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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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