7.2. Strip Problems for Strongly Parabolic Systems

In this section we consider parabolic systems

in the strip 0 ? x ?1, t ?0. At time t=0 we give initial data

As boundary conditions we require n linearly independent relations between the components of u and u _x at each boundary point x=0, x=1; i.e., the boundary conditions have the form

with constant n n matrices L _{j 0}, L _{j 1} . The n 2 n matrix ( L _{j 0}, L _{j 1}) has rank n for j=0 and j=1 since the boundary conditions are linearly independent. The matrix coefficients A, B, C in (7.2.1) are assumed to be C ^?-smooth.

Furthermore, we require that

for some ?> 0; i.e., the system (7.2.1) is symmetric parabolic. For the initial function f we assume that

All functions and matrices are taken as real, for simplicity. The constants c ₁, c ₂, etc. introduced below will depend on the time interval 0 ? t ? T, where T is arbitrary but fixed.

Extensions. The reader can generalize all arguments from the real to the complex case and assume, instead of (7.2.4),

This generalization (to a strongly parabolic system) essentially requires one to replace 2 (u, Au _xx ) by (Au _xx , u)+(u, Au _xx ) in the...