The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions
By H. Takamura, Kyouhei Wakasa. 2010
76 citations recorded by the source corpus.
The final open part of Strauss' conjecture on semilinear wave eqautions 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. \par 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 \[ \exp\left(c\e^{-2}\right)\le T(\e)\le\exp\left(C\e^{-2}\right), \] 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.
Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/e8f8756a57e7865f0f48cf67c25fee32006550d0
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In Applied Mathematics
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Citations in the corpus
- Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data (2017, unknown): “Lai–Takamura–Wakasa proved in [9] that for pF(N) ≤ p p0(N + 2µ) with µ ∈ [0, µ∗2 ), the lifespan of solution uε is estimated above by ε − 2p(p−1) γ(N+2µ;p) .”
- Life-Span of Solutions to Critical Semilinear Wave Equations (2011, methodology): “[15–21] and, [18]), this is the famous Strauss’ conjecture, we only give a brief summary here and refer the reader to [4, 6, 8, 9, 11, 17, 20].”
- Nonexistence of global solutions for the semilinear Moore – Gibson – Thompson equation in the conservative case (2019, background): “(4) The lifespan of solutions to (3) that blow up in finite time has been intensively considered [26, 47, 48, 49, 27, 9, 41, 52, 40, 15].”