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for all φ ∈ C c (Q,R). Here, we write dz = dxdt and W 1,0 2 (Q,R ) is the set of vector-valued functions whose spatial weak derivatives are in L(Q). It is now a classical result (see [10, 16]) for the scalar case (m = 1) that bounded weak solutions to (1.1) are Hölder continuous and if A,F are sufficiently smooth, then bounded weak solutions are classical. On the other hand, in the case of systems of equations (i.e. m > 1), one cannot generally expect bounded weak solutions of (1.1) to be Hölder continuous everywhere (see counterexamples, e.g., in [5]). Partial regularity theory for (1.1), when A is linear in Du, was considered in [2, 3], where it was shown that bounded weak solutions are Hölder continuous on a full measure subset of Q and the Hausdörff dimension of their singular sets could be estimated. In a recent work, see [11, 14], we introduced the so-called linear A-heat approximation technique to investigate partial and everywhere regularity for degenerate parabolic systems. However, as pointed out in [1], it is crucial to establish everywhere Hölder continuity for bounded weak solutions for systems and to control their Hölder norms in order to discuss further important questions in applications (e.g.; see [18]) such as global existence and dynamics of the solutions. Works in these directions were reported in [6, 7, 8, 12, 17, 20] for systems satisfying

Regularity of BMO weak solutions to nonlinear parabolic systems via homotopy