Decay characterization of solutions to the viscous Camassa-Holm equations

In this paper we study the decay characterization in the space H-sigma(K)(R-n) of solutions to the viscous Camassa-Holm equations (VCHE) in the whole space R-n (n = 2, 3, 4), namely, parallel to partial derivative(p)(t)del(m)v(t)parallel to(2) <= C(1+t)(-min(r*+m+n/2+2p,m+n/2+2p+1)) where m + 2p <= K, r* = r* (v(0)) is the decay character of the initial datum v(0) is an element of H-sigma(K) (R-n). We also get the optimal lower bounds for decay rates of solutions to VCHE when -n/2 < r* <= 1. In particular, when v(0) is an element of H-sigma(K) (R-n) boolean AND L-1(R-n) has decay character r* (v(0)) = 0, then we recover the previous results of Bjorland and Schonbek. (2008 Ann. Inst. Henri Poincare Anal. Non Lineaire 25 907-36).