305 research outputs found
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 in 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 and , has an estimate
\exp(c\e^{-2})\le T(\e)\le\exp(C\e^{-2}), where and are positive
constants depending only on and . 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
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