The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions

The final open part of Strauss' conjecture on semilinear wave equations was the blow-up theorem for the critical case in high dimensions. This problem was solved by Yordanov and Zhang in 2006, or Zhou in 2007 independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. In this paper, we refine their theorems and introduce a new iteration argument to get the sharp upper bound of the lifespan. As a result, with the sharp lower bound by Li and Zhou in 1995, the lifespan T(\e) of solutions of $u_{tt}-\Delta u=u^2$ in $\R^4\times[0,\infty)$ with the initial data u(x,0)=\e f(x),u_t(x,0)=\e g(x) of a small parameter \e>0, compactly supported smooth functions $f$ and $g$, has an estimate \exp(c\e^{-2})\le T(\e)\le\exp(C\e^{-2}), where $c$ and $C$ are positive constants depending only on $f$ and $g$. This upper bound has been known to be the last open optimality of the general theory for fully nonlinear wave equations

Blow-up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities

In this work we study the blow-up of solutions of a weakly coupled system of damped semilinear wave equations in the scattering case with power nonlinearities. We apply an iteration method to study both the subcritical case and the critical case. In the subcritical case our approach is based on lower bounds for the space averages of the components of local solutions. In the critical case we use the slicing method and a couple of auxiliary functions, recently introduced by Wakasa-Yordanov, to modify the definition of the functionals with the introduction of weight terms. In particular, we find as critical curve for the pair (p, q) of the exponents in the nonlinear terms the same one as for the weakly coupled system of semilinear wave equations with power nonlinearities

Recent developments on the lifespan estimate for classical solutions of nonlinear wave equations in one space dimension

In this paper, we overview the recent progresses on the lifespan estimates of classical solutions of the initial value problems for nonlinear wave equations in one space dimension. There are mainly two directions of the developments on the model equations which ensure the optimality of the general theory. One is on the so-called "combined effect" of two kinds of the different nonlinear terms, which shows the possibility to improve the general theory. Another is on the extension to the non-autonomous nonlinear terms which includes the application to nonlinear damped wave equations with the time-dependent critical case.Comment: 19 pages. arXiv admin note: text overlap with arXiv:2305.0018