Damped wave equation in the subcritical case

[摘要] We study large time asymptotics of small solutions to the Cauchy problem for the one dimensional nonlinear damped wave equation {upsilon(tt) + upsilon(t) - upsilon(xx) + + upsilon(1+sigma) = 0, x is an element of R, t > 0, upsilon(0,x) = epsilonupsilon(0) (x), upsilon(t) (0,x) = epsilonupsilon(1) (x) (1) in the sub critical case sigmais an element of (2 - epsilon(3),2). We assume that the initial data upsilon(0), (1 + partial derivative(x))(-1)(upsilon1) is an element of L(infinity)boolean ANDL(1,a), a is an element of (0,1) where L-1,L-a = {phi is an element of L-1; parallel tophiparallel to(L1,a) = parallel to<(.)>(a)phiparallel to(L1) = root1+x(2). Also we suppose that the mean value of initial data (R) (upsilon(0) (x) + upsilon(1) (x)) dx > 0. Then there exists a positive value epsilon such that the Cauchy problem (1) has a unique global solution upsilon (t, x) is an element of C ([0, infinity); L-infinity boolean AND L-1,L-a), satisfying the following time decay estimate: parallel toupsilon(t)parallel toL(infinity)less than or equal toCepsilon(-1/a) where large t > 0, here 2-epsilon(3) < sigma < 2. (C) 2004 Elsevier Inc. All rights reserved.

[发布日期] 2004-12-01

[关键词] damped wave equation;subcritical nonlinearity;asymptotic expansion;large time behavior