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Mirjana STOJANOVIC 《数学物理学报(B辑英文版)》2013,33(6):1721-1735
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. 相似文献
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A fluid is injected from a slot into a stream of another fluid.In a simple model this leads to a two-phase two-free-boundaryproblem with the jump relation |u|2 |u+|2 = on the free boundary {u=0}, and |u| = 1 on the free boundary{u > Q}, where u is the stream function and Q isthe flux of the injected fluid. Using the variational theoryof Alt, Caffarelli & Friedman, we prove existence of (,1, u) such that there is a smooth fit for both free boundaries. 相似文献
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