Regularity for the wave equation with a critical nonlinearity. (English) Zbl 0785.35065

Consider a nonlinear wave equation with a self-interacting term of the form \[ \square\varphi+f(\varphi)=0,\;(t,x)\in\mathbb{R}\times\mathbb{R}^ n \tag{1} \] where \(\square=\partial_ t\partial^ t-\partial_ j\partial^ j\) is the D’Alembertian in \(n+1\) dimensions with respect to the Minkowski metric. In order to be able to solve (1), initial data have to be prescribed over some space-like hypersurface, for example at \(t=t_ 0\) \(\varphi(t_ 0,x)=\varphi_ 0(x)\), \(\varphi_{,t}(t_ 0,x)=\varphi_ 1(x)\), \(x\in R^ n\). In the present work it is assumed that the nonlinear potential is repulsive, i.e., \[ F(\varphi):=\int^ \varphi_ 0f(s)ds\geq 0, \tag{2} \] and that it grows like the critical Sobolev exponent, i.e., there exists some positive constant \(C\) such that \[ F(\varphi):=|\varphi|^{{2n\over n-2}}\quad\text{for }|\varphi|>C. \tag{3} \] In order to avoid some technicalities well known from the linear theory, the author assumes for reasons of simplicity that the initial data \(\varphi_ 0\) and \(\varphi_ 1\) as well as \(f\) are \(C^ \infty\). It is the purpose of this work to show that the solution of (1) is in fact \(C^ \infty\), provided that \(3\leq n\leq 5\). Conditions (2), (3), and the smoothness of \(f\) and the initial data can be relaxed.


35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35B65 Smoothness and regularity of solutions to PDEs
Full Text: DOI


[1] Brenner, Math. Z. 145 pp 251– (1975)
[2] Brenner, Math. Z. 176 pp 87– (1981)
[3] Ginibre, Math. Z. 189 pp 487– (1985)
[4] Grillakis, Ann. of Math. 132 pp 485– (1990)
[5] Hörmander, Acta Math. 104 pp 93– (1960)
[6] Jorgens, Math. Z. 77 pp 295– (1961)
[7] Kapitanskii, Leningrad Math. J. 1 pp 693– (1990)
[8] Littman, Bull. Amer. Math. Soc. 83 pp 482– (1956)
[9] Marshal, J. Math. Pures Appl. 59 pp 417– (1980)
[10] Pecher, Math. Z. 150 pp 159– (1976)
[11] Manuscripta Math. 20 pp 227– (1977)
[12] The u5-Klein-Gordon equation, pp. 335–364 in: Pitman Research Notes in Mathematics, Vol. 53, and , eds., Pitman, 1976.
[13] Stein, Trans. AMS 83 pp 482– (1956)
[14] Oscillatory integrals in Fourier analysis, pp. 307–355 in: Beijing Lectures in Harmonic Analysis, Princeton University Press, 1986.
[15] Strauss, CONM AMS 73N (1989)
[16] Strichartz, Trans. AMS 148 (1970)
[17] Strichartz, J. Fun. Anal. 5 pp 218– (1970)
[18] Strichartz, Duke Math. J. 44 pp 705– (1977)
[19] Struwe, Ann. Sc. Norm. Sup. Pisa (Ser 4) 15 pp 495– (1988)
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.