摘要
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 isD_t~αu- div(u(-Δ)^(-σ)u) = f, 0 < σ <1/2.This paper poses the problem over {t ∈ R^+, x ∈ R^n} 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 H¨older continuity for such weak solutions.
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 isDt^αu- div（u（-Δ）^-σu） = f, 0 〈 σ 〈1/2.This paper poses the problem over {t ∈ R^＋, x ∈ R^n} 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 H¨older continuity for such weak solutions.
基金
supported by NSG grant DMS-1303632
NSF grant DMS-1500871,NSF grant DMS-1209420
