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Forum Math. 23 (2011), 1113–1134 DOI 10.1515/FORM.2011.039 Forum Mathematicum © de Gruyter 2011 Almost global existence for quasilinear wave equations with inhomogeneous terms in 3D Yi Zhou and Wei Xu Communicated by Christopher D. Sogge Abstract. This article establishes the almost global existence of solutions for three-dimen- sional nonlinear wave equations with quadratic, divergence-form nonlinearities and time- independent inhomogeneous terms. The approach used here can be applied to the system of homogeneous, isotropic hyperelasticity with time

proof, but also considerable improvement of the earlier results due to John [ 15 ], Klainerman and Sideris [ 19 ] and the second author [ 40 ] concerning almost global existence of small solutions to 3-D nonlinear elastic wave equations. Recall that the standard local existence theorem was established in the Sobolev space H s + 1 × H s {H^{s+1}\times H^{s}} with s > 5 2 {s>\frac{5}{2}} (see Hughes, Kato and Marsden [ 11 ]), which has motivated us to obtain almost global existence results in H 4 × H 3 {H^{4}\times H^{3}} , the lowest integer-order Sobolev space

Journal of Applied Analysis Vol. 7, No. 1 (2001), pp. 61–79 ON ALMOST GLOBAL EXISTENCE FOR THE CAUCHY PROBLEM FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS IN THE LP-FRAMEWORK P. B. MUCHA Received February 23, 2000 and, in revised form, August 11, 2000 Abstract. The almost global in time existence of regular solutions to equations of viscous compressible barotropic fluid motion for the initial value problem in the Lp-framework is shown. To prove the result we apply the Lp-estimate for the linearization of the equations which are obtained at the end of the paper. 1

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1265 Tianyi Ren, Yakun Xi, Cheng Zhang An endpoint version of uniform Sobolev inequalities 1279 Kunio Hidano, Dongbing Zha Space-time L2 estimates, regularity and almost global existence for elastic waves 1291 Maria Cristina Perez-Garcia k-spaces and duals of non-archimedean metrizable locally convex spaces 1309 Alexander Grigor’yan, Yuri Muranov, Vladimir Vershinin, Shing-Tung Yau Path homology theory of multigraphs and quivers 1319

loss, preprint (2011), http://arxiv.org/abs/1103.3908. [4] Y. Colin de Verdiére and B. Parisse, Équilibre instable en régime semi-classique. I. Concentration microlocale, Comm. Partial Differential Equations 19 (1994), 1535– 1563. [5] Y. Du, J. Metcalfe, C. D. Sogge and Y. Zhou, Concerning the Strauss conjecture and almost global existence for nonlinear Dirichlet-wave equations in 4-dimensions, Comm. Partial Differential Equations 33 (2008), 1487–1506. [6] Y. Du and Y. Zhou, The life span for nonlinear wave equation outside of star-shaped obstacle in three space

. 180 (1998), 1–29 [3] Christodoulou D.: Global solutions of nonlinear hyperbolic equations for small initial data. Comm. Pure Appl. Math. 39 (1986), 267–282 [4] Gilbarg D. and Trudinger N.: Elliptic partial di¤erential equations of second order. Springer, Second Ed., Third Printing, 1998 [5] Hidano K.: An elementary proof of global or almost global existence for quasi-linear wave equations; to appear, Tohoku Math. J. [6] Hidano K. and Yokoyama K.: A remark on the almost global existence theorems of Keel, Smith, and Sogge, preprint [7] Hörmander L.: Lectures on

–268. 5. F. John, Blow-up for quasilinear wave equations in three space dimensions. Comm. Pure Appl. Math. 34(1981), No. 1, 29–51. 6. F. John and S. Klainerman, Almost global existence to nonlinear wave equations in three space dimensions. Comm. Pure Appl. Math. 37(1984), No. 4, 443–455. 7. T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations. Comm. Pure Appl. Math. 33(1980), No. 4, 501–505. 8. J. Ginibre, A. Soffer, and G. Velo, The global Cauchy problem for the critical nonlinear wave equation. J. Funct. Anal. 110(1992), No. 1, 96–130. 9. W. A. Strauss

initial boundary-value problems based on special construction of series. Russ. J. Numer. Anal. Math. Modelling (1993) 8, No. 2, 101 -126. 5. F. John, Formatting of singularities in one-dimensional nonlinear wave propagation. Communs Pure Appl. Math. (1976) 29, 649-681. 6. F. John and S. Klainerman, Almost global existence of nonlinear wave equations in three space dimensions. Communs Pure Appl Math. (1984) 37, 443-455. 7. M. M. Khapaev, On the study of stability in the theory of nonlinear oscillations. Mat. Zametki (1968) 3, No.3, 307-319 (in Russian). 8. S. Klainerman

s , Sobolev Spaces, Academic Press, New York, 1975. [2] J. G a w i n e c k i , Global solution to the Cauchy problem in non-linear hyperbolic ther- moelasticity, Math. Method Appl. Sci. 15 (1992), 223-237. [3] F. J o h n , S. K l a i n e r m a n , Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math. 37 (1984), 443-455. [4] S. K a w a s h i m i i , Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Thesis, Kyoto University, 1983. [5] S. K l a i n e r m

: (6.4) Subtracting (6.4) from (6.3), we obtain (2) of Proposition 6.1. Weighted energy estimates for wave equations in exterior domains 1257 Bibliography [1] N. Burq, Decroissance de l’energie locale de l’equation des ondes pour le probleme exterieur et absence de resonance au voisinage du reel, Acta Math. 180 (1998), 1–29. [2] F. Cardoso, G. Popov and G. Vodev, Distribution of resonances and local energy decay in the transmission problem II, Math. Res. Lett. 6 (1999), no. 3–4, 377–396. [3] K. Hidano and K. Yokoyama, A remark on the almost global existence theorems