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UBC Theses and Dissertations

The construction of blow-up solutions for some evolution equations Zhang, Qidi


In this dissertation, we construct blow-up solutions for the critical heat equations and the two-dimensional Landau-Lifshitz-Gilbert equation. In Chapter 2, we construct a radial smooth positive ancient solution for the energy critical semi-linear heat equation in the Euclidean space with a dimension greater or equal to seven. It blows up at the origin with the profile of multiple Aubin-Talenti bubbles in the backward time infinity. In Chapter 3, we consider the Cauchy problem for the four-dimensional energy critical heat equation. We construct a positive infinite time blow-up solution with the blow-up rate ln t as t goes to infinity and study the stability of the infinite time blow-up solution. This gives rigorous proof of the infinite time blow-up predicted by Fila and King. In Chapter 4, we construct finite time blow-up solutions to the Landau-Lifshitz-Gilbert (LLG) equation from the two-dimensional Euclidean space into the unit sphere with dimension two. Given any prescribed N points in the two-dimensional Euclidean space and a small positive constant T, we prove that there exists smooth initial data such that the solution blows up precisely at these points at finite time T, taking around each point the profile of sharply scaled degree 1 harmonic map.

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