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Initial Boundary Value Problem of an Equation from Mathematical Finance

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Chinese Annals of Mathematics,Series B 2016年第3期37卷 465-482页

作者： huashui zhan School of Applied Mathematics Xiamen University of Technology

Consider the initial boundary value problem of the strong degenerate parabolic equation ?_(xx)u + u?_yu-?_tu = f(x, y, t, u),(x, y, t) ∈ Q_T = Ω×(0, T)with a homogeneous boundary condition. By introducing a new kind of entropy solution, according to Oleinik rules, the partial boundary condition is given to assure the well-posedness of the problem. By the parabolic regularization method, the uniform estimate of the gradient is obtained, and by using Kolmogoroff 's theorem, the solvability of the equation is obtained in BV(Q_T) sense. The stability of the solutions is obtained by Kruzkov's double variables method.

关键词： Mathematical finance Oleinik rules Partial boundary condition Entropy solution Kruzkov's double variables method

Solution to Nonlinear Parabolic Equations Related to P-Laplacian

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Chinese Annals of Mathematics,Series B 2012年第5期33卷 767-782页

作者： huashui zhan School of Sciences Jimei University Xiamen 361021 Fujian China. E-mail: hszhan@***

Consider the following Cauchy problem：where 1 〈 p 〈 2, 1 〈 m 〈 p_~11, and # is a a-finite measure in N. By the Moser＇s iteration method, the existence of the weak solution is obtained, provided that （M＋1）N 〈 P. In mN＋l contrast, if 〉 p, there is no solution to the Cauchy problem with an initial value δ（X）, where 5（x） is the classical Dirac function.

关键词： Nonlinear parabolic equation Cauchy problem Existence a-Finite measure

On the Well-Posedness Problem of the Anisotropic Porous Medium Equation with a Variable Diffusion Coefficient

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Journal of Partial Differential Equations 2024年第2期37卷 135-149页

作者： zhan huashui School of Mathematics and Statistics Xiamen University of TechnologyXiamen 361024China

The initial-boundary value problem of an anisotropic porous medium equation■is *** with the usual porous medium equation,there are two different characteristics in this *** lies in its anisotropic property,another one is that there is a nonnegative variable diffusion coefficient a(x,t)*** a(x,t)may be degenerate on the parabolic boundary∂Ω×(0,T),instead of the boundedness of the gradient|∇u|for the usual porous medium,we can only show that∇u∈L^(∞)(0,T;L^(2)_(loc)(Ω)).Based on this property,the partial boundary value conditions matching up with the anisotropic porous medium equation are discovered and two stability theorems of weak solutions can be proved naturally.

关键词： Anisotropic porous medium equation variable diffusion coefficient stability partial boundary condition

On a Nonlinear Heat Equation with Degeneracy on the Boundary

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Journal of Partial Differential Equations 2019年第1期32卷 20-32页

作者： zhan huashui School of Applied Mathematics Xiamen University of Technology Xiamen 361024 China.

The paper studies the stability of weak solutions of a nonlinear heat equation with degenerate on the boundary. A new kind of weak solutions are introduced. By the new weak solution, the stability of weak solutions is... 详细信息

关键词： Nonlinear heat equation weak solution stability, initial value.

Definite Condition of the Evolutionary p(x)−Laplacian Equation

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Analysis in Theory and Applications 2022年第3期38卷 297-321页

作者： huashui zhan Zhaosheng Feng School of Applied Mathematics Xiamen University of TechnologyXiamenFujian 361024China School of Mathematical and Statistical Sciences University of Texas Rio Grande ValleyEdinburgTX 78539USA

For the nonlinear degenerate parabolic equations,how to find an appropriate boundary value condition to ensure the well-posedness of weak solution has been an interesting and challenging *** this paper,we develop the general characteristic function method to study the stability of weak solutions based on a partial boundary value condition.

关键词： Definite condition stability general characteristic function method weak solution Laplacian equation

Parabolic System Related to the P-Laplician with Degeneracy on the Boundary

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Journal of Partial Differential Equations 2019年第3期32卷 281-292页

作者： OU Qnitdong zhan huashui School of Applied Mathematics Xiamen University of TechnologyXiamen 361024China

In this article,we study the system w让h boundary degeneracy wit-div(a(x)|s■ui|Pi^-2■ui)=fi(x,t,u1,u2),(x,t)∈*** the monotone iterattion technique and the regularization method,we get the existence of solution for a regularized ***,under an integral condition on the coefficient function a(x),the existence and the uniqueness of the local solutions of the system is obtained by using a standard limiting ***,the stability of the solutions is proved without any boundary value cond让ion,provided a(x)satisfies another restriction.

关键词： Weak solution boundary degeneracy parabolic system initial boundary value problem existence,stability

Local Solutions to a Class of Parabolic System Related to the P-Laplacian

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Advances in Pure Mathematics 2016年第12期6卷 868-877页

作者： Qitong Ou huashui zhan School of Applied Mathematics Xiamen University of Technology Xiamen China

In this paper, the existence and uniqueness of local solutions to the initial and boundary value problem of a class of parabolic system related to the p-Laplacian are studied. The regularization method is used to construct a sequence of approximation solutions, with the help of monotone iteration technique, then we get the existence of solution of a regularized system. By the use of a standard limiting process, the existence of the local solutions of the system is obtained. Finally, the uniqueness of the solution is also proven.

关键词： Existence Uniqueness Evolution P-Laplacian Parabolic System

The Harmonic Functions on a Complete Asymptotic Flat Riemannian Manifold

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Advances in Pure Mathematics 2011年第2期1卷 5-8页

作者： huashui zhan 不详

Let be a simply connected complete Riemannian manifold with dimension n≥3 . Suppose that the sectional curvature satisfies , where p is distance function from a base point of M, a, b are constants and . Then there ex... 详细信息

关键词： Harmonic Function Riemannian Manifold Negative Sectional Curvature

The Manifolds with Ricci Curvature Decay to Zero

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Advances in Pure Mathematics 2012年第1期2卷 36-38页

作者： huashui zhan 不详

The paper quotes the concept of Ricci curvature decay to zero. Base on this new concept, by modifying the proof of the canonical Cheeger-Gromoll Splitting Theorem, the paper proves that for a complete non-compact Riemannian manifold M with Ricci curvature decay to zero, if there is a line in M, then the isometrically splitting M = R × N is true.

关键词： Cheeger-Gromoll Theorem Busemann Function Complete Riemannian Manifold Ricci Curvature Decay to Zero