REGULARITY OF SOLUTIONS TO NONLINEAR TIME FRACTIONAL DIFFERENTIAL EQUATION

We find an upper viscosity solution and give a proof of the existence-uniqueness in the space C^∞（t∈（0,∞）;H2^s＋2（R^n））∩C^0（t∈[0,∞）;H^s（R^n））,s∈R,to the nonlinear time fractional equation of distributed order with spatial Laplace operator subject to the Cauchy conditions ∫0^2p（β）D＊^βu（x,t）dβ=△xu（x,t）＋f（t,u（t,x））,t≥0,x∈R^n,u（0,x）=φ（x）,ut（0,x）=ψ（x）,（0.1） where △xis the spatial Laplace operator,D＊^β is the operator of fractional differentiation in the Caputo sense and the force term F satisfies the Assumption 1 on the regularity and growth. For the weight function we take a positive-linear combination of delta distributions concentrated at points of interval （0, 2）, i.e., p（β） =m∑k=1bkδ（β-βk）,0〈βk〈2,bk〉0,k=1,2,…,m.The regularity of the solution is established in the framework of the space C^∞（t∈（0,∞）;C^∞（R^n））∩C^0（t∈[0,∞）;C^∞（R^n））when the initial data belong to the Sobolev space H2^8（R^n）,s∈R.     

EXISTENCE OF MINIMISERS FOR A CLASS OF FREE DISCONTINUITY PROBLEMS IN THE HEISENBERG GROUP Hn

The purpose of this paper is to prove existence of minimisers of the functional J(K,u):=∫Ω\K f(Lu)dx α∫Ω\K |u - g|qdx βSQ-1d(K∩Ω),where Ω is an open set of the Heisenberg group Hn, K runs over all closed sets of Hn, u varies in C1H(Ω\K), α,β＞0,q ≥ 1,g ∈ Lq(Ω) ∩L∞(Ω) and f: R2n → R is a convex function satisfying some structure conditions (H1)(H2)(H3) (see below).