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The heat equation,
is the prototype for PDE's of parabolic type. We study the pure initial-value
problem associated with (para.1), augmented with -periodic boundary
conditions and subject to initial conditions
We can solve this equation using Fourier transform which yields
It reflects the dissipative effect
(= the rapid decay of the amplitudes
, , as functions of the high wavenumbers, ),
which is the
essential feature of parabolicity.
As before, we study the manner in which the solution depends on its
initial data.
- : the principal of superposition holds.
- : the solution is uniquely determined for t > 0
by the explicit
formula
- for large enough set of admissible initial data:
bounded initial data f(x) can be prescribed
(and even f's with ), and the corresponding
solution is
- in fact
u(x,t > 0) is analytic because of exponential decay in Fourier space.
- : follows directly from the
representation of u(x,t) as a convolution of f(x) with
the unit mass positive kernel Q(z).
- : as in the hyperbolic case
we may proceed in one of two ways: Fourier analysis and the energy method.
Eitan Tadmor
Thu Jan 22 19:07:34 PST 1998