An extremum principle for the generalized solution of an equation of mixed type with lowest terms

1973 ◽  
Vol 14 (1) ◽  
pp. 158-163 ◽  
Author(s):  
L. I. Kovalenko
Keyword(s):  
Mixed Type ◽  
Generalized Solution ◽  
Extremum Principle ◽  
Equation Of Mixed Type

scholarly journals On the solvability of a boundary value problem for a quasilinear equation of mixed type with two degeneration lines

2021 ◽  
Vol 2070 (1) ◽  
pp. 012002
Author(s):  
Xaydar R. Rasulov
Keyword(s):  
Boundary Value Problem ◽  
Mixed Type ◽  
Boundary Value ◽  
Quasilinear Equation ◽  
Generalized Solution ◽  
Order Partial Differential Equation ◽  
Existence Of A Solution ◽  
First Order ◽  
Weighted Norms ◽  
Equation Of Mixed Type

Abstract The article investigates the existence of a generalized solution to one boundary value problem for an equation of mixed type with two lines of degeneration in the weighted space of S.L. Sobolev. In proving the existence of a generalized solution, the spaces of functions U(Ω) and V (Ω) are introduced, the spaces H1(Ω) and H 1 * (Ω) are defined as the completion of these spaces of functions, respectively, with respect to the weighted norms, including the functions K(y) and N(x). Using an auxiliary boundary value problem for a first order partial differential equation, Kondrashov’s theorem on the compactness of the embedding of W 2 1 (Ω) in L2(Ω) and Vishik’s lemma, the existence of a solution to the boundary value problem is proved.

scholarly journals Interaction of homoclinic solutions and Hopf points in functional differential equations of mixed type

2009 ◽  
Vol 139 (1) ◽  
pp. 123-155 ◽  
Author(s):  
Marc Georgi
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Homoclinic Orbit ◽  
Mixed Type ◽  
Functional Differential Equations ◽  
Homoclinic Orbits ◽  
Homoclinic Solution ◽  
Equations Of Mixed Type ◽  
Functional Differential ◽  
Equation Of Mixed Type

We study a homoclinic bifurcation in a general functional differential equation of mixed type. More precisely, we investigate the case when the asymptotic steady state of a homoclinic solution undergoes a Hopf bifurcation. Bifurcations of this kind are diffcult to analyse due to the lack of Fredholm properties. In particular, a straightforward application of a Lyapunov–Schmidt reduction is not possible.As one of the main results we prove the existence of centre-stable and centre-unstable manifolds of steady states near homoclinic orbits. With their help, we can analyse the bifurcation scenario similar to the case for ordinary differential equations and can show the existence of solutions which bifurcate near the homoclinic orbit, are decaying in one direction and oscillatory in the other direction. These solutions can be visualized as an interaction of the homoclinic orbit and small periodic solutions that exist on account of the Hopf bifurcation, for exactly one asymptotic direction t→8 or t→−∞.

scholarly journals A Transport Equation of Mixed Type

1998 ◽  
Vol 150 (1) ◽  
pp. 188-202 ◽  
Author(s):  
Jorge Aarão
Keyword(s):  
Transport Equation ◽  
Mixed Type ◽  
Equation Of Mixed Type
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