The generalized combined effect for one dimensional wave equations with semilinear terms including product type

Abstract

We are interested in the so-called "combined effect" of two different kinds of nonlinear terms for semilinear wave equations in one space dimension. Recently, the first result with the same formulation as in the higher dimensional case has been obtained if and only if the total integral of the initial speed is zero, namely Huygens' principle holds. In this paper, we extend the nonlinear term to the general form including the product type. Such model equations are extremely meaningful only in one space dimension because the most cases in higher dimensions possess the global-in-time existence of a classical solution in the general theory for nonlinear wave equations. It is also remarkable that our results on the lifespan estimates are partially better than those of the general theory. This fact tells us that there is a possibility to improve the general theory which was expected complete more than 30 years ago.

arXiv commentThe first version has a wrong citation of the result of the general theory by [12,13] at the last line in (2.24). This part is corrected. As a result, the possibility to improve the general theory may happen only for the combined effected case as stated this version. This kind of error is also found in the previous work in arXiv:2205.07198

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Cited by in corpus (2)

• Blow-up of classical solutions of quasilinear wave equations in one space dimension
• Slicing method for nonlinear integral inequalities related to critical nonlinear wave equations