Geodesic
On the convergence of the formal Fourier solution of the wave equation with a summable potential
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 10, pp. 1795-1809

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The convergence of the formal Fourier solution to a mixed problem for the wave equation with a summable potential is analyzed under weaker assumptions imposed on the initial position $u(x,0)=\varphi(x)$ than those required for a classical solution up to the case $\varphi(x)\in L_p[0,1]$ for $p>1$. It is shown that the formal solution series always converges and represents a weak solution of the mixed problem. 
@article{ZVMMF_2016_56_10_a10,
     author = {A. P. Khromov},
     title = {On the convergence of the formal {Fourier} solution of the wave equation with a summable potential},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1795--1809},
     publisher = {mathdoc},
     volume = {56},
     number = {10},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_10_a10/}
}
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A. P. Khromov. On the convergence of the formal Fourier solution of the wave equation with a summable potential. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 10, pp. 1795-1809. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_10_a10/