In this paper we establish general well-posedeness results for a wide class of weakly parabolic 2 × 2 systems in a bounded domain of RN . Our results cover examples arising in sulphation of marbles and chemotaxis, when the density of one chemical component is not diffusing. We show that, under quite general assumptions, uniform L∞ estimates are sufficient to establish the global existence and stability of solutions, even if in general the nonlinear terms in the equations depend also on the gradient of the solutions. Applications are presented and discussed.
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