Classical-regular solvability of initial boundary value problems of nonlinear wave equations with time-dependent differential operator and Dirichlet boundary conditions
| dc.contributor.author | Jawad, Salih | |
| dc.date.accessioned | 2022-08-17T17:48:37Z | |
| dc.date.available | 2022-08-17T17:48:37Z | |
| dc.date.issued | 2017-10-31 | |
| dc.description.abstract | This 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. | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 18 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.citation | Jawad, 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. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/16072 | |
| dc.language.iso | en | |
| dc.publisher | Texas State University, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2017, San Marcos, Texas: Texas State University and University of North Texas. | |
| dc.subject | Initial-boundary value problem | |
| dc.subject | Hyperbolic equation | |
| dc.subject | Semilinear second-order | |
| dc.subject | Existence problem | |
| dc.subject | Classical solution | |
| dc.title | Classical-regular solvability of initial boundary value problems of nonlinear wave equations with time-dependent differential operator and Dirichlet boundary conditions | |
| dc.type | Article |