Global dynamics of blow-up profiles in one-dimensional reaction diffusion equations

We consider reaction diffusion equations of the prototype form ut u(xx) + lambda u + vertical bar u vertical bar(P-1) u on the interval 0 < x < pi, with p > 1 and lambda > m(2). We study the global blow-up dynamics in the in-dimensional fast unstable manifold of the trivial equilibrium u equivalent to 0. In particular, sign-changing solutions are included. Specifically, we find initial conditions such that the blow-up profile u (t, x) at blow-up time t = T possesses in + 1 intervals of strict monotonicity with prescribed extremal values u 1,..., u, Since u(k) = +/-infinity at blow-up time t = T, for some k, this exhausts the dimensional possibilities of trajectories in the m-dimensional fast unstable manifold. Alternatively, we can prescribe the locations x=x(1),...,x(m), of the extrema, at blow-up time, up to a one-dimensional constraint. The proofs are based on an elementary Brouwer degree argument for maps which encode the shapes of solution profiles via their extremal values and extremal locations, respectively. Even in the linear case, such an "interpolation of shape" was not known to us. Our blow-up result generalizes earlier work by Chen and Matano (1989), J Diff Eq. 78, 160-190, and Merle (1992), Commun. Pure Appl. Math. 45(3), 263-300 on multi-point blow-up for positive solutions, which were not constrained to possess global extensions for all negative times. Our results are based on continuity of the blow-up time, as proved by Merle (1992), Commun. Pure Appl. Math. 45(3), 263-300, and Quittner (2003), Houston J Math. 29(3), 757-799, and on a refined variant of Merle's continuity of the blow-up profile, as addressed in the companion paper Matano and Fiedler (2007) (in preparation).