subjected to nonlocal implicit initial conditions.

    After preliminaries on nonlinear evolution equations governed by dissipative operators, the book gives a thorough study of the existence, uniqueness, and asymptotic behavior of global bounded solutions for differential equations with delay and local initial conditions. It then focuses on two important nonlocal cases: autonomous and quasi-autonomous. The authors next discuss sufficient conditions for the existence of almost periodic solutions, describe evolution systems with delay and nonlocal initial conditions, examine delay evolution inclusions, and extend some results to the multivalued case of reaction-diffusion systems. The book concludes with results on viability for nonlocal evolution inclusions.

    Preliminaries
    Topologies on Banach spaces
    A Lebesgue-type integral for vector-valued functions
    The superposition operator
    Compactness theorems
    Multifunctions
    C0-semigroups
    Mild solutions
    Evolutions governed by m-dissipative operators
    Examples of m-dissipative operators
    Strong solutions
    Nonautonomous evolution equations
    Delay evolution equations
    Integral inequalities
    Brezis–Browder Ordering Principle
    Bibliographical notes and comments

    Local Initial Conditions
    An existence result for ODEs with delay
    An application to abstract hyperbolic problems
    Local existence: The case f Lipschitz
    Local existence: The case f continuous
    Local existence: The case f compact
    Global existence
    Examples
    Global existence of bounded C0-solutions
    Three more examples
    Bibliographical notes and comments

    Nonlocal Initial Conditions: The Autonomous Case
    The problem to be studied
    The case f and g Lipschitz
    Proofs of the main theorems
    The transport equation in Rd
    The damped wave equation with nonlocal initial conditions
    The case f Lipschitz and g continuous
    Parabolic problems governed by the p-Laplacian
    Bibliographical notes and comments

    Nonlocal Initial Conditions: The Quasi-Autonomous Case
    The quasi-autonomous case with f and g Lipschitz
    Proofs of Theorems 4.1.1, 4.1.2
    Nonlinear diffusion with nonlocal initial conditions
    Continuity with respect to the data
    The case f continuous and g Lipschitz
    An example involving the p-Laplacian
    The case f Lipschitz and g continuous
    The case A linear, f compact, and g nonexpansive
    The case f Lipschitz and compact, g continuous
    The damped wave equation revisited
    Further investigations in the case ℓ = ω
    The nonlinear diffusion equation revisited
    Bibliographical notes and comments

    Almost Periodic Solutions
    Almost periodic functions
    The main results
    Auxiliary lemmas
    Proof of Theorem 5.2.1
    The w-limit set
    The transport equation in one dimension
    An application to the damped wave equation
    Bibliographical notes and comments

    Evolution Systems with Nonlocal Initial Conditions
    Single-valued perturbed systems
    The main result
    The idea of the proof
    An auxiliary lemma
    Proof of Theorem 6.2.1
    Application to a reaction-diffusion system in L2(Ω)
    Nonlocal initial conditions with linear growth
    The idea of the proof
    Auxiliary results
    Proof of Theorem 6.7.1
    A nonlinear reaction-diffusion system in L1(Ω)
    Bibliographical notes and comments

    Delay Evolution Inclusions
    The problem to be studied
    The main results and the idea of the proof
    Proof of Theorem 7.2.1
    A nonlinear parabolic differential inclusion
    The nonlinear diffusion in L1(Ω)
    The case when F has affine growth
    Proof of Theorem 7.6.1
    A differential inclusion governed by the p-Laplacian
    A nonlinear diffusion inclusion in L1(Ω)
    Bibliographical notes and comments

    Multivalued Reaction-Diffusion Systems
    The problem to be studied
    The main result
    Idea of the proof of Theorem 8.2.1
    A first auxiliary lemma
    The operator ΓE
    Proof of Theorem 8.2.1
    A reaction-diffusion system in L1(Ω)
    A reaction-diffusion system in L2(Ω)
    Bibliographical notes and comments

    Viability for Nonlocal Evolution Inclusions
    The problem to be studied
    Necessary conditions for viability
    Sufficient conditions for viability
    A sufficient condition for null controllability
    The case of nonlocal initial conditions
    An approximate equation
    Proof of Theorem 9.5.1
    A comparison result for the nonlinear diffusion
    Bibliographical notes and comments

    Bibliography

    Index

    Biography

    Monica-Dana Burlică is an associate professor in the Department of Mathematics and Informatics at the “G. Asachi” Technical University of Iaşi. She received her doctorate in mathematics from the University “Al. I. Cuza” of Iaşi. Her research interests include differential inclusions, reaction-diffusion systems, viability theory, and nonlocal delay evolution equations.

    Mihai Necula is an associate professor in the Faculty of Mathematics at the University "Al. I. Cuza” of Iaşi. He received his doctorate in mathematics from the University “Al. I. Cuza” of Iaşi. His research interests include differential inclusions, viability theory, and nonlocal delay evolution equations.

    Daniela Roşu is an associate professor in the Department of Mathematics and Informatics at the “G. Asachi” Technical University of Iaşi. She received her doctorate in mathematics from the University "Al. I. Cuza” of Iaşi. Her research interests include evolution equations, viability theory, and nonlocal delay evolution equations.

    Ioan I. Vrabie is a full professor in the Faculty of Mathematics at the University "Al. I. Cuza” of Iaşi and a part-time senior researcher at the "O. Mayer" Mathematical Institute of the Romanian Academy. He received his doctorate in mathematics from the University “Al. I. Cuza” of Iaşi. He has been a recipient of several honors, including The First Prize of the Balkan Mathematical Union and the “G. Ţiţeica” Prize of the Romanian Academy. His research interests include evolution equations, viability theory, and nonlocal delay evolution equations.

    "This book will be useful to researchers and graduate students interested in delay evolution equations and inclusions subjected to nonlocal initial conditions." - Sotiris K. Ntouyas (Ioannina)