scholarly journals Positively invariant subset for non-densely defined Cauchy problems

2021 ◽  
Vol 494 (2) ◽  
pp. 124600
Author(s):  
Pierre Magal ◽  
Ousmane Seydi ◽  
Feng-Bin Wang
Keyword(s):  
Cauchy Problems ◽  
Invariant Subset ◽  
Densely Defined

scholarly journals Complex powers of nondensely defined operators

2011 ◽  
Vol 90 (104) ◽  
pp. 47-64 ◽  
Author(s):  
Marko Kostic
Keyword(s):  
Mild Solutions ◽  
Acute Angle ◽  
Analytic Semigroups ◽  
Cauchy Problems ◽  
Growth Order ◽  
Fractional Powers ◽  
Complex Powers ◽  
Cosine Functions ◽  
Abstract Cauchy Problems ◽  
Densely Defined

The power (?A)b, b ? C is defined for a closed linear operator A whose resolvent is polynomially bounded on the region which is, in general, strictly contained in an acute angle. It is proved that all structural properties of complex powers of densely defined operators with polynomially bounded resolvent remain true in the newly arisen situation. The fractional powers are considered as generators of analytic semigroups of growth order r > 0 and applied in the study of corresponding incomplete abstract Cauchy problems. In the last section, the constructed powers are incorporated in the analysis of the existence and growth of mild solutions of operators generating fractionally integrated semigroups and cosine functions.

scholarly journals Variation of constants formula and exponential dichotomy for nonautonomous non-densely defined Cauchy problems

2020 ◽  
pp. 1-43
Author(s):  
Pierre Magal ◽  
Ousmane Seydi
Keyword(s):  
Exponential Dichotomy ◽  
Homogeneous Problem ◽  
Cauchy Problems ◽  
Small Perturbations ◽  
Variation Of Constants Formula ◽  
Variation Of Constants ◽  
Age Structured ◽  
Necessary And Sufficient ◽  
Age Structured Model ◽  
Densely Defined

Abstract In this paper, we extend to the non-Hille–Yosida case a variation of constants formula for a nonautonomous and nonhomogeneous Cauchy problems first obtained by Gühring and Räbiger. By using this variation of constants formula, we derive a necessary and sufficient condition for the existence of an exponential dichotomy for the evolution family generated by the associated nonautonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally, we illustrate our results by considering two examples. The first example is a parabolic equation with nonlocal and nonautonomous boundary conditions, and the second example is an age-structured model that is a hyperbolic equation.

scholarly journals Fractional Cauchy Problems for Infinite Interval Case-II

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1165
Author(s):  
Mohammed Al Horani ◽  
Mauro Fabrizio ◽  
Angelo Favini ◽  
Hiroki Tanabe
Keyword(s):  
Existence And Uniqueness ◽  
Direct Problem ◽  
Abstract Cauchy Problem ◽  
Continuous Functions ◽  
Cauchy Problems ◽  
Fractional Powers ◽  
Uniqueness Of Solutions ◽  
Infinite Intervals ◽  
Abstract Cauchy Problems ◽  
Densely Defined

We consider fractional abstract Cauchy problems on infinite intervals. A fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces is considered. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Required conditions on spaces are also given, guaranteeing the existence and uniqueness of solutions. The fractional powers of the involved operator B X have been investigated in the space which consists of continuous functions u on [ 0 , ∞ ) without assuming u ( 0 ) = 0 . This enables us to refine some previous results and obtain the required abstract results when the operator B X is not necessarily densely defined.

scholarly journals Hopf bifurcation for non-densely defined Cauchy problems

2010 ◽  
Vol 62 (2) ◽  
pp. 191-222 ◽  
Author(s):  
Zhihua Liu ◽  
Pierre Magal ◽  
Shigui Ruan
Keyword(s):  
Hopf Bifurcation ◽  
Cauchy Problems ◽  
Densely Defined

scholarly journals A Parameter Property of Classical Solutions of Cauchy Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Min He
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Semigroup Theory ◽  
Classical Solutions ◽  
Parabolic Partial Differential Equation ◽  
Cauchy Problems ◽  
Integrated Semigroup ◽  
The Cauchy Problem ◽  
Abstract Cauchy Problems ◽  
Densely Defined

This work is concerned with the abstract Cauchy problems that depend on parameters. The goal is to study continuity in the parameters of the classical solutions of the Cauchy problems. The situation considered in this work is when the operator of the Cauchy problem is not densely defined. By applying integrated semigroup theory and the results on continuity in the parameters ofC0-semigroup and integrated semigroup, we obtain the results on the existence and continuity in parameters of the classical solutions of the Cauchy problems. The application of the obtained abstract results in a parabolic partial differential equation is discussed in the last section of the paper.

Global stability result for parabolic Cauchy problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mourad Choulli ◽  
Masahiro Yamamoto
Keyword(s):  
New York ◽  
Partial Differential Equations ◽  
Global Stability ◽  
Classical Problem ◽  
Stability Result ◽  
Cauchy Problems ◽  
Smooth Solutions ◽  
Linear Partial Differential Equations ◽  
Ill Posed ◽  
The Stability

AbstractUniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953], these kind of problems are known to be ill-posed and even severely ill-posed. Until now, there are only few partial results concerning the quantification of the stability of parabolic Cauchy problems. We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.

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