open access

https://doi.org/10.14943/84116

https://hdl.handle.net/2115/69776

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

英語

2010-09-24

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

969

1-18

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 [17], or Zhou [20] 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 [9], the lifespan T(ε) of solutions of utt − Δu = u2 in R4 × [0,∞) with the initial data u(x, 0) = εf(x), ut(x, 0) = εg(x) of a small parameter ε > 0, compactly supported smooth functions f and g, has an estimate exp ( cε −2) ≤ T(ε) ≤ exp ( Cε −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.

紀要論文

Version of Record