Jawad, Salih2022-08-172022-08-172017-10-31Jawad, S. (2017). Classical-regular solvability of initial boundary value problems of nonlinear wave equations with time-dependent differential operator and Dirichlet boundary conditions. <i>Electronic Journal of Differential Equations, 2017</i>(271), pp. 1-18.1072-6691https://hdl.handle.net/10877/16072This article concerns the nonlinear wave equation utt - n∑i,j=1 ∂/∂xi {αij(t, x) ∂u/∂xj} + c(t, x)u + λu + F′(|u|2)u + ζu = 0, t ∈ [0, ∞), x ∈ Ω̅ u(0, x) = ϕ, ut(0, x) = ψ, u|∂Ω = 0. Essentially this article ascertains and proves the important mapping property M : D(A(k″0+1/2(0)) → D(Ak″0/2(0)), D(A(0)) = H1 0(Ω) ∩ H2(Ω), as well as the associated Lipschitz condition ∥Ak″0/2(0)(Mu - Mv)∥ ≤ k(∥A(k″0+1)/2(0)u∥ + ∥Ak″0+1)/2(0)v∥) ∥Ak″0+1)/2(0) (u - v)∥, where A(t) ≔ - n∑i,j=1 ∂/∂xi {αij(t, x) ∂/∂xj} + c(t, x) + λ, Mu ≔ F (|u|2)u + ζu, k″ ∈ ℕ, k″ > n/2 + 1, k″0 ≔ min{k″}, and k(⋅) ∈ C0 loc (ℝ⁺, ℝ⁺⁺) is monotonically increasing. Here are ℝ⁺ = [0, ∞), ℝ⁺⁺ = (0, ∞). This mapping property is true for the dimensions n ≤ 5. But we investigate only the case n = 5 because the problem is already solved for n ≤ 4, however, without the mapping property. With the proof of the mapping property and the associated Lipschitz condition, the problem becomes considerably comparable with a paper from von Wahl, who investigated the same problem as Cauchy problem and solved it for the dimensions n ≤ 6, i.e. without boundary condition. In the case of the Cauchy problem there are no difficulties with regard to the mapping property.Text18 pages1 file (.pdf)enAttribution 4.0 InternationalInitial-boundary value problemHyperbolic equationSemilinear second-orderExistence problemClassical solutionArticle