Global singularity structure of weak solutions to 3-D semilinear dispersive wave equations with discontinuous initial data.
The commutator technique introduced in \textit{J.-M. Bony's} paper [Taniguchi Symposium, Katata, 11--49 (1984; Zbl 0669.35073)] and Strichartz' inequalities are used to describe the singularity set for the solutions of the three dimensional Cauchy problem for the semilinear wave equation \(\partial_{t}^{2}{u}-\Delta_{x}u+g(u)=0\) with discontinuous Cauchy data \((0, u_{1}(x))\), \(u_{1}\in{C^{\infty}(\overline{B(0,1)})}\) and \(u_{1}(x)=0\) for \(|{x}|>1\). It is assumed that \(g(0)=0\), \(\int^{u}_{0}{g(s)\,ds}\geq0\), \(|{g^{(j)}(u)}|\leq{C_{j}(1+|{u}|)^{p-j}}\), \(1<p\leq5\). One proves that the weak solution is smooth outside \(\Sigma_{1}\cup\Sigma_{2}\), where \(\Sigma_{1,2}=\{(t,x)\); \(t\geq{0}\), \((t\pm1)^{2}=|{x}|^{2}\}\).
scientific article; zbMATH DE number 2033004
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