The Continuous Dependence on the Nonlinearities of Solutions of Fast Diffusion Equations

In this paper, we consider the Cauchy problem {u(t) = Delta(u(m)), x epsilon R-N, t > 0, N >= 3, u(x, 0) = u(0)(x), x epsilon R-N We will prove that (i) for m(c) < m, m(0) < 1, vertical bar u(x, t, m) - u(x, t, m(0))vertical bar -> 0 as m -> m(0) uniformly on every compact subset of R-N x R+, where m(c) = (N-2)+/N; (ii) there is a C* that explicitly depends on m such that parallel to u(. , . , m) - u(. , . , 1)parallel to(L2(RNxR+)) <= C* vertical bar m - 1 vertical bar.