scholarly journals Mixed problem for quasilinear hyperbolic system with coefficients functionally dependent on solution

2017 ◽  
Vol 25 (3) ◽  
pp. 215
Author(s):  
Małgorzata Zdanowicz ◽  
Zbigniew Peradzyński
Keyword(s):  
Banach Space ◽  
Initial Data ◽  
Mixed Problem ◽  
Hyperbolic System ◽  
Quasilinear Hyperbolic System ◽  
Nonlinear Operators ◽  
Uniqueness And Existence ◽  
Class C ◽  
Additional Assumptions

Abstract The mixed problem for quasilinear hyperbolic system with coefficients functionally dependent on the solution is studied. We assume that the coefficients are continuous nonlinear operators in the Banach space C1(ℝ) satisfying some additional assumptions. Under these assumptions we prove the uniqueness and existence of local in time C1 solution, provided that the initial data are also of class C1.

scholarly journals Existence and uniqueness of a finite energy solution for the mixed value problem of porous thermoelastic bodies

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
M. Marin ◽  
S. Vlase ◽  
C. Carstea
Keyword(s):  
Initial Data ◽  
Mixed Problem ◽  
Existence And Uniqueness ◽  
Finite Energy ◽  
Theory Of Elasticity ◽  
Existence Of A Solution ◽  
Porous Bodies ◽  
Finite Energy Solution ◽  
Uniqueness And Existence ◽  
Dipolar Structure

AbstractWe consider the mixed problem with boundary and initial data in thermoelasticity of porous bodies with dipolar structure. By generalizing some known results developed by Dafermos in a more simple case of the classical theory of elasticity, we prove new theorems in which we address the issues regarding the uniqueness and existence of a solution with finite energy of the respective problem after we define this type of solution.

scholarly journals Positive coincidence points for a class of nonlinear operators and their applications to matrix equations

2020 ◽  
Vol 18 (1) ◽  
pp. 858-872
Author(s):  
Imed Kedim ◽  
Maher Berzig ◽  
Ahdi Noomen Ajmi
Keyword(s):  
Banach Space ◽  
Positive Solution ◽  
Positive Definite ◽  
Positive Cone ◽  
Matrix Equations ◽  
Ordered Banach Space ◽  
Coincidence Points ◽  
Nonlinear Operators ◽  
Nonlinear Matrix

Abstract Consider an ordered Banach space and f,g two self-operators defined on the interior of its positive cone. In this article, we prove that the equation f(X)=g(X) has a positive solution, whenever f is strictly \alpha -concave g-monotone or strictly (-\alpha ) -convex g-antitone with g super-homogeneous and surjective. As applications, we show the existence of positive definite solutions to new classes of nonlinear matrix equations.

scholarly journals Local Exponential $H^2$ Stabilization of a $2\times2$ Quasilinear Hyperbolic System Using Backstepping

2013 ◽  
Vol 51 (3) ◽  
pp. 2005-2035 ◽  
Author(s):  
Jean-Michel Coron ◽  
Rafael Vazquez ◽  
Miroslav Krstic ◽  
Georges Bastin
Keyword(s):  
Hyperbolic System ◽  
Quasilinear Hyperbolic System

scholarly journals Breakdown of Regularity of Scattering for Mass-Subcritical NLS

2020 ◽  
Author(s):  
Gyu Eun Lee
Keyword(s):  
Initial Data ◽  
Wave Operator ◽  
Schrodinger Equation ◽  
Scattering Problem ◽  
Asymptotic Completeness ◽  
Nonlinear Schrodinger Equation ◽  
Mild Form ◽  
Scattering State ◽  
Subcritical Nonlinearity ◽  
Class C

Abstract We study the scattering problem for the nonlinear Schrödinger equation $i\partial _t u + \Delta u = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that asymptotic completeness in $L^2$ with initial data in $\Sigma$ holds and the wave operator is well defined on $\Sigma$. We show that there exists $0<\beta <p$ such that the wave operator and the data-to-scattering-state map do not admit extensions to maps $L^2\to L^2$ of class $C^{1+\beta }$ near the origin. This constitutes a mild form of ill-posedness for the scattering problem in the $L^2$ topology.

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