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1531-3492
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Discrete & Continuous Dynamical Systems - B
July 2016 , Volume 21 , Issue 5
Special issue dedicated to Lishang Jiang on his 80th birthday
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2016, 21(5): i-ii
doi: 10.3934/dcdsb.201605i
+[Abstract](1753)
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Abstract:
We dedicate this volume of the Journal of Discrete and Continuous Dynamical Systems-B to Professor Lishang Jiang on his 80th birthday. Professor Lishang Jiang was born in Shanghai in 1935. His family had migrated there from Suzhou. He graduated from the Department of Mathematics, Peking University, in 1954. After teaching at Beijing Aviation College, in 1957 he returned to Peking University as a graduate student of partial differential equations under the supervision of Professor Yulin Zhou. Later, as a professor, a researcher and an administrator, he worked at Peking University, Suzhou University and Tongji University at different points of his career. From 1989 to 1996, Professor Jiang was the President of Suzhou University. From 2001 to 2005, he was the Chairman of the Shanghai Mathematical Society.
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We dedicate this volume of the Journal of Discrete and Continuous Dynamical Systems-B to Professor Lishang Jiang on his 80th birthday. Professor Lishang Jiang was born in Shanghai in 1935. His family had migrated there from Suzhou. He graduated from the Department of Mathematics, Peking University, in 1954. After teaching at Beijing Aviation College, in 1957 he returned to Peking University as a graduate student of partial differential equations under the supervision of Professor Yulin Zhou. Later, as a professor, a researcher and an administrator, he worked at Peking University, Suzhou University and Tongji University at different points of his career. From 1989 to 1996, Professor Jiang was the President of Suzhou University. From 2001 to 2005, he was the Chairman of the Shanghai Mathematical Society.
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2016, 21(5): 1389-1400
doi: 10.3934/dcdsb.2016001
+[Abstract](2200)
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Abstract:
This paper investigates the positive solutions for second order linear elliptic equation in unbounded cylinder with zero boundary condition. We prove there exist two special positive solutions with exponential growth at one end while exponential decay at the other, and all the positive solutions are linear combinations of these two.
This paper investigates the positive solutions for second order linear elliptic equation in unbounded cylinder with zero boundary condition. We prove there exist two special positive solutions with exponential growth at one end while exponential decay at the other, and all the positive solutions are linear combinations of these two.
2016, 21(5): 1401-1420
doi: 10.3934/dcdsb.2016002
+[Abstract](2170)
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Abstract:
In this paper we discuss the optimal liquidation over a finite time horizon until the exit time. The drift and diffusion terms of the asset price are general functions depending on all variables including control and market regime. There is also a local nonlinear transaction cost associated to the liquidation. The model deals with both the permanent impact and the temporary impact in a regime switching framework. The problem can be solved with the dynamic programming principle. The optimal value function is the unique continuous viscosity solution to the HJB equation and can be computed with the finite difference method.
In this paper we discuss the optimal liquidation over a finite time horizon until the exit time. The drift and diffusion terms of the asset price are general functions depending on all variables including control and market regime. There is also a local nonlinear transaction cost associated to the liquidation. The model deals with both the permanent impact and the temporary impact in a regime switching framework. The problem can be solved with the dynamic programming principle. The optimal value function is the unique continuous viscosity solution to the HJB equation and can be computed with the finite difference method.
2016, 21(5): 1421-1434
doi: 10.3934/dcdsb.2016003
+[Abstract](2287)
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Abstract:
The following type of parabolic Barenblatt equations
min {$\partial_t V - \mathcal{L}_1 V, \partial_t V-\mathcal{L}_2 V$} = 0
is studied, where $\mathcal{L}_1$ and $\mathcal{L}_2$ are different elliptic operators of second order. The (unknown) free boundary of the problem is a divisional curve, which is the optimal insured boundary in our stochastic control problem. It will be proved that the free boundary is a differentiable curve.
To the best of our knowledge, this is the first result on free boundary for Barenblatt Equation. We will establish the model and verification theorem by the use of stochastic analysis. The existence of classical solution to the HJB equation and the differentiability of free boundary are obtained by PDE techniques.
The following type of parabolic Barenblatt equations
min {$\partial_t V - \mathcal{L}_1 V, \partial_t V-\mathcal{L}_2 V$} = 0
is studied, where $\mathcal{L}_1$ and $\mathcal{L}_2$ are different elliptic operators of second order. The (unknown) free boundary of the problem is a divisional curve, which is the optimal insured boundary in our stochastic control problem. It will be proved that the free boundary is a differentiable curve.
To the best of our knowledge, this is the first result on free boundary for Barenblatt Equation. We will establish the model and verification theorem by the use of stochastic analysis. The existence of classical solution to the HJB equation and the differentiability of free boundary are obtained by PDE techniques.
