ON WELL-POSEDNESS OF ABSTRACT HYPERBOLIC PROBLEMS IN FUNCTION SPACES

2009 ◽  
Author(s):  
A. ASHYRALYEV ◽  
M. MARTINEZ ◽  
J. PASTOR ◽  
S. PISKAREV
Keyword(s):  
Function Spaces ◽  
Well Posedness ◽  
Hyperbolic Problems

scholarly journals The Concepts of Well-Posedness and Stability in Different Function Spaces for the 1D Linearized Euler Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Stefan Balint ◽  
Agneta M. Balint
Keyword(s):  
Euler Equation ◽  
Function Space ◽  
Euler Equations ◽  
Small Perturbation ◽  
Function Spaces ◽  
Well Posedness ◽  
Small Solution ◽  
Null Solution ◽  
Linearized Euler Equations ◽  
The Stability

This paper considers the stability of constant solutions to the 1D Euler equation. The idea is to investigate the effect of different function spaces on the well-posedness and stability of the null solution of the 1D linearized Euler equations. It is shown that the mathematical tools and results depend on the meaning of the concepts “perturbation,” “small perturbation,” “solution of the propagation problem,” and “small solution, that is, solution close to zero,” which are specific for each function space.

Well-posedness of Third Order Differential Equations in Hölder Continuous Function Spaces

2018 ◽  
Vol 62 (4) ◽  
pp. 715-726
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Function Spaces ◽  
Linear Operators ◽  
Well Posedness ◽  
Third Order ◽  
Hölder Continuous ◽  
Vector Valued ◽  
Closed Linear Operators

AbstractIn this paper, by using operator-valued ${\dot{C}}^{\unicode[STIX]{x1D6FC}}$-Fourier multiplier results on vector-valued Hölder continuous function spaces, we give a characterization of the $C^{\unicode[STIX]{x1D6FC}}$-well-posedness for the third order differential equations $au^{\prime \prime \prime }(t)+u^{\prime \prime }(t)=Au(t)+Bu^{\prime }(t)+f(t)$, ($t\in \mathbb{R}$), where $A,B$ are closed linear operators on a Banach space $X$ such that $D(A)\subset D(B)$, $a\in \mathbb{C}$ and $0<\unicode[STIX]{x1D6FC}<1$.

scholarly journals Weak and Strong Solutions for a Strongly Damped Quasilinear Membrane Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Jin-soo Hwang
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Continuous Dependence ◽  
Dirichlet Boundary ◽  
Strong Solutions ◽  
Function Spaces ◽  
Well Posedness ◽  
Continuous Dependence On Data ◽  
Strongly Damped ◽  
Weak And Strong Solutions

We consider a strongly damped quasilinear membrane equation with Dirichlet boundary condition. The goal is to prove the well-posedness of the equation in weak and strong senses. By setting suitable function spaces and making use of the properties of the quasilinear term in the equation, we have proved the fundamental results on existence, uniqueness, and continuous dependence on data including bilinear term of weak and strong solutions.

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