Local solutions to Darboux problem with a discontinuous right-hand side

Abstract

The existence of a local solution to the Darboux problem uxy (x, y) = g (u (x, y)), u (x, 0) = u (0, y) = 0, where g is Lebesgue measurable and has at most polynomial growth, is proved.Доведено iснування локального розв’язку проблеми Дарбу uxy (x, y) = g (u (x, y)), u (x, 0) = u (0, y) = 0, де g є вимiрна за Лебегом функцiя, що росте не швидше, нiж полiном.