Global existence for systems of nonlinear wave equations in two space dimensions. II
[For part I see Zbl 0830.35083.] The author considers the Cauchy problem for the system of nonlinear wave equations \[ (\partial^2_t-\Delta)u_i= F_i(u,u',u'') \qquad\text{in}\quad (0,\infty)\times\mathbb{R}^2, \quad i=1,\dots,N, \] with initial data \(u_i(0,x)= \varepsilon\phi_i\), \(\partial_t u_i(0,x)=\varepsilon \psi_i(x)\), where \(F_i\) are smooth functions of degree 3 near the origin \((u,u',u'')=0\), \(\phi_i,\psi_i\in C_0^\infty(\mathbb{R}^2)\), and \(\varepsilon\) is a small positive parameter. \(F_i\) are assumed to be independent of \(u_j''\) for any \(j\neq i\). The author also discussed such a Cauchy problem in his previous paper. But in this paper, he alleviates the restriction of his conclusion and obtains the global existence provided that Klainerman's null condition on the cubic part of the nonlinear terms is satisfied.
scientific article; zbMATH DE number 878494
scientific article