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Dynamic theory of quasilinear parabolic equations. II: Reaction-diffusion systems. (English) Zbl 0729.35062
[For part I, see Nonlinear Anal., Theory Methods Appl. 12, No.9, 895-919 (1988; Zbl 0666.35043).]
The author considers the initial boundary value problem of the following reaction-diffusion system $\partial_ tu+A(u)u=f(.,u,\partial u)\text{ in } \Omega \times (0,\infty),\quad B(u)u=0\text{ on } \partial \Omega \times (0,\infty),\quad u(,0)=u_ 0\text{ on } \Omega.$ He proves that the mapping $$(t,u_ 0)\to u(t,u_ 0)$$ is smooth, bounded and relatively compact in a suitable subspace of Sobolev space under assumption that (A(u),B(u)) is normally elliptic. Applications to concrete examples are also given. The methods used here are those of evolution equations in semigroup theory. [For part III see Math. Z. 202, No.2, 219-250 (1989; Zbl 0702.35125)].

##### MSC:
 35K57 Reaction-diffusion equations 35K55 Nonlinear parabolic equations 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations
##### Keywords:
quasilinear parabolic equations; normally elliptic