Initial Value Problems of Parabolic Type

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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.

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 ), and the corresponding solution is - 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.

 
• The heat equation -- Fourier analysis and the energy method
• Parabolic systems