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5.7 Well-posed problems

Well-posed problems consist of a PDE in a domain together with a set of initial and/or
boundary conditions (or other auxiliary conditions) that enjoy the fol-lowing fundamental
properties:

(i) Existence: There exists at least one solution u(x, t) satisfying all these conditions.

(ii) Uniqueness: There is at most one solution.

(iii) Stability: The unique solution u(x, t) depends in a stable manner on the data of
the problem. This means that if the data are changed a little, the corresponding
solution changes only a little.

For a physical problem modelled by a PDE, the scientist normally tries to formulate
physically realistic auxiliary conditions which all together make a well-posed problem.
The mathematician tries to prove that a given problem is or is not well-posed. If too
few auxiliary conditions are imposed, then there maybe more than one solution (non-
uniqueness) and the problem is called underdetermined. If, on the other hand, there are
too many auxiliary conditions, there may be no solution at all (nonexistence) and the
problem is called overdetermined.
The stability property (iii) is normally required in models of physical problems. This is
because you could never measure the data with mathematical precision but only up to some
number of decimal places. You cannot distinguish a set of data from a tiny perturbation
of it. The solution ought not be significantly affected by such tiny perturbations, so it
should change very little.
Let us take an example. We know that a vibrating string with an external force, whose
ends are moved in a specified way, satisfies the problem

Tu tt − ρu xx = f(x, t),
u(x, 0) = φ(x), u t (x, 0) = ψ(x), (5.9)
u(0, t) = g(t), u(L, t) = h(t)

for 0 < x < L. The data for this problem consist of the five functions f(x, t), φ(x),
ψ(x), g(t), and h(t). Existence and uniqueness would mean that there is exactly one
solution u(x, t) for arbitrary (differentiable) functions f, φ, ψ, g, h. Stability would mean
that if any of these five functions are perturbed, then u is also changed only slightly. To
make this precise requires a definition of the ”nearness” of functions. Mathematically, this
requires the concept of a ”distance”, ”metric”, ”norm”, or ”topology” in function space.
The problem (5.9) is indeed well-posed if we make the appropriate choice of ”nearness”.
As a second example, consider the diffusion equation. Given an initial condition
u(x, 0) = f(x), we expect a unique solution, in fact, well-posedness, for t > 0. But
consider the backwards problem! Given f(x), find u(x, t) for t < 0. What past behavior
could have led up to the concentration f(x) at time 0? Any chemist knows that diffusion
is a smoothing process since the concentration of a substance tends to flatten out. Go-
ing backward (”antidiffusion’), the situation becomes more and more chaotic. Therefore,

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