## Classical-regular solvability of initial boundary value problems of nonlinear wave equations with time-dependent differential operator and Dirichlet boundary conditions

##### Date

2017-10-31

##### Authors

Jawad, Salih

##### Journal Title

##### Journal ISSN

##### Volume Title

##### Publisher

Texas State University, Department of Mathematics

##### 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.

##### Description

##### Keywords

Initial-boundary value problem, Hyperbolic equation, Semilinear second-order, Existence problem, Classical solution

##### 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.