On the solvability of one multidimensional version of the first Darboux problem for some nonlinear wave equations

One multidimensional version of the Darboux first problem for one class of semilinear second order hyperbolic systems

“…In regard to the multidimensional problem (1.1), (1.2) for scalar case, i.e. when N = 1, in the case of nonlinearity of the form f (u) = λ|u| p u, as it is shown in the paper [18], depending on the sign of the parameter λ and the values of power exponent p, in some cases the problem (1.1), (1.2) is globally solvable, while in other cases it is not globally solvable. [19].…”

“…Indeed, after scalar multiplication of the both parts of the vector equation (3.1) by 2 ∂u ∂t and integration in the domain D τ , 0 < τ ≤ T , and simple transformations by help of the equalities (3.2) and integration by parts we receive the equality [18], [24, p. 116…”

On the Solution of NBVP for Multidimensional Hyperbolic Equations

“…In paper [ 15 ], the existence or nonexistence of global solutions of a multidimensional version of the first Darboux problem for wave equations with power nonlinearity in the conic domain was investigated.…”

The first Darboux problem for wave equations with nonlinear dissipative term

“…Note that on questions of existence, uniqueness and blow-up of solutions of initial, mixed, nonlocal and other problems posed for nonlinear hyperbolic type equations there are devoted a number of papers (see, e.g. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]). In linear case, i. e. when g(x, t, u) = g(x, t), problem (1.1), (1.2), as it is known, is wellposed and a global solvability takes place in corresponding function spaces Multiplying the both sides of equality (2.5) by u nt and integrating the received in domain D τ := (x, t) ∈ D T : 0 < t < τ , 0 < τ ≤ T we have…”