Lq-decay rates of solutions for Benjamin-Bona-Mahony-Burgers equations

This paper studies the asymptotic behavior of solutions for the Benjamin-Bona-Mahony Burgers equations u(t) - u(xxt) - alpha u(xx) + beta u(x) + u(p)u(x) = 0, x is an element of R, t greater than or equal to 0, with the initial data u\(t = 0) = u(0)(x) --> 0 as x --> +/- infinity. Under the restrictions integral(-infinity)(infinity) u(0)(x) dx = 0 and integral(-infinity)(x) u(0)(y) dy is an element of W-2p+1,W- 1, we obtain more results on the energy decay rates of the solutions in the forms that if p greater than or equal to 1, then //partial derivative(x)(j)u(t)//(L2) = O(1) t(-(2j+3)/4) for j = 0, 1, ..., 2p - 1, and //partial derivative(x)(j)u(t)//(Lq) = O(1) t(-((j+2)q-1)/(2q)), for 2 < q less than or equal to infinity and j = 0, 1, ..., 2p - 2; furthermore, if p greater than or equal to 2, then //partial derivative(x)(j)u(t)//(Lq) = O(1) t(-((j+4)q-1)/(2q)) for 1 less than or equal to q < 2, j = 0, 1, ..., 2p - 3, and //partial derivative(x)(j)u(t)(t)//(Lq) = O(1) t(-((j+4)q-1)/(2q)) for 2 less than or equal to q less than or equal to infinity, j = 0, 1, ..., 2p - 3, which are optimal. The proof is dependent on the Fourier transform method, the energy method and the point wise method of the Green function. (C) 1999 Academic Press.