Nonlinear wave equations of carrier type on manifolds

Abstract

In this paper, we investigate the existence and uniqueness of a solution for differential equations of the carrier type on lateral boundary † of the cylinder Q, cf. (1). The main point is to transform this initial value problem into a differential operator equation of the type u 00 C M ÂZ cf. (6). The operator A is defined in Section 2, and it acts in Sobolev spaces on , boundary of . The initial value problem (6) is investigated in Section 3 by the method of Faedo-Galerkin. Thus, we obtain the existence of a weak solution for (6), and in Section 4, we prove its uniqueness.Ã Au Cˇu 0ˇ u 0 D f , and we can apply the usual methodology for the initial value problem (6).

On nonlinear wave equations of Carrier type

“…The above mixed problem (*) was investigated from mathematical point of view in [2], [8], [9] among others. We fixe our attention on the results contained in [2], [9] with the nonlinearity δ ∂u ∂t ∂u ∂t ρ with the restriction ρ > 1.…”

“…We fixe our attention on the results contained in [2], [9] with the nonlinearity δ ∂u ∂t ∂u ∂t ρ with the restriction ρ > 1. For a motivation of this type of nonlinearity look [10] p.38, when M (λ) = 1.…”

“…The authors in [8] analised also the case ρ = 1. The main difficulty in [2], [8] is because the proofs depends of Young-Hölder inequality which is true only for ρ > 1.…”