Pukach P. Ya.
Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables
Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1523–1531
We investigate the first mixed problem for a quasilinear hyperbolic equation of the second order with power nonlinearity in a domain unbounded with respect to space variables. We consider the case of an arbitrary number of space variables. We obtain conditions for the existence and uniqueness of the solution of this problem independent of the behavior of solution as $|x| \rightarrow +\infty$. The indicated classes of the existence and uniqueness are defined as spaces of local integrable functions. The dimension of the domain in no way limits the order of nonlinearity.
Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 454–456
We indicate the classes for which a solution of the problem for a certain nonlinear degenerating parabolic system without initial conditions exists and is unique. The uniqueness conditions are established both in the case where restrictions are imposed on the behavior of solutions as $t \rightarrow -\infty$ and in the case where they are not imposed. The existence is proved for the arbitrary behavior of the right-hand side of the system as $t \rightarrow -\infty$.