The damped nonlinear wave equation, also known as the nonlinear telegraph equation, is studied within the framework of semigroups and eigenfunction approximation. The linear semigroup assumes a central role: it is bounded on the domain of its generator for all time t > 0. This permits eigenfunction approximation within the semigroup framework as a tool for the study of weak solutions. The semigroup convolution formula, known to be rigorous on the generator domain, is extended to the interpretation of weak solution on an arbitrary time interval. A separate approximation theory can be developed by using the invariance of the semigroup on eigenspaces of the Laplacian as the system evolves. For (locally) bounded continuous L^2 forcing, this permits a natural derivation of a maximal solution, which can logically include a constraint on the solution as well. Operator forcing allows for the incorporation of concurrent physical processes. A significant feature of the proof in the nonlinear case is verification of successive approximation without standard fixed point analysis.

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多维阻尼波方程：通过半群和近似的非线性强迫的最大弱解

阻尼非线性波动方程，也称为非线性电报方程，是在半群和特征函数逼近的框架内研究的。线性半群扮演着核心角色：它在所有时间 t > 0 内都在其生成器的域上有界。这允许在半群框架内进行特征函数逼近，作为研究弱解的工具。已知在生成器域上严格的半群卷积公式被扩展到任意时间间隔上的弱解的解释。随着系统的演化，可以通过在拉普拉斯算子的特征空间上使用半群的不变性来开发一个单独的近似理论。对于（局部）有界连续 L^2 强迫，这允许最大解的自然推导，这在逻辑上也可以包含对解决方案的约束。运算符强制允许合并并发物理过程。非线性情况下证明的一个重要特征是在没有标准不动点分析的情况下验证逐次逼近。