Qualitative Properties Of Models About Chemotaxis And Executive Stock Options

A chemotaxis model describes the movement of cells or some species reacting to the density of a chemical substance. The executive stock options model is looking for the best exercise strategy for managers. These two models help us understanding the phenomenon and making reasonable decisions. Hence they are important research topics.The qualitative properties of these two models are investigated in this paper.Firstly, one-dimensional chemotaxis model with volume-filling effect is developed.???????????????τut=(ux- k f(u)vx)x, x ∈(0, ?), t > 0vt= vxx- v + g(u), x ∈(0, ?), t > 0ux= vx= 0, x = 0, ?, t > 0∫?0u dx = m, t 0Properties of the steady state solution are studied as the chemotaxis parameter k is large enough. Transform the differential equations into the equivalent algebra system. Existence and uniqueness of the steady state solution are established by the implicit function theorem and the contraction mapping principle. The limit and the property of decaying exponentially of the solution are given. And stability of the solution is shown by the negative upper bound. Especially, the specific expressions of eigenpairs are given when the chemotaxis parameter k is su?ciently large and the time relaxation parameter τ is zero. In the end, asymptotic expansions of the steady state solution and the solution to the algebra system are shown by the method of matching in the interfacial region.Secondly, the derivative eigenvalue problem arising from the chemotaxis model is considered.???????-(pwx)x= μpw, x ∈(0, ?)wx= 0, x = 0, ?Divide the interval into the sub-intervals according to the limit of the principal eigenvalue to the adjoint operator problem with the homogeneous Neumann boundary condition.Perpendicular functions mutually are obtained by modifying the eigenfunctions to the sub-problems. The first few eigenvalues are calculated by the eigenfunctions with the Hermite polynomials. Taking the special base functions and using the variational char-acter of eigenvalues, the first four eigenvalues of the derivative eigenvalue problem are taken. Split the operation of the original system into several linear operators. Compared the time relaxation parameter with the second eigenvalue of derivation of the chemotaxis model, the principal eigenvalue of the original eigenvalue problem is estimated by the operator properties.Finally, the perpetual executive stock options model is discussed.???????min{Raφa- φzz- νφz+ φ2z, φa- g+}= 0,(z, a) ∈ R ×(0, ∞)φ(z, 0) = 0, z ∈ R Change the system into the regular obstacle problem. The variational inequality including the degenerate parabolic operator is written as the parabolic equation by the reformation of the penalty method. Existence and uniqueness of the classical solutions to the approximated problems are obtained by Schauder fixed point theorem, Gronwall inequality and upper and lower solutions. Also, the solutions decay exponentially. By the maximal principle and Hopf lemma, a prior estimations are shown. Existence of the solution to the perpetual executive stock options model is provided via Arzela-Ascoli theorem and the imbedding theorem. Continuity and monotonicity of the free boundary and existence and uniqueness of the classical solution are given. Asymptotic behaviors of the solution and the free boundary are supplied.