Large-time behaviour of the higher-dimensional logarithmic diffusion equation

Let n >= 3 and let psi lambda(0) be the radially symmetric solution of Delta log psi + 2 beta psi + beta x . del psi = 0 in R-n, psi(0) = lambda(0), for some constants lambda(0) > 0, beta > 0. Suppose u(0) >= 0 satisfies u(0) - psi lambda(0) is an element of L-1(R-n) and u(0)(x) approximate to (2(n - 2)/beta)(log vertical bar x vertical bar/vertical bar x vertical bar(2)) as vertical bar x vertical bar -> infinity. We prove that the rescaled solution (u) over tilde (x, t) = e(2 beta t)u(e(beta t)x, t) of the maximal global solution u of the equation u(t) =Delta log u in R-n x (0, infinity), u(x, 0) = u(0)(x) in R-n, converges uniformly on every compact subset of R-n and in L-1(R-n) to psi lambda(0) as t -> infinity. Moreover, parallel to(u) over tilde(. , t) - psi lambda(0)parallel to(L1(Rn)) <= e(-(n-2)beta t)parallel to u(0) - psi lambda(0)parallel to(L1(Rn)) for all t >= 0.