A priori estimates for the solution of the first boundary-value problem for a class of second-order parabolic systems

2001 ◽  
Vol 65 (4) ◽  
pp. 705-726
Author(s):  
L I Kamynin ◽  
B N Khimchenko
Keyword(s):  
Boundary Value Problem ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Parabolic Systems ◽  
Second Order ◽  
Priori Estimates

A priori estimates of the positive solution of the two-point boundary value problem for one second-order nonlinear differential equation

2019 ◽  
pp. 38-48
Author(s):  
Edelkhan Abduragimov
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Nonlinear Differential Equation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Second Order ◽  
Point Boundary ◽  
Priori Estimates

A priori estimates of the positive solution of the two-point boundary value problem are obtained $y^{\prime\prime}=-f(x,y)$, $0<x<1$, $y(0)=y(1)=0$ assuming that $f(x,y)$ is continuous at $x \in [0,1]$, $y \in R$ and satisfies the condition $a_0 x^{\gamma}y^p \leq f(x,y) \leq a_1 y^p$, where $a_0>0$, $a_1>0$, $p>1$, $\gamma \geq 0$ -- constants.

scholarly journals A Boundary Value Problem with Multivariables Integral Type Condition for Parabolic Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-13
Author(s):  
A. L. Marhoune ◽  
F. Lakhal
Keyword(s):  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Continuous Dependence ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Analysis Method ◽  
Type Condition ◽  
Integral Type ◽  
Priori Estimates

We study a boundary value problem with multivariables integral type condition for a class of parabolic equations. We prove the existence, uniqueness, and continuous dependence of the solution upon the data in the functional wieghted Sobolev spaces. Results are obtained by using a functional analysis method based on two-sided a priori estimates and on the density of the range of the linear operator generated by the considered problem.

scholarly journals A Result on the Existence and Uniqueness of Stationary Solutions for a Bioconvective Flow Model

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
Aníbal Coronel ◽  
Luis Friz ◽  
Ian Hess ◽  
Alex Tello
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Flow Model ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Stationary Solutions ◽  
Local Concentration ◽  
Priori Estimates ◽  
Bioconvective Flow

In this note, we prove the existence and uniqueness of weak solutions for the boundary value problem modelling the stationary case of the bioconvective flow problem. The bioconvective model is a boundary value problem for a system of four equations: the nonlinear Stokes equation, the incompressibility equation, and two transport equations. The unknowns of the model are the velocity of the fluid, the pressure of the fluid, the local concentration of microorganisms, and the oxygen concentration. We derive some appropriate a priori estimates for the weak solution, which implies the existence, by application of Gossez theorem, and the uniqueness by standard methodology of comparison of two arbitrary solutions.

scholarly journals Solvability of a Third-Order Singular Generalized Left Focal Problem in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-18
Author(s):  
Youwei Zhang
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Banach Spaces ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Multiple Positive Solutions ◽  
Third Order ◽  
The Third ◽  
Priori Estimates

We consider the existence of positive solution for a third-order singular generalized left focal boundary value problem with full derivatives in Banach spaces. Green’s function and its properties, explicit a priori, estimates will be presented. By means of the theories of the fixed point in cones, we establish some new and general results on the existence of single and multiple positive solutions to the third-order singular generalized left focal boundary value problem. Our results are generalizations and extensions of the results of the focal boundary value problem. An example is included to illustrate the results obtained.

scholarly journals Solvability of a multi-point boundary value problem of Neumann type

1999 ◽  
Vol 4 (2) ◽  
pp. 71-81 ◽  
Author(s):  
Chaitan P. Gupta ◽  
Sergei Trofimchuk
Keyword(s):  
Boundary Value Problem ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Type A ◽  
Point Boundary ◽  
Existence Of A Solution ◽  
Priori Estimates ◽  
Neumann Type ◽  
Existence Conditions

Letf:[0,1]×ℝ2→ℝbe a function satisfying Carathéodory's conditions ande(t)∈L1[0,1]. Letξi∈(0,1),ai∈ℝ,i=1,2,…,m−2,0<ξ1<ξ2<⋯<ξm−2<1be given. This paper is concerned with the problem of existence of a solution for them-point boundary value problemx″(t)=f(t,x(t),x′(t))+e(t),0<t<1;x(0)=0,x′(1)=∑i=1m−2ai x′(ξi). This paper gives conditions for the existence of a solution for this boundary value problem using some new Poincaré type a priori estimates. This problem was studied earlier by Gupta, Ntouyas, and Tsamatos (1994) when all of theai∈ℝ,i=1,2,…,m−2, had the same sign. The results of this paper give considerably better existence conditions even in the case when all of theai∈ℝ,i=1,2,…,m−2, have the same sign. Some examples are given to illustrate this point.

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