On the dynamics of energy-critical focusing wave equations

Abstract : In this thesis we study the global behavior of solutions of the energy-criticalfocusing nonlinear wave equation, with a special emphasis on the description of the dynamics in the energy space. We develop a new approach, based on the energy method, to constructing unstable type II blow-up solutions. Next, we give the first example of a radial two-bubble solution of the energy-critical wave equation. By implementing this construction in the case of the equivariant wave map equation, we obtain bubble-antibubble solutions in equivariance classes k > 2. We also study the relationship between the speed of a type II blow-up and the weak limit of the solution at the blow-up time. Finally, we prove that there are no pure radial two-bubbles with opposite signs for the energy-critical wave equation.
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Jacek Jendrej. On the dynamics of energy-critical focusing wave equations. Analysis of PDEs [math.AP]. Université Paris-Saclay, 2016. English. ⟨NNT : 2016SACLX029⟩. ⟨tel-01474613⟩

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