Higher integrability for parabolic systems of p-Laplacian type

Higher integrability for parabolic systems with non-standard growth and degenerate diffusions

“…Therefore our result ensures that weak solutions of (1.1) belong to a slightly higher Sobolev space than the natural energy space determined uniquely by the growth of the vector-field a and therefore obey a certain self-improving property of integrability. This is the p(z)-analogue of the higher integrability result of Kinunnen and Lewis [29] concerning parabolic systems with a standard p-growth condition. Moreover, it generalizes the result of Antontsev and Zhikov [6] for homogeneous p(z)-Laplacean equations to the case of general systems considered in (1.1).…”

“…It is a well know fact that the first step towards a higher integrability result for solutions to elliptic and parabolic systems is a suitable Caccioppoli-type inequality (see for example [29], [30], [7], [8]). Such inequalities when combined with Sobolev-type estimates yield a reverse Hölder-type inequality on intrinsic parabolic cylinders.…”

“…Note that this case is much easier since it falls into the realm of non-degenerate problems which can be treated without using the intrinsic geometry. The case 2n n+2 < p = 2 goes back to the work of Kinnunen and Lewis [29], and for a global version we refer to [38]. The restriction on the growth exponent from below is unavoidable and already occurs in the parabolic regularity theory for p-Laplacean systems and equations; see [14].…”

Self-improving property of nonlinear higher order parabolic systems near the boundary

“…Nevertheless, Giaquinta and Struwe proved a first parabolic analogue for systems with linear growth in [22]. Higher integrability for more general parabolic systems with non-linear growth conditions remained open for some time: The first positive result for degenerate and singular second order parabolic p-growth systems was obtained by Kinnunen and Lewis [26]. The proof employs the method of intrinsic scaling with respect to the gradient of the solution.…”

“…Our basic strategy follows the guidelines of the local result in [26]. Indeed, we first derive a reverse Hölder inequality on intrinsic cylinders up to the boundary and then use a covering argument to extend the estimates to the whole space-time cylinder.…”