Global existence and nonexistence of solutions for a system of nonlinear damped wave equations |
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Authors: | Hiroshi Takeda |
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Institution: | aMathematical Institute, Tohoku University, Sendai 980-8578, Japan |
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Abstract: | We consider the Cauchy problem for the system of semilinear damped wave equations with small initial data: We show that a critical exponent which classifies the global existence and the finite time blow up of solutions indeed coincides with the one to a corresponding semilinear heat systems with small data. The proof of the global existence is based on the Lp–Lq estimates of fundamental solutions for linear damped wave equations K. Nishihara, Lp–Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003) 631–649; K. Marcati, P. Nishihara, The Lp–Lq estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations 191 (2003) 445–469; T. Hosono, T. Ogawa, Large time behavior and Lp–Lq estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004) 82–118; T. Narazaki, Lp–Lq estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan 56 (2004) 585–626]. And the blow-up is shown by the Fujita–Kaplan–Zhang method Q. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001) 109–114; F. Sun, M. Wang, Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping, Nonlinear Anal. 66 (12) (2007) 2889–2910; T. Ogawa, H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. 70 (10) (2009) 3696–3701]. |
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Keywords: | Nonlinear damped wave equations Critical exponent Global solution Finite time blow up |
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