In this paper, we verify the Glassey conjecture in the radial case for all spatial dimensions, which states that, for the nonlinear wave equations of the form $\Box u=|\nabla u|^p$, the critical exponent to admit global small solutions is given by $p_c=1+\frac{2}{n-1}$. Moreover, we are able to prove the existence results with low regularity assumption on the initial data and extend the solutions to the sharp lifespan. The main idea is to exploit the trace estimates and KSS type estimates.Comment: 28 page

We discuss the global existence of small solutions to the Cauchy problem for systems of quasilinear wave equations in three space dimensions, when their nonlinear terms have quadratic nonlinearity. A global existence theorem is established on the null condition which is extended to the condition for systems of wave equations with di¨erent propagation speeds.

This paper investigates the combined effects of two distinctive power-type nonlinear terms (with parameters p, q > 1) in the lifespan of small solutions to semilinear wave equations. We determine the full region of ( p, q) to admit global existence of small solutions, at least for spatial dimensions n = 2, 3. Moreover, for many ( p, q) when there is no global existence, we obtain sharp lower bound of the lifespan, which is of the same order as the upper bound of the lifespan. Mathematics Subject ClassificationIn memory of Rentaro Agemi.The authors are very grateful to the referees for their helpful comments.

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Abstract. We give a new proof of temporally global existence of small solutions for systems of semi-linear wave equations. Our proof uses the Klainerman-Sideris inequality and a space-time L 2 -estimate. We also discuss whether the scale-invariant version of the space-time L 2 -estimates can hold, and obtain some related estimates.Among other things, we prove that the Keel-Smith-Sogge estimate actually holds in all space dimensions.