TY - JOUR
T1 - Random travelling waves for the KPP equation with noise
AU - Mueller, Carl
AU - Sowers, Richard B.
PY - 1995/3
Y1 - 1995/3
N2 - Consider the stochastic partial differential equation [formula] where Ẇ =Ẇ(t, x) is two-parameter while noise. Assume that u0 is a continuous function taking values in [0, 1] such that for some constant a > 0, we have (C1) u0(x) = 1 for x < -a.(C2) u0(x) = 0 for x > a. Let the wavefront b(t) = sup(x ∈ R: u(t, x) > 0). We show that for ε(lunate) small enough and with probability 1, • limt→∞b(t)/t exists and lies in (0, ∞). This limit depends only on ε(lunate). •The law of v(t, x) ≡ u(t, b(t) + x) tends toward a stationary limit as t → ∞. We also analyze the length of the region [a(t), b(t)], which is the smallest closed interval containing the points x at which 0 < u(t, x) < 1. We show that the length of this region tends toward a stationary distribution. Thus, the wavefront does not degenerate.
AB - Consider the stochastic partial differential equation [formula] where Ẇ =Ẇ(t, x) is two-parameter while noise. Assume that u0 is a continuous function taking values in [0, 1] such that for some constant a > 0, we have (C1) u0(x) = 1 for x < -a.(C2) u0(x) = 0 for x > a. Let the wavefront b(t) = sup(x ∈ R: u(t, x) > 0). We show that for ε(lunate) small enough and with probability 1, • limt→∞b(t)/t exists and lies in (0, ∞). This limit depends only on ε(lunate). •The law of v(t, x) ≡ u(t, b(t) + x) tends toward a stationary limit as t → ∞. We also analyze the length of the region [a(t), b(t)], which is the smallest closed interval containing the points x at which 0 < u(t, x) < 1. We show that the length of this region tends toward a stationary distribution. Thus, the wavefront does not degenerate.
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U2 - 10.1006/jfan.1995.1038
DO - 10.1006/jfan.1995.1038
M3 - Article
AN - SCOPUS:0038101626
SN - 0022-1236
VL - 128
SP - 439
EP - 498
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 2
ER -