# zbMATH — the first resource for mathematics

Weak and yet weaker solutions of semilinear wave equations. (English) Zbl 0831.35109
This interesting paper deals with the existence of low regularity local solutions to the Cauchy problem $\square u = \lambda |u |^\sigma u,\;u(0,x) = \varphi (x),\;u_t(0,x) = \psi (x)\;(t \geq 0,\;x \in \mathbb{R}^n),$ where $$\lambda \in \mathbb{R}$$, $$\sigma > 0$$ and $$n \geq 3$$. It is well known that, for all $$\sigma \geq 0$$, this problem is locally well-posed in $${\mathcal H}^r \equiv H^r \times H^{r - 1}$$ (i.e. for any $$(\varphi, \psi) \in {\mathcal H}^r$$ there is a unique solution $$u \in C ([0,T], H^r) \cap C^1 ([0,T], H^{r - 1})$$, for some $$T > 0)$$ as soon as $$r \geq n/2$$. Such a result is based on the energy estimates for the linearized equation and on the Gagliardo-Nirenberg inequalities for the nonlinear term, and holds true for any second order hyperbolic operator. On the other hand, using the Strichartz estimates for the linear wave equation, J. Ginibre, G. Velo and L. Kapitanski were able to prove that the problem is locally well posed in $${\mathcal H}^1$$ for all $$\sigma \leq 2 \cdot (n/2 - 1)^{-1}$$. Thus, fixed $$r > 0$$, it is natural to try to determine the largest value of $$\sigma$$ for which there is a well-posedness in $${\mathcal H}^r$$.
Here, the author considers the critical value $$\sigma_* (n,r) = 2 \cdot (n/2 - r)^{-1}$$ for the rate of growth of the nonlinearity: he proves that, if $$1/2\leq r\leq 1$$ and $$\sigma\leq\sigma_* (n,r)$$, the above problem is well-posed in $${\mathcal H}^r$$ and conjectures that the same conclusion holds true for $$1 < r < n/2$$. As to the range $$0 < r < 1/2$$, he finds a different bound $$\sigma_{**} (n,r)$$ (with $$\sigma_{**} < \sigma_*)$$. Other results in this direction have been obtained by H. Lindblad and C. Sogge [J. Funct. Anal. 130, 357-426 (1995)].

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35L15 Initial value problems for second-order hyperbolic equations 35D05 Existence of generalized solutions of PDE (MSC2000)
Full Text:
##### References:
 [1] Bergh J., Interpolation Spaces (1976) [2] DOI: 10.1007/BF01215290 · Zbl 0321.35052 · doi:10.1007/BF01215290 [3] DOI: 10.1007/BF01258907 · Zbl 0457.35059 · doi:10.1007/BF01258907 [4] DOI: 10.1007/BF01258601 · Zbl 0696.35153 · doi:10.1007/BF01258601 [5] DOI: 10.1007/BFb0086749 · doi:10.1007/BFb0086749 [6] DOI: 10.1016/0362-546X(90)90023-A · Zbl 0706.35127 · doi:10.1016/0362-546X(90)90023-A [7] Colombeau J.F., New generalized functions and multiplication of distributions (1985) · Zbl 0761.46021 [8] DOI: 10.1070/RM1990v045n05ABEH002683 · Zbl 0754.46034 · doi:10.1070/RM1990v045n05ABEH002683 [9] DOI: 10.1016/0022-1236(92)90044-J · Zbl 0813.35054 · doi:10.1016/0022-1236(92)90044-J [10] DOI: 10.1007/BF01168155 · Zbl 0549.35108 · doi:10.1007/BF01168155 [11] Ginibre J., Ann. Inst. H.Poincare, Analyse Non Lineaire 6 pp 15– (1989) [12] J.Ginibre, G.Velo, Regularity of solutions of critical and subcritical non linear wave equations, Preprint LPTHE 92/06. · Zbl 0831.35108 [13] DOI: 