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Regularity results for nonlinear wave equations. (English) Zbl 0836.35096
We report on some recent progress on the Cauchy problem for semilinear wave equations \(u_{tt} - \Delta u + g(u) = 0\) in \(\mathbb{R}^n \times \mathbb{R}\) for smooth nonlinearities \(g\) of critical growth \(g(u) = u |u |^{2^* - 2}\) for large \(|u |\), where \(2^* = (2n)/(n - 2)\), and related problems. Our aim in this article is twofold. First we intend to present a simplified proof of the known regularity results for \(3 \leq n \leq 5\) that extends to \(n = 6\) and 7. As will become apparent, these results follow almost directly from the “classical” Strichartz- Ginibre-Velo a priori estimates. Our second goal is to apply these results to show regularity for \((2 + 1)\)-dimensional equivariant harmonic maps into rotationally symmetric surfaces, otherwise called \(\sigma\)- models, assuming small initial energy, but no convexity on the range. In this respect our results extend the work of J. Shatah and A. Tahvildar-Zadeh [Commun. Pure. Appl. Math. 45, No. 8, 947-971 (1992; Zbl 0769.58015)].

MSC:
35L70 Second-order nonlinear hyperbolic equations
35B65 Smoothness and regularity of solutions to PDEs
35L15 Initial value problems for second-order hyperbolic equations
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