Combined effects of two nonlinearities in lifespan of small solutions to semi-linear wave equations

Abstract

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 semi-linear 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.

Final version. Presentation polished and references updated. To appear in Mathematische Annalen. 22 pages, 5 figures. The final publication is available at Springer via http://dx.doi.org/10.1007/s00208-015-1346-1

References in corpus (2)

• Improved Kato's lemma on ordinary differential inequality and its application to semilinear wave equations
• The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions

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• Global existence for a system of quasi-linear wave equations in $3$D satisfying the weak null condition
• The generalized combined effect for one dimensional wave equations with semilinear terms including product type