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Variation of constants formula and exponential dichotomy for nonautonomous non-densely defined Cauchy problems
- Source :
- Canadian Journal of Mathematics. 73:1347-1389
- Publication Year :
- 2020
- Publisher :
- Canadian Mathematical Society, 2020.
-
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.
- Subjects :
- General Mathematics
Exponential dichotomy
010102 general mathematics
Mathematical analysis
Mathematics::Analysis of PDEs
Cauchy distribution
Variation of parameters
01 natural sciences
010101 applied mathematics
Homogeneous
Boundary value problem
0101 mathematics
Persistence (discontinuity)
Hyperbolic partial differential equation
Mathematics
Subjects
Details
- ISSN :
- 14964279 and 0008414X
- Volume :
- 73
- Database :
- OpenAIRE
- Journal :
- Canadian Journal of Mathematics
- Accession number :
- edsair.doi...........6392eb6e8e23df3f99f938da4ca8e1e8