The existence of the critical curve for p-q systems for nonlinear wave equations was already established by D. Del Santo, V. Georgiev, and E. Mitidieri [1997, Global existence of the solutions and formation of singularities for a class of hyperbolic systems, in``Geometric Optics and Related Topics'' (F. Colombini and N. Lerner, Eds.), Progress in Nonlinear Differential Equations and Their Applications, Vol. 32, pp. 117 139, Birkha user, Basel] except for the critical case. Our main purpose is to prove a blow-up theorem for which the nonlinearity ( p, q) is just on the critical curve in three space dimensions. Moreover, the lower and upper bounds of the lifespan of solutions are precisely estimated, including the sub-critical case. 2000Academic Press

The blow-up for semilinear wave equations with the scale invariant damping has been well-studied for sub-Fujita exponent. However, for super-Fujita exponent, there is only one blow-up result which is obtained in 2014 by Wakasugi in the case of non-effective damping. In this paper we extend his result in two aspects by showing that: (I) the blow-up will happen for bigger exponent, which is closely related to the Strauss exponent, the critical number for non-damped semilinear wave equations; (II) such a blow-up result is established for a wider range of the constant than the known non-effective one in the damping term.

In our previous two works, we studied the blow-up and lifespan estimates for damped wave equations with a power nonlinearity of the solution or its derivative, with scattering damping independently. In this work, we are devoted to establishing a similar result for a combined nonlinearity. Comparing to the result of wave equation without damping, one can say that the scattering damping has no influence.

The surgical results of foramen magnum decompression in seven patients with syringomyelia associated with Chiari I malformations are reported. The patients were two men and five women, ranging in age from 23 to 54 years (mean, 30 yr). A bony foramen magnum decompression combined with the removal of the outer layer of the dura mater was performed in seven patients. In an average postoperative follow-up period of 2 years (range, 1-3 yr), neurological symptoms and signs improved in six of seven patients and were unchanged in one patient. Postoperative magnetic resonance imaging scans taken a few weeks after surgery showed a decrease in the size of the syrinx or a disappearance of the syrinx in five of seven patients; a few months were required for a similar reduction in the remaining two patients. In all seven patients, the decompression of the foramen magnum proved to be sufficient, as disclosed by postoperative magnetic resonance imaging. Foramen magnum decompression by our surgical technique is advantageous because all the procedures are extradural and there are, therefore, fewer postoperative complications than the foramen magnum decompression techniques previously reported. Foramen magnum decompression by the authors' surgical technique is effective as the initial surgical treatment for syringomyelia associated with Chiari I malformations.