It is shown that general quasilinear parabolic systems possess unique maximal classical solutions for sufficiently smooth initial values, provided the boundary conditions are âtime-independentâ. Moreover it is shown that, in the autonomous case, these equations generate local semiflows on appropriate Sobolev spaces. Our results apply, in particular, to the case of prescribed boundary values (Dirichlet boundary conditions).
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Herbert Amann (1986) studied this question.