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Global Dynamics of Blow-up Profiles in One-dimensional Reaction Diffusion Equations
Authors:Bernold Fiedler  Hiroshi Matano
Institution:(1) Institute of Mathematics, Free University of Berlin, Arnimallee 2–6, Berlin, 14199, Germany;(2) Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan
Abstract:We consider reaction diffusion equations of the prototype form u t = u xx + λ u + |u| p-1 u on the interval 0 < x < π, with p > 1 and λ > m 2. We study the global blow-up dynamics in the m-dimensional fast unstable manifold of the trivial equilibrium u ≡ 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 m + 1 intervals of strict monotonicity with prescribed extremal values u 1, . . . ,u m . Since u k = ± ∞ 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). Dedicated to Palo Brunovsky on the occasion of his birthday.
Keywords:Blow-up  parabolic partial differential equations  reaction diffusion equations  Sturm property  nodal properties  sign changing solutions  topological methods in analysis  trigonometric interpolation
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