Uniform Convergence for Approximate Traveling Waves in Linear Reaction–Diffusion–Hyperbolic Systems
Tóm tắt
In this paper we study linear reaction–hyperbolic systems of the form
$$\varepsilon (\partial_t + v_i \partial_x) p_i = \sum_{j=0}^n k_{ij} p_j$$
, (i = 1, 2, ..., n) for x > 0, t > 0 coupled to a diffusion equation for p
0 = p
0(x, y, θ, t) with “near-equilibrium” initial and boundary data. This problem arises in a model of transport of neurofilaments in axons. The matrix (k
ij
) is assumed to have a unique null vector
$$(\lambda_0,\lambda_1,\ldots,\lambda_n)$$
with positive components summed to 1 and the v
j
are arbitrary velocities such that
$$v \equiv \frac{1}{1-\lambda_0} \sum_{j=1}^n \lambda_j v_j > 0$$
. We prove that as
$$\varepsilon \to 0$$
the solution converges to a traveling wave with velocity v and a spreading front, and that the convergence rate in the uniform norm is
$$O(\varepsilon^{(1-\alpha)/2})$$
, for any small positive α.
Từ khóa
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