2016, 21(5): 1435-1444
doi: 10.3934/dcdsb.2016004
+[Abstract](2340)
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Based on the optimal estimate of convergence rate $O(\Delta x)$ of the value function of an explicit finite difference scheme for the American put option problem in [6], an $O(\sqrt{\Delta x})$ rate of convergence of the free boundary resulting from a general compatible numerical scheme to the true free boundary is proven. A new criterion for the compatibility of a generic numerical scheme to the PDE problem is presented. A numerical example is also included.
Based on the optimal estimate of convergence rate $O(\Delta x)$ of the value function of an explicit finite difference scheme for the American put option problem in [6], an $O(\sqrt{\Delta x})$ rate of convergence of the free boundary resulting from a general compatible numerical scheme to the true free boundary is proven. A new criterion for the compatibility of a generic numerical scheme to the PDE problem is presented. A numerical example is also included.
2016, 21(5): 1445-1454
doi: 10.3934/dcdsb.2016005
+[Abstract](2103)
+[PDF](388.6KB)
Abstract:
In this note, we remove the technical assumption $\gamma>0$ imposed by Dai et. al. [SIAM J. Control Optim., 48 (2009), pp. 1134-1154] who consider the optimal investment and consumption decision of a CRRA investor facing proportional transaction costs and finite time horizon. Moreover, we present an estimate on the resulting optimal consumption.
In this note, we remove the technical assumption $\gamma>0$ imposed by Dai et. al. [SIAM J. Control Optim., 48 (2009), pp. 1134-1154] who consider the optimal investment and consumption decision of a CRRA investor facing proportional transaction costs and finite time horizon. Moreover, we present an estimate on the resulting optimal consumption.
2016, 21(5): 1455-1468
doi: 10.3934/dcdsb.2016006
+[Abstract](2625)
+[PDF](338.3KB)
Abstract:
Recent years have seen a dramatic increase in the number and variety of new mathematical models describing biological processes. Some of these models are formulated as free boundary problems for systems of PDEs. Relevant biological questions give rise to interesting mathematical questions regarding properties of the solutions. In this review we focus on models whose formulation includes Stokes equations. They arise in describing the evolution of tumors, both at the macroscopic and molecular levels, in wound healing of cutaneous wounds, and in biofilms. We state recent results and formulate some open problems.
Recent years have seen a dramatic increase in the number and variety of new mathematical models describing biological processes. Some of these models are formulated as free boundary problems for systems of PDEs. Relevant biological questions give rise to interesting mathematical questions regarding properties of the solutions. In this review we focus on models whose formulation includes Stokes equations. They arise in describing the evolution of tumors, both at the macroscopic and molecular levels, in wound healing of cutaneous wounds, and in biofilms. We state recent results and formulate some open problems.
2016, 21(5): 1469-1481
doi: 10.3934/dcdsb.2016007
+[Abstract](2240)
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Abstract:
To capture the impact of spatial heterogeneity of environment and available resource of the public health system on the persistence and extinction of the infectious disease, a simplified spatial SIS reaction-diffusion model with allocation and use efficiency of the medical resource is proposed. A nonlinear space dependent recovery rate is introduced to model impact of available public health resource on the transmission dynamics of the disease. The basic reproduction numbers associated with the diseases in the spatial setting are defined, and then the low, moderate and high risks of the environment are classified. Our results show that the complicated dynamical behaviors of the system are induced by the variation of the use efficiency of medical resources, which suggests that maintaining appropriate number of public health resources and well management are important to control and prevent the temporal-spatial spreading of the infectious disease. The numerical simulations are presented to illustrate the impact of the use efficiency of medical resources on the control of the spreading of infectious disease.
To capture the impact of spatial heterogeneity of environment and available resource of the public health system on the persistence and extinction of the infectious disease, a simplified spatial SIS reaction-diffusion model with allocation and use efficiency of the medical resource is proposed. A nonlinear space dependent recovery rate is introduced to model impact of available public health resource on the transmission dynamics of the disease. The basic reproduction numbers associated with the diseases in the spatial setting are defined, and then the low, moderate and high risks of the environment are classified. Our results show that the complicated dynamical behaviors of the system are induced by the variation of the use efficiency of medical resources, which suggests that maintaining appropriate number of public health resources and well management are important to control and prevent the temporal-spatial spreading of the infectious disease. The numerical simulations are presented to illustrate the impact of the use efficiency of medical resources on the control of the spreading of infectious disease.
2016, 21(5): 1483-1505
doi: 10.3934/dcdsb.2016008
+[Abstract](2089)
+[PDF](538.9KB)
Abstract:
This paper introduces a new class of optimal switching problems, where the player is allowed to switch at a sequence of exogenous Poisson arrival times, and the underlying switching system is governed by an infinite horizon backward stochastic differential equation system. The value function and the optimal switching strategy are characterized by the solution of the underlying switching system. In a Markovian setting, the paper gives a complete description of the structure of switching regions by means of the comparison principle.
This paper introduces a new class of optimal switching problems, where the player is allowed to switch at a sequence of exogenous Poisson arrival times, and the underlying switching system is governed by an infinite horizon backward stochastic differential equation system. The value function and the optimal switching strategy are characterized by the solution of the underlying switching system. In a Markovian setting, the paper gives a complete description of the structure of switching regions by means of the comparison principle.
