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Abstract
We consider the wave equation ε2(∂t2+Δ)u+f(u)=0 for 0 < ε≪ 1 , where f is the derivative of a balanced, doublewell potential, the model case being f(u) = u u^{3}. For equations of this form, we construct solutions that exhibit an interface of thickness O(ε) that separates regions where the solution is O(ε^{k}) close to ± 1 , for k≥ 1 , and that is close to a timelike hypersurface of vanishing Minkowskian mean curvature. This provides a Minkowskian analog of the numerous results that connect the Euclidean Allen–Cahn equation and minimal surfaces or the parabolic Allen–Cahn equation and motion by mean curvature. Compared to earlier results of the same character, we develop a new constructive approach that applies to a larger class of nonlinearities and yields much more precise information about the solutions under consideration.
Original language  English 

Pages (fromto)  9711009 
Number of pages  39 
Journal  Communications in Mathematical Physics 
Volume  373 
Early online date  5 Dec 2019 
DOIs  
Publication status  Published  28 Feb 2020 
ASJC Scopus subject areas
 Statistical and Nonlinear Physics
 Mathematical Physics
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 1 Active

Concentration phenomena in nonlinear analysis
Engineering and Physical Sciences Research Council
27/04/20 → 31/03/24
Project: Research council