Resonance-type behaviour in a system of nonlinear wave equations
This paper is concerned with the Cauchy problem for a system of wave equations: \[ (*)\quad \partial^ 2_ tu_ i-c_ i^ 2\Delta u_ i=F_ i(\partial_ tu,\partial_ 1u,...,\partial_ nu),\quad u(0,x)=\epsilon f_ 0(x),\quad \partial_ tu(0,x)=\epsilon f_ 1(x) \] where \(u=(u_ 1,...,u_ m)\) is a vector function of \(t>0\) and \(x\in {\mathbb{R}}^ n\) \((n=2\) or \(n=3)\), \(\epsilon\) is a small parameter, the \(c_ i's\) are constants and \(f_ 0\), \(f_ 1\) are \(C_ 0^{\infty}\). The author assumes that for u satisfying \(\sup_{i,R}\{| u_ i|\), \(| \partial_ Ru_ i| \}\leq 1\) the \(F_ i\) can be written as \(F_ i=\sum_{a+b\geq 0}g_{ab_ i}\partial_ au_ b(\partial_ 0=\partial_ t)\) where \(g_{ab_ i}\in C^{\infty}\) and \(| \frac{\partial^ j}{\partial (\partial u)^ j}g_{ab_ i}| \leq C\sum^{n}_{R=0}| \partial_ ku|^{p-1\quad -j}\), \(0\leq j\leq p-1\). Here \(p=2\) if \(n=3\) and \(p=3\) if \(n=2\). Also he assumes that \(\partial g_{ab_ i}/\partial (\partial_ cub)=0\) for all a,b,c,i and \(\sum_{i,j}| \partial_ iu_ j| \leq 1.\) In the above conditions and assuming that the m numbers \(c_ 1,...,c_ m\) are all real and distinct, the author proves that there exists an \(\epsilon_ 0>0\), depending only on \(f_ 0,f_ 1,F_ i\) and the \(c_ i's\) such that for all \(0\leq \epsilon \leq \epsilon_ 0\) system (*) has a solution which is \(C^{\infty}\) for all \(t\leq 0\), \(x\in {\mathbb{R}}^ n\).
scientific article; zbMATH DE number 4122487
scientific article