Abstract
The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional and is of Caputo-type, which takes into account “memory”. The precise model is
This paper poses the problem over {t ∈ R+, x ∈ Rn} with nonnegative initial data u(0, x) ≥ 0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x) have exponential decay at infinity is proved. The main result is Hölder continuity for such weak solutions.
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Dedicated to Haim Brezis on the occasion of his 70th birthday
This work was supported by NSG grant DMS-1303632, NSF grant DMS-1500871 and NSF grant DMS-1209420.
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Allen, M., Caffarelli, L. & Vasseur, A. Porous medium flow with both a fractional potential pressure and fractional time derivative. Chin. Ann. Math. Ser. B 38, 45–82 (2017). https://doi.org/10.1007/s11401-016-1063-4
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DOI: https://doi.org/10.1007/s11401-016-1063-4