×

zbMATH — the first resource for mathematics

Blow-up for solutions of \(\square u=| u| ^ p\) with small initial data. (English) Zbl 0712.35018
The author proves precise asymptotic estimates on the lifespan \(T_{\epsilon}\) for solutions of the Cauchy problem \[ \square u(t,x)=u(t,x)^ 2,\quad u(0,x)=\epsilon f(x),\quad \partial_ tu(0,x)=\epsilon g(x), \] where \(f\in C^ 3_ 0({\mathbb{R}}^ 3)\), \(g\in C^ 2_ 0({\mathbb{R}}^ 3)\), \(\square =\partial_ t^ 2- \sum^{3}_{i=1}\partial^ 2_{x_ i}\).
Reviewer: S.G.Krantz

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1090/S0002-9947-1986-0849476-3
[2] Caffarelli L., Arch. Rational Mech. Anal. 91 pp 83– · Zbl 0593.35055
[3] DOI: 10.1098/rspa.1962.0162 · Zbl 0106.41501
[4] DOI: 10.1098/rspa.1964.0111 · Zbl 0117.43904
[5] DOI: 10.1007/BF01162066 · Zbl 0438.35045
[6] DOI: 10.1007/BF01262042 · Zbl 0451.35039
[7] DOI: 10.1007/BFb0077745
[8] DOI: 10.1007/BF01647974 · Zbl 0406.35042
[9] DOI: 10.1002/cpa.3160400104 · Zbl 0662.35070
[10] DOI: 10.1002/cpa.3160330403 · Zbl 0421.35053
[11] DOI: 10.1002/cpa.3160100404 · Zbl 0090.31802
[12] DOI: 10.1002/cpa.3160380305 · Zbl 0635.35059
[13] DOI: 10.1002/cpa.3160330104 · Zbl 0405.35056
[14] Schaeffer, Proc. Roy .Soc. Edinburgh pp 31– (1985) · Zbl 0592.35080
[15] DOI: 10.1016/0022-0396(84)90169-4 · Zbl 0555.35091
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.