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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
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