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

Nonlinear wave equations

The analysis of nonlinear wave equations has experienced a dramatic growth in the last ten years or so. The key factor in this has been the transition from linear analysis, first to the study of bilinear and multilinear wave interactions, useful in the analysis of semilinear equations, and next to the study of nonlinear wave interactions, arising in fully nonlinear equations. The dispersion phenomena plays a crucial role in these problems. The purpose of this article is to highlight a few recent ideas and results, as well as to present some open problems and possible future directions in this field

Blow-Up of Positive Solutions to Wave Equations in High Space Dimensions

This paper is concerned with the Cauchy problem for the semilinear wave equation: u_{tt}-\Delta u=F(u) \ \mbox{in} \ R^n\times[0, \infty), where the space dimension $n \ge 2$, $F(u)=|u|^p$ or $F(u)=|u|^{p-1}u$ with $p>1$. Here, the Cauchy data are non-zero and non-compactly supported. Our results on the blow-up of positive radial solutions (not necessarily radial in low dimensions $n=2, 3$) generalize and extend the results of Takamura(1995) and Takamura, Uesaka and Wakasa(2011). The main technical difficulty in the paper lies in obtaining the lower bounds for the free solution when both initial position and initial velocity are non-identically zero in even space dimensions.Comment: 16page

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