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**Higher integrability for parabolic systems of \(p\)-Laplacian type.**
*(English)*
Zbl 0994.35036

From the introduction: In this work, we study regularity of solutions to second-order parabolic systems:
\[
{\partial u_i\over \partial t}= \text{div} A_i(x,t, \nabla u)+B_i(x,t,\nabla u), \quad i=1,\dots,N. \tag{1}
\]
In particular, we are interested in systems of \(p\)-Laplacian type. The principal prototype that we have in mind is the \(p\)-parabolic system \({\partial u_i\over \partial t}=\text{div}(|\nabla u|^{p-2} \nabla u_i)\), \(i=1,\dots,N\), with \(1<p <\infty\). The purpose of this work is to obtain higher integrability results in the \(p\)-parabolic setting. We prove that the gradient of a weak solution to (1) satisfies a reverse Hölder inequality for \(p>2n/(n+2)\). One of the difficulties in proving our main result is that a solution does not remain a solution under multiplication by a constant that is neither 0 nor 1. Since reverse Hölder inequalities are invariant under multiplication by a constant, we have to choose a class of cylinders whose side lengths depend on the size of the function in order to obtain a reverse Hölder inequality and then higher integrability.

### MSC:

35D10 | Regularity of generalized solutions of PDE (MSC2000) |

35K55 | Nonlinear parabolic equations |

35K40 | Second-order parabolic systems |

### Keywords:

reverse Hölder inequality
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\textit{J. Kinnunen} and \textit{J. L. Lewis}, Duke Math. J. 102, No. 2, 253--271 (2000; Zbl 0994.35036)

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### References:

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