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UBC Theses and Dissertations

Long-time dynamics for the energy-critical harmonic map heat flow and nonlinear heat equation Roxanas, Dimitrios


The main focus of this thesis is on critical parabolic problems, in particular, the harmonic map heat from the plane to S2, and nonlinear focusing heat equations with an algebraic nonlinearity. The focus of this work has been on long-time dynamics, stability and singularity formation, and the investigation of the role of special, soliton-like, solutions to the asymptotic behaviour of solutions. Harmonic Map Heat Flow: Flow: we consider m-corotational solutions to the harmonic map heat flow from R2 to S2. We first work in a class of maps with trivial topology and energy of the initial data below two times the energy of the stationary harmonic map solutions. We give a new proof of global existence and decay. The proof is based on the "concentration-compactness plus rigidity" approach of Kenig and Merle and relies on the dissipation of the energy and a profile decomposition. We also treat m-corotational maps (m greater than 3) with non-trivial topology and energy of the initial data less than three times the energy of the stationary harmonic map solutions. Through a new stability argument we rule out finite-time blow-up and show that the global solution asymptotically converges to a harmonic map. Nonlinear Heat Equation: we also study solutions of the focusing energy-critical nonlinear heat equation. We show that solutions emanating from initial data with energy and kinetic energy below those of the stationary solutions are global and decay to zero. To prove that global solutions dissipate to zero we rely on a refined small data theory, L2-dissipation and an approximation argument. We then follow the "concentration-compactness plus rigidity" roadmap of Kenig and Merle (and in particular the approach taken by Kenig and Koch for Navier-Stokes) to exclude finite-time blow-up.

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