scholarly journals Existence of solutions of a non-variational bi-harmonic system via fixed point theory

Filomat ◽  
2018 ◽  
Vol 32 (12) ◽  
pp. 4113-4130 ◽  
Author(s):  
Idir Mechai ◽  
Metib Alghamdi ◽  
Habib Yazidi
Keyword(s):  
Numerical Solutions ◽  
Existence Of Solutions ◽  
A Priori Estimates ◽  
Fixed Point Theory ◽  
A Priori ◽  
Point Theory ◽  
Estimates Of Solutions ◽  
Priori Estimates ◽  
Theoretical Results ◽  
Harmonic System

We prove existence of a positive solution for a system of non-variational bi-harmonic equations. Furthermore, we give some a priori estimates of solutions and a non-existence result. In addition we compute numerical solutions to illustrate the theoretical results.

scholarly journals Method of Monotone Solutions for Reaction-Diffusion Equations

2017 ◽  
Vol 63 (3) ◽  
pp. 437-454
Author(s):  
V Volpert ◽  
V Vougalter
Keyword(s):  
Existence Of Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Systems ◽  
Estimates Of Solutions ◽  
Priori Estimates ◽  
Monotone Solutions

Existence of solutions of reaction-diffusion systems of equations in unbounded domains is studied by the Leray-Schauder (LS) method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces. We identify some reactiondiffusion systems for which there exist two subclasses of solutions separated in the function space, monotone and non-monotone solutions. A priori estimates and existence of solutions are obtained for monotone solutions allowing to prove their existence by the LS method. Various applications of this method are given.

scholarly journals The Existence of Solutions for Local Dirichlet (r(u),s(u))-Problems

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 237
Author(s):  
Calogero Vetro
Keyword(s):  
Polynomial Growth ◽  
Existence Of Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Point Theory ◽  
Continuous Functions ◽  
Space Structure ◽  
Laplacian Operator ◽  
Dirichlet Problems ◽  
Priori Estimates

In this paper, we consider local Dirichlet problems driven by the (r(u),s(u))-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents r,s are real continuous functions and we have dependence on the solution u. The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the space structure. In this case, we use a priori estimates and asymptotic analysis of regularized auxiliary problems to establish the existence and uniqueness theorems via a fixed-point argument.

Existence of bistable waves for a nonlocal and nonmonotone reaction-diffusion equation

2019 ◽  
Vol 150 (2) ◽  
pp. 721-739
Author(s):  
Sergei Trofimchuk ◽  
Vitaly Volpert
Keyword(s):  
Diffusion Equation ◽  
A Priori Estimates ◽  
Travelling Waves ◽  
A Priori ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Nonlocal Nonlinearity ◽  
Estimates Of Solutions ◽  
Priori Estimates

AbstractReaction-diffusion equation with a bistable nonlocal nonlinearity is considered in the case where the reaction term is not quasi-monotone. For this equation, the existence of travelling waves is proved by the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions in properly chosen weighted spaces.

A priori estimates of solutions of nonlinear boundary value problems for singular in a phase variable second order differential inequalities

2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Ivan Kiguradze
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Second Order ◽  
Phase Variable ◽  
Differential Inequalities ◽  
Nonlinear Boundary Value Problems ◽  
Estimates Of Solutions ◽  
Priori Estimates

Abstract.For singular in a phase variable second order differential inequalities, a priori estimates of positive solutions, satisfying nonlinear nonlocal boundary conditions, are established.

scholarly journals Estimates of Solutions of Elliptic Differential-Difference Equations with Degeneration

2018 ◽  
Vol 64 (1) ◽  
pp. 131-147
Author(s):  
V A Popov
Keyword(s):  
Difference Equations ◽  
A Priori Estimates ◽  
Difference Operator ◽  
A Priori ◽  
Difference Operators ◽  
Elliptic Differential Operator ◽  
Friedrichs Extension ◽  
Estimates Of Solutions ◽  
Priori Estimates ◽  
Differential Difference Equation

We consider a second-order differential-difference equation in a bounded domain Q ⊂ Rn. We assume that the differential-difference operator contains some difference operators with degeneration corresponding to differentiation operators. Moreover, the differential-difference operator under consideration cannot be expressed as a composition of a difference operator and a strongly elliptic differential operator. Degenerated difference operators do not allow us to obtain the G˚arding inequality. We prove a priori estimates from which it follows that the differential-difference operator under consideration is sectorial and its Friedrichs extension exists. These estimates can be applied to study the spectrum of the Friedrichs extension as well. It is well known that elliptic differential-difference equations may have solutions that do not belong even to the Sobolev space W 1(Q). However, using the obtained estimates, we can prove some smoothness of solutions, though not in the whole domain Q, but inside some subdomains Qr generated by the shifts of the boundary, where U Qr = Q.

scholarly journals Numerical-analytical method for solving boundary value problem for the generalized moisture transport equation

2021 ◽  
Vol 31 (1) ◽  
pp. 19-34
Author(s):  
M.A. Kerefov ◽  
S.Kh. Gekkieva
Keyword(s):  
Moisture Transfer ◽  
A Priori Estimates ◽  
A Priori ◽  
Method Of Lines ◽  
Numerical Tests ◽  
Rates Of Change ◽  
Priori Estimates ◽  
The Method Of Lines ◽  
Numerical Analytical Method ◽  
Theoretical Results

The paper studies qualitatively new equations of moisture transfer, which generalize the Aller and Aller-Lykov equations. The generalization contributes to revealing in the original equations the specific features of the studied massifs, their structure, physical properties, processes occurring in them through the introduction of the notion of the rates of change of the fractal dimension. We have obtained solutions to the constant coefficient difference equations as a system arising when using the method of lines for the equations with a Riemann-Liouville time fractional derivative with boundary conditions of the first kind. A priori estimates are obtained that imply convergence of the obtained solutions to systems of ordinary differential equations with variable fractional coefficients. Numerical tests have been carried out to confirm theoretical results of the study.

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