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It is well-known that small, regular, spherically symmetric characteristicinitial data to the Einstein-scalar-field system which are decaying towards(future null) infinity give rise to solutions which are foward-in-time global(in the sense of future causal geodesic completeness). We construct a class ofspherically symmetric solutions which are global but the initial norms areconsistent with initial data not decaying towards infinity. This gives thefollowing consequences: (1) We prove that there exist foward-in-time globalsolutions with arbitrarily large (and in fact infinite) initial boundedvariation (BV) norms and initial Bondi masses. (2) While general solutions withnon-decaying data do not approach Minkowski spacetime, we show using theresults of Luk--Oh that if a sufficiently strong asymptotic flatness conditionis imposed on the initial data, then the solutions we construct (with large BVnorms) approach Minkowski spacetime with a sharp inverse polynomial rate. (3)Our construction can be easily extended so that data are posed at past nullinfinity and we obtain solutions with large BV norms which are causallygeodesically complete both to the past and to the future. Finally, we discussapplications of our method to construct global solutions for other nonlinearwave equations with infinite critical norms.

Solutions to the Einstein-scalar-field system in spherical symmetry with large bounded variation norms