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Blowup for systems of semilinear wave equations in two space dimensions. (English) Zbl 1146.35061

The authors investigate blowup for solutions of the following initial-boundary value problem for a system of semilinear wave equations in two space dimensions,
\[ \begin{aligned} &(\partial_{t}^{2}-c_{i}^2\triangle)u_{i} =F_{i}(u),\quad (x,t)\in {\mathbb R}^{2}\times [0,\infty),\\ &u_{i}(x,0)=\varepsilon \varphi_{i}(x),\quad \partial u_{i}(x,0)=\varepsilon \psi _{i}(x)\in {\mathbb R}^{2}, \end{aligned} \]
where \(i=1,\dots,m,\;c_{i}>0,\;u=(u_{1},\dots,u_{m})\) and \(\varepsilon>0 \) is a small parameter. The typical example of \(F_{i}\) is
\[ F_{i}=\sum_{1\leq j<k\leq m}A_{i}^{jk}|u_{j}|^{p_{ijk}} |u_{k}|^{\alpha -p_{ijk}}, \]
where \(A_{i}^{jk}\in {\mathbb R}\), \(\alpha \geq 2\) and \(1\leq p_{ijk}\leq \alpha -1\), or a further simpler one for \(m=2\) is
\[ F_{1}(u)=|u_{1}|^{p_{1}}|u_{2}|^{q_{1}}\quad \text{and}\quad F_{2}(u)=|u_{1}|^{p_{2}}|u_{2}|^{q_{2}}, \tag \(*\) \]
where \(\alpha := p_{1}+q_{1}= p_{2}+q_{2}\).
The authors say that small data global existence holds for any \(\varphi_{i}, \psi _{i}\) there exists a constant \(\varepsilon _{0}>0\) such that the Cauchy problem admits a global solution for any \(\varepsilon \in (0,\varepsilon _{0})\). They say that small data blowup occurs if small data global existence does not hold. Their main goal is to show that small data blowup occurs with some additional assumptions for \(F_{i}\) when \(n=2\) and \(2\leq \alpha \leq 3\) even if \(c_{1}\neq c_{2}\). On the other hand when \(n=3\) and \(m=2\) small data global existence holds with \((*)\) for any \(\alpha \geq 2\) if \(c_{1}\neq c_{2}\). The difference comes from the fact that Huygens’ principle holds or not.
A short history is put in the introduction of this paper, from which readers will be get useful knowledge on the blowup problem.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
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