scholarly journals Steady states of the reaction-diffusion equations. Part III: Questions of multiplicity and uiqueness of solutions

1985 ◽  
Vol 27 (1) ◽  
pp. 88-110 ◽  
Author(s):  
J. G. Burnell ◽  
A. A. Lacey ◽  
G. C. Wake
Keyword(s):  
Existence And Uniqueness ◽  
Multiple Solutions ◽  
Reaction Diffusion ◽  
Elliptic Partial Differential Equations ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Linear Boundary ◽  
Nonlinear Elliptic ◽  
Equivalent Parameter ◽  
Exothermic Chemical Reaction

AbstractIn earlier papers (Parts I and II) existence and uniqueness of the solutions to a coupled pair of nonlinear elliptic partial differential equations with linear boundary conditions was considered. These equations arise when material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant. In this paper we establish the existence of multiple solutions for many different values of the parameters not considered in the earlier parts. It is shown that the case, also omitted in earlier parts, with perfect thermal and mass transfer on the boundary (the double-Dirichlet case) does have a unique solution for sufficiently large values of the exothermicity or an equivalent parameter. The methods of solution provide specific bounds on the region of existence of multiple solutions.

scholarly journals Steady states of the reaction-diffusion equations. Part 1: Questions of existence and continuity of solution branches

1983 ◽  
Vol 24 (4) ◽  
pp. 374-391 ◽  
Author(s):  
J. G. Burnell ◽  
A. A. Lacey ◽  
G. C. Wake
Keyword(s):  
Existence Of Solutions ◽  
Reaction Diffusion ◽  
Elliptic Partial Differential Equations ◽  
Reaction Diffusion Equations ◽  
Steady State Regime ◽  
Nonlinear Elliptic ◽  
Minimal And Maximal Solutions ◽  
Exothermic Chemical Reaction ◽  
State Regime ◽  
Special Case

AbstractWhen material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant, the steady-state regime is governed by a coupled pair of nonlinear elliptic partial differential equations with linear boundary conditions. In this paper we consider questions of existence of solutions to these equations. It is shown that, with the exception of the special case in which the mass-transfer is uninhibited on the boundary, a solution always exists, whereas in this special case a solution exists only for sufficiently low values of the exothermicity. Bounds are established for the solutions and the occurrence of minimal and maximal solutions is shown for some cases. Finally the behaviour of the solution set with respect to one of the parameters is studied.

scholarly journals Steady states of the reaction-diffusion equations. Part II: Uniqueness of solutions and some special cases

1983 ◽  
Vol 24 (4) ◽  
pp. 392-416 ◽  
Author(s):  
J. G. Burnell ◽  
A. A. Lacey ◽  
G. C. Wake
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Uniqueness Of Solutions ◽  
One Dimensional ◽  
Nonlinear Elliptic ◽  
Special Cases ◽  
Exothermic Chemical Reaction ◽  
Dimensional Shape ◽  
The One ◽  
Special Case

AbstractIn an earlier paper (Part I) the existence and some related properties of the solution to a coupled pair of nonlinear elliptic partial differential equations was considered. These equations arise when material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant. In this paper we consider the range of parameters for which the uniqueness of solution is assured and we also investigate the converse question of multiple solutions. The special case of the one dimensional shape of the infinite slab is investigated in full as this case admits to solution by integration.

scholarly journals Existence and uniqueness of propagating terraces

2019 ◽  
Vol 22 (06) ◽  
pp. 1950055
Author(s):  
Thomas Giletti ◽  
Hiroshi Matano
Keyword(s):  
Spatial Structure ◽  
Existence And Uniqueness ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Stationary States ◽  
One Dimensional ◽  
Periodic Environment ◽  
Insight Into

This work focuses on dynamics arising from reaction-diffusion equations, where the profile of propagation is no longer characterized by a single front, but by a layer of several fronts which we call a propagating terrace. This means, intuitively, that transition from one equilibrium to another may occur in several steps, that is, successive phases between some intermediate stationary states. We establish a number of properties on such propagating terraces in a one-dimensional periodic environment under very wide and generic conditions. We are especially concerned with their existence, uniqueness, and their spatial structure. Our goal is to provide insight into the intricate dynamics arising from multistable nonlinearities.

scholarly journals Existence and uniqueness of the solution to granuloma model in visceral leishmaniasi

Filomat ◽  
2019 ◽  
Vol 33 (11) ◽  
pp. 3561-3575
Author(s):  
Chunyan Liu ◽  
Xuemei Wei
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Reaction Diffusion ◽  
Local Solution ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Banach Fixed Point Theorem ◽  
Extension Method ◽  
Uniqueness Of The Solution ◽  
Coupled Reaction

In this paper we study a granuloma model in visceral leishmaniasis, which contains eleven coupled reaction diffusion equations. The existence and uniqueness of the model in the local solution is proved by using the Banach Fixed Point Theorem and theory of parabolic equation. Then, the existence and uniqueness of the global solution are obtained by using the extension method.

On a Nonlinear System of Reaction-Diffusion Equations

2006 ◽  
Vol 11 (2) ◽  
pp. 115-121 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Weak Solution ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic System ◽  
Positive Parameter ◽  
Nonlinear Elliptic ◽  
Positive Weak Solution

The aim of this article is to study the existence of positive weak solution for a quasilinear reaction-diffusion system with Dirichlet boundary conditions,− div(|∇u1|p1−2∇u1) = λu1α11u2α12... unα1n,   x ∈ Ω,− div(|∇u2|p2−2∇u2) = λu1α21u2α22... unα2n,   x ∈ Ω, ... , − div(|∇un|pn−2∇un) = λu1αn1u2αn2... unαnn,   x ∈ Ω,ui = 0,   x ∈ ∂Ω,   i = 1, 2, ..., n,  where λ is a positive parameter, Ω is a bounded domain in RN (N > 1) with smooth boundary ∂Ω. In addition, we assume that 1 < pi < N for i = 1, 2, ..., n. For λ large by applying the method of sub-super solutions the existence of a large positive weak solution is established for the above nonlinear elliptic system.

Sign in / Sign up

Export Citation Format

Share Document