Electron. J. Differential Equations, Vol. 2017 (2017), No. 271, pp. 1-18.

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

Salih Jawad

Abstract:
This article concerns the nonlinear wave equation
$$ \eqalign{ & u_{tt}-\sum_{i,j=1}^{n}\frac{\partial}{\partial x_{i}} \big\{ a_{ij}(t,x)\frac{\partial u}{\partial x_{j}} \big\} + c(t,x)u +\lambda u \cr &+ \mathbf{F}'\bigl( | u |^2\bigr)u +\zeta u=0, \quad t\in [0,\infty),\;x \in \bar{\Omega}\cr u(0,x) =\varphi, \quad u_{t}(0,x)=\psi, \quad u|_{\partial \Omega}=0.\cr }$$
Essentially this article ascertains and proves the important mapping property
$$ \mathbf{M}: D\bigl(\mathbf{A}^{(k''_0+1)/2}(0)\bigr) \to D\bigl(\mathbf{A}^{k''_0/2}(0)\bigr), \quad D(\mathbf{A}(0)) = H_0^{1}(\Omega) \cap H^2(\Omega), $$
as well as the associated Lipschitz condition
$$\eqalign{ &\| \mathbf{A}^{k''_0/2}(0)(\mathbf{M} u -\mathbf{M} v) \|\cr &\leq k \Bigl( \| \mathbf{A}^{(k''_0+1)/2}(0)u \| + \| \mathbf{A}^{(k''_0+1)/2}(0)v \| \Bigr) \bigl \| \mathbf{A}^{(k''_0+1)/2}(0)(u-v) \|, }$$
where
$$ \eqalign{ \mathbf{A}(t) := - \sum_{i,j=1}^{n}\frac{\partial}{\partial x_{i}} \big\{ a_{ij}(t,x)\frac{\partial}{\partial x_{j}}\big\} + c(t,x) + \lambda,\cr \mathbf{M} u := \mathbf{F}'\bigl( | u |^2\bigr)u+\zeta u,\cr k'' \in \mathbb{N}, \quad k'' > \frac{n}{2}+1, \quad k''_0:= \min\{ k''\},\cr }$$
and $k(\cdot) \in C^{0}_{\rm loc}\bigl(\mathbb{R}^{+},\mathbb{R}^{++}\bigr)$ is monotonically increasing. Here are $\mathbb{R}^{+}=[0, \infty)$, $\mathbb{R}^{++}=(0,\infty \bigr)$. This mapping property is true for the dimensions $ n \leq 5 $. But we investigate only the case $ n=5 $ because the problem is already solved for $ n \leq 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 \leq 6 $, i.e. without boundary condition. In the case of the Cauchy problem there are no difficulties with regard to the mapping property.

Submitted October 19, 2016. Published October 31, 2017.
Math Subject Classifications: 35A01, 35A09, 35L05, 35L71, 34B15, 35L10, 35L20, 35L70, 58D25
Key Words: Initial-boundary value problem; hyperbolic equation; semilinear second-order; existence problem; classical solution.

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Salih Jawad
Elisenstrase 17
30451 Hannover, Germany
email: jassir83@yahoo.de

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