arXiv:2611.06959v1 Announce Type: new Abstract: This Ph.D. thesis concerns the blow-up analysis, and more precisely, the asymptotic decomposition into profiles of solutions to nonlinear partial differential equations. In Chapter 1, we present a unifying framework for finding profile decompositions in Hilbert spaces using the notion of dislocation spaces. The next chapters are then devoted to my publications. In the second chapter, we construct solutions to the focusing, energy-critical, nonlinear wave equation in dimension 4 and 5, exhibiting finite-time Type II blow-up precisely at the origin with a single profile and a prescribed polynomial blow-up rate of order $-1-\nu$, where $\nu > 1$ in 4D and $\nu > 3$ in 5D. The major difference between dimension 4 and 5 consists in the renormalization procedure. In 4D, we essentially follow the Krieger-Schlag-Tataru scheme developed for the 3-dimensional equation, where the polynomial structure of the nonlinearity permits the use of simple algebraic manipulations to control error terms. By contrast, the 5D case requires a modified setup due to the lower regularity of the nonlinearity, which complicates the treatment of nonlinear error terms. In Chapter 3, we provide a modern presentation of classical Lp and Schauder estimates for parabolic equations. Then, in Chapter 4, we consider finite-time and $k$-equivariant solutions to the harmonic map heat flow from a disk to a sphere. Under general time-dependent, $k$-equivariant and smooth boundary data, we prove that the profile decomposition at the singular time contains only one bubble. The method relies on the maximum and comparison principle. We also prove a global existence criterion and exhibit solutions blowing up in infinite time for any degree of equivariance. Finally, the appendix presents the profile decomposition for the harmonic map heat flow proved by Struwe and Qing using energy arguments and the Palais-Smale condition.
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