Geodesic
On global behavior of solutions to an inverse problem for semi-linear hyperbolic equations
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 36, Tome 318 (2004), pp. 120-134

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This paper is concerned with global in time behavior of solutions for a semi-linear hyperbolic inverse source problem. We prove two types of results, the first is a global nonexistence result for smooth solutions when the data is chosen appropriately. The second type of result is on asymptotic stability of solutions when the integral constraint vanishes as $t$ goes to infinity. 
@article{ZNSL_2004_318_a6,
     author = {A. Eden and V. K. Kalantarov},
     title = {On global behavior of solutions to an inverse problem for semi-linear hyperbolic equations},
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     publisher = {mathdoc},
     volume = {318},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_318_a6/}
}
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A. Eden; V. K. Kalantarov. On global behavior of solutions to an inverse problem for semi-linear hyperbolic equations. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 36, Tome 318 (2004), pp. 120-134. http://geodesic.mathdoc.fr/item/ZNSL_2004_318_a6/