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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
Zbl 0769.58015
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