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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)
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