  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.
