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Hyperbolic and parabolic equations are the two most important categories of
time-dependent problems whose evolution process is well-posed. Thus, consider
the initial value problem

We assume that a large enough class of admissible initial data

there exists a unique solution, *u*(*x*,*t*). This defines a solution operator,
which describes the evolution of the problem

Hoping to compute such solutions, we need that the solutions will depend
continuously in their initial data, i.e.,

In view of linearity, this amounts to having the a priori estimate
(boundedness)

which includes the hyperbolic and parabolic cases.

__: (Hadamard) By Cauchy-Kowalewski, the system
has a unique solution for arbitrary __

__
__

__
Finally, we note that a well-posed problem is stable against
perturbations of inhomogeneous data in view of the following
__

__
__

__
. The solution of the inhomogeneous problem
__

is given by

__
Indeed, a straightforward substitution yields
This implies the a priori stability estimate
as asserted.
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Thu Jan 22 19:07:34 PST 1998