scholarly journals Initial boundary value problems for coupled nerve fibres

1985 ◽  
Vol 28 (2) ◽  
pp. 249-269
Author(s):  
P. Grindrod
Keyword(s):  
General Model ◽  
Action Potentials ◽  
Geometric Approach ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Electrical Behaviour ◽  
Nerve Fibres ◽  
Membrane Action ◽  
Image Position ◽  
Initial Boundary Value

In this paper we analyse the electrical behaviour within systems of long and short coupled nerve axons by using a geometric approach to obtain a priori bounds on solutions. In [4[ we developed a general model for a bundle of n-uniform unmylinated nerve fibres. If FitzHugh-Nagumo dynamics, [3[ are used to describe the ionic membrane currents, then the model takes the formHere W=(w1,…wn)T denotes the membrane action potentials for each fibre in the bundle and Z=(Z1,…Zn)T represents the recovery variables for each fibre, which control the return to the resting equilibrium after any transmission of signals.

On the convolution operators arising in the study of abstract initial boundary value problems

1996 ◽  
Vol 126 (3) ◽  
pp. 515-539 ◽  
Author(s):  
I. Alonso-Mallo ◽  
C. Palencia
Keyword(s):  
Banach Space ◽  
Boundary Value Problems ◽  
Infinitesimal Generator ◽  
Boundary Value ◽  
Continuous Mapping ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Convolution Operators ◽  
Image Position ◽  
Initial Boundary Value

We consider convolution operators arising in the study of abstract initial boundary value problems. These operators are of the formwhere {S(t)}t ≧0 is a C0-semigroup in a Banach space X,, with infinitesimal generator A0,: D(A0), ⊂ X, → X, and K(z): Y → X is a linxear, continuous mapping defined in another Banach space Y., We study the continuity of T between the spaces Lp([0, + ∞), Y), and Lq([0, + ∞), X), 1 ≦ p, q, ≦ + ∞. We give several examples of the applicability of the results to some familiar initial boundary value problems, including both parabolic and hyperbolic cases.

Bounds for the solution of a non-linear parabolic initial-boundary value problem

1977 ◽  
Vol 82 (1) ◽  
pp. 131-145
Author(s):  
M. R. Carter
Keyword(s):  
Steady State ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
The Past ◽  
Existence And Stability ◽  
Image Position ◽  
Initial Boundary Value ◽  
Steady State Solutions

A number of papers have appeared over the past decade or so which study questions of the existence and stability of positive steady-state solutions for parabolic initial-boundary value problems of the general form

Energy and pointwise bounds in some non-standard parabolic problems

2004 ◽  
Vol 134 (1) ◽  
pp. 1-9 ◽  
Author(s):  
K. A. Ames ◽  
L. E. Payne ◽  
P. W. Schaefer
Keyword(s):  
Maximum Principle ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Parabolic Problems ◽  
Differential Inequalities ◽  
Initial Boundary Value Problems ◽  
Auxiliary Condition ◽  
Image Position ◽  
Initial Boundary Value

We study a class of initial-boundary-value problems for which an auxiliary condition of the form is prescribed. We determine bounds on an energy expression by means of differential inequalities and derive pointwise bounds for the solution and its gradient by use of a parabolic maximum principle.

On the zeros of solutions to nonlinear hyperbolic equations

1987 ◽  
Vol 106 (1-2) ◽  
pp. 121-129 ◽  
Author(s):  
Norio Yoshida
Keyword(s):  
Initial Value Problems ◽  
Hyperbolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Nonlinear Hyperbolic Equations ◽  
Initial Value ◽  
Zero Characteristic ◽  
Zeros Of Solutions ◽  
Image Position ◽  
Initial Boundary Value

SynopsisWe consider the hyperbolic equation uxy + c(x, y, u) =f(x, y) and the wave equationWe show that, under suitable conditions, there are bounded domains in which every solution to certain problems has a zero. Characteristic initial value problems and initial boundary value problems are considered.

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