scholarly journals A degenerate bifurcation from simple eigenvalue theorem

2021 ◽  
Vol 30 (1) ◽  
pp. 116-125
Author(s):  
Ping Liu ◽  
◽  
Junping Shi ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Functional Equations ◽  
Simple Eigenvalue ◽  
Partial Differential ◽  
Nonlinear Functional ◽  
The Stability ◽  
Degenerate Bifurcation

<abstract><p>A new bifurcation from simple eigenvalue theorem is proved for general nonlinear functional equations. It is shown that in this bifurcation scenario, the bifurcating solutions are on a curve which is tangent to the line of trivial solutions, while in typical bifurcations the curve of bifurcating solutions is transversal to the line of trivial ones. The stability of bifurcating solutions can be determined, and examples from partial differential equations are shown to demonstrate such bifurcations.</p></abstract>

scholarly journals A spectral mapping theorem for semigroups solving PDEs with nonautonomous past

2003 ◽  
Vol 2003 (16) ◽  
pp. 933-951 ◽  
Author(s):  
Genni Fragnelli
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Partial Differential ◽  
The Stability

We prove a spectral mapping theorem for semigroups solving partial differential equations with nonautonomous past. This theorem is then used to give spectral conditions for the stability of the solutions of the equations.

scholarly journals Stability of observations of partial differential equations under uncertain perturbations

2017 ◽  
Vol 24 (1) ◽  
pp. 45-61 ◽  
Author(s):  
Martin Lazar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Wave Operator ◽  
System Dynamic ◽  
Schrödinger Equations ◽  
Partial Differential ◽  
Main System ◽  
Different Types ◽  
The Stability ◽  
Observability Estimates

We demonstrate the stability of observability estimates for solutions to wave and Schrödinger equations subjected to additive perturbations. This work generalises recent averaged observability/control results by allowing for systems consisting of operators of different types. We also consider the simultaneous observability problem by which one tries to estimate the energy of each component of a system under consideration. Our analysis relies on microlocal defect tools, in particular on standard H-measures when the main system dynamic is governed by the wave operator, and parabolic H-measures in the case of the Schrödinger operator.

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