| The class of partial differential equations is given by.;v(,t) = (eta)v(,xx) + (sigma)u - (gamma)v.;where (eta), (sigma), (alpha) and (gamma) are positive constants, t (epsilon) (0,(INFIN)), x (epsilon) (0,L) where L is finite or infinite. The boundary conditions are given by:;u(0,t) = g(t), v(0,t) = h(t) for t (epsilon) {0,(INFIN)).;(*(,L))...u(,t) = u(,xx) - (alpha)u - v + f(u).;u(,x)(L,t) = v(,x)(L,t) = 0 for t (epsilon) (0,(INFIN)).;where the functions g and h are smooth with compact support. The second condition is omitted when L is infinite. The initial conditions are given by u(,0)(x) = u(x,0), v(,0)(x) = v(x,0). The nonlinearity f(u) = u('2)(1 + (alpha) - u) is the remainder after linearization of the usual cubic u(1 - u)(u - (alpha)), where 0 < (alpha) (LESSTHEQ) 1. We refer to (*(,L)) together with the boundary conditions as in the FitzHugh-Nagumo equations, FN. The homogeneous linearized FitzHugh-Nagumo equations {HLFN} consist of (*(,L)) with f(u) omitted, together with the corresponding homogeneous boundary conditions.;The partial differential equations and boundary conditions of HLFN give rise to certain operators. Bounds are given on the spectra of the operators and the operators are shown to generate holomorphic semigroups on the spaces C('0)(0,L;R('2)) and L('p)(O,L;R('2)) for 1 < p < (INFIN). For L < (INFIN), an explicit representation of the semigroups is given. Also for L 0. Explicit representations of resolvents as matrix-valued integral kernels are developed for 0 < L (LESSTHEQ) (INFIN). Decay estimates are found for (VBAR)(VBAR)U('L) - U(VBAR)(VBAR) where U('L) and U are solutions of HLFN for L finite and L finite, respectively. The norm for C('0)(0,m;R('2)) or C('0)(0,(xi)L;R('2)), (xi) (epsilon) (0,1), m < L, is used.;To apply theory developed for HLFN in FN, we employ a change of variable to produce homogeneous boundary conditions. Global existence, uniqueness, boundedness and smoothness of solutions are developed using methods of Rauch-Smoller and of Henry. For g and h sufficiently small, solutions W('L), W of FN are shown to decay as t (--->) (INFIN). Gronwall estimates show that (VBAR)(VBAR)W('L) - W(VBAR)(VBAR) decays as L (--->) (INFIN). |
