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Regularity for the wave equation with a critical nonlinearity. (English) Zbl 0785.35065
Consider a nonlinear wave equation with a self-interacting term of the form \[ \square\varphi+f(\varphi)=0,\;(t,x)\in\mathbb{R}\times\mathbb{R}^ n \tag{1} \] where \(\square=\partial_ t\partial^ t-\partial_ j\partial^ j\) is the D’Alembertian in \(n+1\) dimensions with respect to the Minkowski metric. In order to be able to solve (1), initial data have to be prescribed over some space-like hypersurface, for example at \(t=t_ 0\) \(\varphi(t_ 0,x)=\varphi_ 0(x)\), \(\varphi_{,t}(t_ 0,x)=\varphi_ 1(x)\), \(x\in R^ n\). In the present work it is assumed that the nonlinear potential is repulsive, i.e., \[ F(\varphi):=\int^ \varphi_ 0f(s)ds\geq 0, \tag{2} \] and that it grows like the critical Sobolev exponent, i.e., there exists some positive constant \(C\) such that \[ F(\varphi):=|\varphi|^{{2n\over n-2}}\quad\text{for }|\varphi|>C. \tag{3} \] In order to avoid some technicalities well known from the linear theory, the author assumes for reasons of simplicity that the initial data \(\varphi_ 0\) and \(\varphi_ 1\) as well as \(f\) are \(C^ \infty\). It is the purpose of this work to show that the solution of (1) is in fact \(C^ \infty\), provided that \(3\leq n\leq 5\). Conditions (2), (3), and the smoothness of \(f\) and the initial data can be relaxed.

MSC:
35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35B65 Smoothness and regularity of solutions to PDEs
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