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Initial Value Problems of Parabolic Type


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 tex2html_wrap_inline11001-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 , tex2html_wrap_inline11079, as functions of the high wavenumbers, tex2html_wrap_inline11081), which is the essential feature of parabolicity.

As before, we study the manner in which the solution depends on its initial data.

  1. : the principal of superposition holds.
  2. : the solution is uniquely determined for t > 0 by the explicit formula
  3. for large enough set of admissible initial data: bounded initial data f(x) can be prescribed (and even f's with tex2html_wrap_inline11089), and the corresponding solution is tex2html_wrap_inline11091 - in fact u(x,t > 0) is analytic because of exponential decay in Fourier space.
  4. : follows directly from the representation of u(x,t) as a convolution of f(x) with the unit mass positive kernel Q(z).
  5. : 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