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Stability and unconditional uniqueness of solutions for energy critical wave equations in high dimensions. (English) Zbl 1332.35007

From the abstract: In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimension \(\mathbb R\times\mathbb R^d\) with \(d \geq 6\). We prove the stability of solutions under the weak condition that the perturbation of the linear flow is small in certain space-time norms. As a by-product of our stability analysis, we also prove local well-posedness of solutions for which we only assume the smallness of the linear evolution. These results provide essential technical tools that can be applied towards obtaining the extension to high dimensions of the analysis of C. E. Kenig and F. Merle [Acta Math. 201, No. 2, 147–212 (2008; Zbl 1183.35202)] of the dynamics of the focusing NLW below the energy threshold. By employing refined paraproduct estimates we also prove unconditional uniqueness of solutions for \( d \geq 6 \) in the natural energy class. This extends an earlier result by F. Planchon [Math. Z. 244, No. 3, 587–599 (2003; Zbl 1023.35079)].

MSC:

35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35L71 Second-order semilinear hyperbolic equations
35B35 Stability in context of PDEs
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