10.2307/1971427 · Zbl 0736.35067 · doi:10.2307/1971427 [14] Grillakis M.G., In: Nonlinear hyperbolic equations and field theory 253 pp 110– (1992) [15] M.G.Grillakis, Regularity for the wave equation with critical nonlinearit (1991) (to appear). [16] DOI: 10.1007/BF01180181 · Zbl 0111.09105 · doi:10.1007/BF01180181 [17] Kapitanski L.V., Leningrad Math.J. 1 pp 693– (1990) [18] DOI: 10.1007/BF01671936 · Zbl 0759.35014 · doi:10.1007/BF01671936 [19] DOI: 10.1007/BF01671000 · doi:10.1007/BF01671000 [20] DOI: 10.1007/BF01102635 · doi:10.1007/BF01102635 [21] L.V. Kapitanski, Global and unique weak solutions to a semilinear wave equation, in preparation. · Zbl 0841.35067 [22] L.V.Kapitanski, Global weak and more regular solutions to a semilinear wave equation, in preparation. · Zbl 0829.35014 [23] Kato T., Proc. Sympos. Pure Math. 45 pp 9– (1986) [24] C.E. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Preprint (1992). · Zbl 0787.35090 [25] S. Klainerman, M. Machedon, On the regularity properties of the wave equation, Preprint (1992). · Zbl 0833.35093 [26] S. Klainerman, M. Machedon, Space-time estimates for null forms and the local existence theorem, Preprint (1992). · Zbl 0803.35095 [27] S. Klainerman, M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Preprint (1992). · Zbl 0818.35123 [28] Ladyzhenskaya O.A., Mixed problem for hyperbolic equations (in Russian) (1953) [29] Ladyzhenskaya O.A., Applied mathematical sciences 49 (1985) [30] Levine H.A., Trans. Amer. Math. Soc. 192 pp 1– (1974) [31] H. Lindblad, personal communication (Spring, 1992). [32] H. Lindblad, A sharp counter example to local existence of low regularity solutions to nonlinear wave equations, submitted to the Duke Mathematical Journal (1993). · Zbl 0797.35123 [33] H. Lindblad, personal communication (Spring, 1993). [34] Lions J.L., Rev. Roumaine Math. Pures Appl. 9 pp 11– (1964) [35] Lions J.L., Quelques méhtodes de résolution des problémes qux limites non linéaires (1969) [36] H.P. McKean, K. Vaninsky, Statistical mecarics of nonlinear wave equations (1992) (to appear). · Zbl 0834.35108 [37] Morawetz, C.S. 1968.Time decay for the nonlinear Klein-Gordon equation, Vol. A. 306, 291–296. Proc.Roy.Soc. · Zbl 0157.41502 [38] DOI: 10.1007/BF01215233 · Zbl 0318.35054 · doi:10.1007/BF01215233 [39] Petrowsky I.G., Matem. Sbornik 2 pp 815– (1937) [40] Segal I.E., Bull.Soc.Math.France 91 pp 129– (1963) [41] J. Shatah, M. Struwe, Regularity result for nonlinear wave equations, Preprint (1991). · Zbl 0836.35096 [42] Strauss W.A., An.Acad.Brasil.Cienc 42 pp 645– (1970) [43] Strauss W.A., Nonlinear wave equations (1989) [44] DOI: 10.1090/S0002-9947-1970-0256219-1 · doi:10.1090/S0002-9947-1970-0256219-1 [45] DOI: 10.1016/0022-1236(70)90027-3 · Zbl 0189.40701 · doi:10.1016/0022-1236(70)90027-3 [46] Struwe M., Ann. Sc. Nor. Sup. Pisa 15 pp 495– (1988) [47] DOI: 10.1090/S0273-0979-1992-00225-2 · Zbl 0767.35045 · doi:10.1090/S0273-0979-1992-00225-2 [48] DOI: 10.1007/978-3-0346-0416-1 · Zbl 1235.46002 · doi:10.1007/978-3-0346-0416-1
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.