The Multidimensional Damped Wave Equation:
Maximal Weak Solutions for Nonlinear Forcing via Semigroups and
Approximation
By: Joseph W. Jerome
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, there is 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.
This paper has appeared
in the Journal of Numerical Functional Analysis and Optimization: vol. 41
(2020), 1970-1989.
https://doi.org/10.1080/01630563.2020.1813759.