Almost global solutions of semilinear wave equations with the critical exponent in high dimensions

We are interested in the “almost” global-in-time existence of classical solutions in the general theory for nonlinear wave equations. All the three such cases are known to be sharp due to blow-up results in the critical case for model equations. However, it is known that we have a possibility to get the global-in-time existence for two of them in low space dimensions if the nonlinear term is of derivatives of the unknown function and satisfies the so-called null condition, or non-positive condition. But another one for the quadratic term in four space dimensions is out of the case as the nonlinear term should include a square of the unknown function itself.In this paper, we get one more example guaranteeing the sharpness of the almost global-in-time existence in four space dimensions. It is also the first example of the blow-up of classical solutions for non-single and indefinitely signed term in high dimensions. Such a term arises from the neglect of derivative-loss factors in Duhamel’s formula for positive and single nonlinear term. This fact may help us to describe a criterion to get the global-in-time existence in this critical situation.