Blow-up for solutions of \(\square u=| u| ^ p\) with small initial data. (English) Zbl 0712.35018

Lindblad, Hans

Commun. Partial Differ. Equations 15, No. 6, 757-821 (1990).

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

Keywords:

lifespan; Cauchy problem

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