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Hölder regularity for singular parabolic systems of \(p\)-Laplacian type. (English) Zbl 1327.35228

The paper under review deals with parabolic systems of \(p\)-Laplacian type \[ \partial_t u^i-\text{div\,}(|Du|^{p-2}Du^i)= \text{div\,}(|g|^{p-2}g^i)+F ^i(t,x,Du),\;i=1,\dots,n, \] in the cylinder \(Q=(0,T)\times\Omega,\) where \(\Omega\subset \mathbb{R}^m,\) \(m\geq2,\) is a bounded domain. Under suitable integrability and growth conditions on the data \(g^i\) and \(F^i,\) the authors prove local Hölder continuity of weak solutions in the singular case \[ \frac{2m}{m+2}<p<2. \] The proofs are based on the Campanato direct approach combined with standard intrinsic scaling to the evolutionary \(p\)-Laplace operator.

MSC:

35K92 Quasilinear parabolic equations with \(p\)-Laplacian
35D30 Weak solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35K67 Singular parabolic equations
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Full Text: Euclid