Free-Boundary Problem And Inverse Problem For A Singular Diffusion Equation With Convection

In this paper,we are concerned with a free-boundary problem and its inverse problem associated with the following singular diffusion equation with convection whereψ(0)=0,ψ’(s)>0,s≠0,(?)ψ(s)<+∞,(?)ψ(s)>-∞,B(s) satisfies some condition on the structure,μ(f)∈Lloc∞([0,+∞)),u satisfies some initial(boundary)value conditions.The free-boundary problem is employed to describe and analyze the solution and the evolution of its jump points.First we consider the following Cauchy problem with discontinuous initial value where f(0)=ρ>0,f(x)∈C1([0,+∞)).The entropy condition should hold along the jump line λ(t),thus we can formulate the following free-boundary problem The main body of this paper can be divided into two parts:(i) for any givencoefcient μ(t) of the convection term, solve the free-boundary problem associatedwith the singular difusion equation and analyze the relationship between the jumpline λ(t) and the coefcient μ(t);(ii) for any given curve λ(t), solve the inverse prob-lem of the free-boundary problem to determine the coefcient μ(t) of the convectionterm, such that λ(t) is the jump line of the corresponding free-boundary problem.Existence, uniqueness, and regularity results are obtained for the free-boundaryproblem associated with the foresaid Cauchy problem and some other initial (bound-ary) value problems. There are also existence and regularity results for the inverseproblem of the free-boundary problems.Both the free-boundary problem and its inverse problem associated with thesingular difusion equation with convection have been considered in this paper, whichshows the efect of the convection term on the jump line of the solution, as well asthe feasibility of controlling the evolution of the jump line by modulating the con-vection term.