Geodesic
On a~class of quasilinear hyperbolic equations
Sbornik. Mathematics, Tome 25 (1975) no. 1, pp. 145-158

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In the bounded cylinder $Q=\Omega\times[0,T]$ with arbitrary fixed $T>0$ the mixed problem with Dirichlet boundary conditions is considered for the quasilinear hyperbolic equation $$ u_{tt}+(-1)^m\cdot a\biggl(\int_\Omega|\nabla^mu|^2\,dx\biggr)\cdot\Delta^mu=f. $$ A particular class of functions is introduced in which there is an existence and uniqueness theorem for solutions of this problem. A theorem on the unique solvability of the Cauchy problem for a certain nonlinear differential equation in Hilbert space is first proved. This problem is a very simple abstract analogue of the indicated mixed problem for the quasilinear hyperbolic equation. Bibliography: 2 titles. 
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     author = {S. I. Pokhozhaev},
     title = {On a~class of quasilinear hyperbolic equations},
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S. I. Pokhozhaev. On a~class of quasilinear hyperbolic equations. Sbornik. Mathematics, Tome 25 (1975) no. 1, pp. 145-158. http://geodesic.mathdoc.fr/item/SM_1975_25_1_a8/