We establish higher integrability up to the boundary for the gradient of solutions to porous medium type systems, whose model case is given bywhere m > 1. More precisely, we prove that under suitable assumptions the spatial gradient D(|u| m−1 u) of any weak solution is integrable to a larger power than the natural power 2. Our analysis includes both the case of the lateral boundary and the initial boundary.
We consider evolutionary problems associated with a convex integrand f : T × R N n → [0, ∞), which is α-Hölder continuous with respect to the x-variable and satisfies a non-standard p, q-growth condition. We prove the existence of weak solutions u : T → R N , which solve ∂ t u − div ∂ ζ f (x, t, Du) = 0 weakly in T. Therefore, we use the concept of variational solutions, which exist under a mild assumption on the gap q − p, namely 2n n + 2 < p ≤ q < p + 1. For 2n n + 2 < p ≤ q < p + min{2, p} α n + 2 , we prove that the spatial derivative Du of a variational solution u admits a higher integrability and is accordingly a weak solution.
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