scholarly journals Structure of the sets of weak solutions of an ordinary differential equation in a Banach space

1984 ◽  
Vol 44 (1) ◽  
pp. 67-72
Author(s):  
Ireneusz Kubiaczyk
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Ordinary Differential Equation ◽  
Weak Solutions

Qualitative properties of nonlinear ordinary differential equations

1977 ◽  
Vol 79 (1-2) ◽  
pp. 79-85 ◽  
Author(s):  
Nelson Onuchic ◽  
Plácido Z. Táboas
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Ordinary Differential Equation ◽  
Point Of View ◽  
Nonlinear Ordinary Differential Equations ◽  
Qualitative Properties ◽  
Image Position

SynopsisThe perturbed linear ordinary differential equationis considered. Adopting the same approach of Massera and Schäffer [6], Corduneanu states in [2] the existence of a set of solutions of (1) contained in a given Banach space. In this paper we investigate some topological aspects of the set and analyze some of the implications from a point of view ofstability theory.

scholarly journals Uniqueness of weak solutions to a prion equation with polymer joining

Analysis ◽  
2017 ◽  
Vol 37 (2) ◽  
Author(s):  
Elena Leis ◽  
Christoph Walker
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlinear Differential Equation ◽  
Weak Solutions ◽  
Reaction Rates ◽  
Global Weak Solutions ◽  
Polymer Joining

AbstractWe consider a model for prion proliferation that includes prion polymerization, polymer splitting, and polymer joining. The model consists of an ordinary differential equation for the prion monomers and a hyperbolic nonlinear differential equation with integral terms for the prion polymers and was shown to possess global weak solutions for unbounded reaction rates [

Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Wave Equation

1980 ◽  
Vol 32 (3) ◽  
pp. 631-643 ◽  
Author(s):  
G. F. Webb
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Ordinary Differential Equation ◽  
Asymptotic Behavior ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Smooth Bounded Domain ◽  
Initial Boundary Value ◽  
Strongly Damped

In this paper we study the nonlinear initial boundary value problem(1.1) ωtt— αΔ ωt— Δω= f(ω), t> 0ω(x, 0) = ϕ(x), x∈ Ωωt(x, 0) = ψ (x), x∈ Ωω(x, t ) = 0, x ∈ ∂Ω, t ≥ 0.In (1.1) Ω is a smooth bounded domain in Rn, n = 1, 2, 3, α > 0, and f ∈ C1(R;R) with f‘(x) ≦ co for all x ∈ R (where c0 is a nonnegative constant), lim sup|x|→+∞f(x)/x ≦0, and f(0) = 0. Our objective will be to establish the existence of unique strong global solutions to (1.1) and investigate their behavior as t→ +∞.Our approach takes advantage of the semilinear character of (1.1) and reformulates the problem as an abstract ordinary differential equation in a Banach space.

scholarly journals Global classical and weak solutions of the prion proliferation model in the presence of chaperone in a banach space

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kapil Kumar Choudhary ◽  
Rajiv Kumar ◽  
Rajesh Kumar
Keyword(s):  
Differential Equation ◽  
System Theory ◽  
Banach Space ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Weak Compactness ◽  
Evolution System ◽  
Global Classical Solution ◽  
Global Weak Solutions ◽  
Degradation Rates

<p style='text-indent:20px;'>The present work is based on the coupling of prion proliferation system together with chaperone which consists of two ODEs and a partial integro-differential equation. The existence and uniqueness of a positive global classical solution of the system is proved for the bounded degradation rates by the idea of evolution system theory in the state space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R} \times \mathbb{R} \times L_{1}(Z,zdz). $\end{document}</tex-math></inline-formula> Moreover, the global weak solutions for unbounded degradation rates are discussed by weak compactness technique.</p>

scholarly journals Semi-discrete optimal transport methods for the semi-geostrophic equations

2022 ◽  
Vol 61 (1) ◽  
Author(s):  
David P. Bourne ◽  
Charlie P. Egan ◽  
Beatrice Pelloni ◽  
Mark Wilkinson
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Weak Solutions ◽  
Compact Support ◽  
Optimal Transport ◽  
Constructive Proof ◽  
3 Dimensional ◽  
Eulerian Coordinates ◽  
Numerical Solver ◽  
Explicit Relation

AbstractWe give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.

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