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Global small amplitude solutions to systems of nonlinear wave equations with multiple speeds

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  • Mathematics

Abstract

Katayama, S. and Yokoyama, K. Osaka J. Math. 43 (2006), 283–326 GLOBAL SMALL AMPLITUDE SOLUTIONS TO SYSTEMS OF NONLINEAR WAVE EQUATIONS WITH MULTIPLE SPEEDS SOICHIRO KATAYAMA and KAZUYOSHI YOKOYAMA1 (Received February 4, 2005) Abstract We give a global existence theorem to systems of quasilinear wave equations in three space dimensions, especially for the multiple-speed cases. It covers a wide class of quadratic nonlinearities which may depend on unknowns as well as their first and second derivatives. Our proof is achieved through total use of pointwise and L2-estimates concerning unknowns and their first and second derivatives. 1. Introduction Let u = u(t; x) = (u i (t; x))m i=1 be an Rm-valued unknown function, and set �i = � 2 t � 2 i 1 x with some positive constants i (i = 1; : : : ; m). We consider the following system of nonlinear wave equations (1.1) � i u i (t; x) = F i (u; �u;r x �u) for t > 0 and x 2 R3 (1 � i � m) with initial data (1.2) u i (0; x) = ' i (x); � t u i (0; x) = i (x) for x 2 R3 (1 � i � m): We use the notation �0 = �t = �=�t and �j = �=�xj for 1 � j � 3 throughout this paper. �u and r x �u are R4m-valued and R12m-valued functions, whose compo- nents are � � u i (1 � i � m, 0 � � � 3) and � j � � u i (1 � i � m, 1 � j � 3, 0 � � � 3), respectively. F (u; v;w) = (F i (u; v;w))1�i�m is a given function of (u; v;w) 2 Rm � R4m � R12m. The components of u, v and w are denoted by u i , v i;� and w i;j� , respectively, where 1 � i � m, 1 � j � 3 and 0 � � � 3. Here v i;� cor- responds to � � u i , and w i;j� to � j � � u i . We suppose that ' = (' i )m i=1 and = ( i)mi=1 in (1.2) are rapidly decreasing functions. We assume that F (u; v;w) is linear with respect to w and satisfies (1.3) F (u; v;w) = O �juj2 + jvj2 + jwj2� near (u; v;w) = (0; 0; 0): 2000 Mathematics Subject Classification. Primary 35L70; Secondary 35L05, 35L15, 35L55. 1This researc

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