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Accelerating fronts in semilinear wave equations

We study dynamics of interfaces in solutions of the equation $ε\Box u + \frac 1 εf_ε(u)=0$, for $f_ε$ of the form $f_ε(u) = (u^2-1)(2u- εκ)$, for $κ\in {\mathbb R}$, as well as more general, but qualitatively similar, nonlinearities. We consider equations of this form both in $(1+n)$-dimensional Minkowski space, $n\ge 1$, and on certain more general Lorentzian manifolds, and we prove that for suitable initial data, solutions exhibit interfaces that sweep out timelike hypersurfaces of mean curvature proportional to $κ$. In particular, in 1 dimension these interfaces behave like a relativistic point particle subject to constant acceleration.