2016, 21(5): 1507-1523
doi: 10.3934/dcdsb.2016009
+[Abstract](2746)
+[PDF](415.2KB)
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This paper is concerned with a coupled Navier-Stokes/Allen-Cahn system describing a diffuse interface model for two-phase flow of viscous incompressible fluids with different densities in a bounded domain $\Omega\subset\mathbb R^N$($N=2,3$). We establish a criterion for possible break down of such solutions at finite time in terms of the temporal integral of both the maximum norm of the deformation tensor of velocity gradient and the square of maximum norm of gradient of phase field variable in 2D. In 3D, the temporal integral of the square of maximum norm of velocity is also needed. Here, we suppose the initial density function $\rho_0$ has a positive lower bound.
This paper is concerned with a coupled Navier-Stokes/Allen-Cahn system describing a diffuse interface model for two-phase flow of viscous incompressible fluids with different densities in a bounded domain $\Omega\subset\mathbb R^N$($N=2,3$). We establish a criterion for possible break down of such solutions at finite time in terms of the temporal integral of both the maximum norm of the deformation tensor of velocity gradient and the square of maximum norm of gradient of phase field variable in 2D. In 3D, the temporal integral of the square of maximum norm of velocity is also needed. Here, we suppose the initial density function $\rho_0$ has a positive lower bound.
2016, 21(5): 1525-1566
doi: 10.3934/dcdsb.2016010
+[Abstract](2410)
+[PDF](602.9KB)
Abstract:
We show that solutions of equations of the form \[ -u_t+D_{11}u+(x^1)D_{22}u = f \] (and also more general equations in any number of dimensions) satisfy simple Hölder estimates involving their derivatives. We also examine some pointwise properties for these solutions. Our results generalize those of Daskalopoulos and Lee, and Hong and Huang.
We show that solutions of equations of the form \[ -u_t+D_{11}u+(x^1)D_{22}u = f \] (and also more general equations in any number of dimensions) satisfy simple Hölder estimates involving their derivatives. We also examine some pointwise properties for these solutions. Our results generalize those of Daskalopoulos and Lee, and Hong and Huang.
2016, 21(5): 1567-1586
doi: 10.3934/dcdsb.2016011
+[Abstract](2105)
+[PDF](479.3KB)
Abstract:
In this paper we consider the following equation $$ u_t=(u^m)_{xx}+(u^n)_x, \ \ (x, t)\in \mathbb{R}\times(0, \infty) $$ with a Dirac measure as initial data, i.e., $u(x, 0)=\delta(x)$. The solution of the Cauchy problem is well-known as source-type solution. In the recent work [11] the author studied the existence and uniqueness of such kind of singular solutions and proved that there exists a number $n_0=m+2$ such that there is a unique source-type solution to the equation when $0 \leq n < n_0$. Here our attention is focused on the nonexistence and asymptotic behavior near the origin for a short time. We prove that $n_0$ is also a critical number such that there exits no source-type solution when $n \geq n_0$ and describe the short time asymptotic behavior of the source-type solution to the equation when $0 \leq n < n_0$. Our result shows that in the case of existence and for a short time, the source-type solution of such equation behaves like the fundamental solution of the standard porous medium equation when $0 \leq n < m+1$, the unique self-similar source-type solution exists when $n = m+1$, and the solution does like the nonnegative fundamental entropy solution in the conservation law when $m+1 < n < n_0$, while in the case of nonexistence the singularity gradually disappears when $n \geq n_0$ that the mass cannot concentrate for a short time and no such a singular solutions exists. The results of previous work [11] and this paper give a perfect answer to such topical researches.
In this paper we consider the following equation $$ u_t=(u^m)_{xx}+(u^n)_x, \ \ (x, t)\in \mathbb{R}\times(0, \infty) $$ with a Dirac measure as initial data, i.e., $u(x, 0)=\delta(x)$. The solution of the Cauchy problem is well-known as source-type solution. In the recent work [11] the author studied the existence and uniqueness of such kind of singular solutions and proved that there exists a number $n_0=m+2$ such that there is a unique source-type solution to the equation when $0 \leq n < n_0$. Here our attention is focused on the nonexistence and asymptotic behavior near the origin for a short time. We prove that $n_0$ is also a critical number such that there exits no source-type solution when $n \geq n_0$ and describe the short time asymptotic behavior of the source-type solution to the equation when $0 \leq n < n_0$. Our result shows that in the case of existence and for a short time, the source-type solution of such equation behaves like the fundamental solution of the standard porous medium equation when $0 \leq n < m+1$, the unique self-similar source-type solution exists when $n = m+1$, and the solution does like the nonnegative fundamental entropy solution in the conservation law when $m+1 < n < n_0$, while in the case of nonexistence the singularity gradually disappears when $n \geq n_0$ that the mass cannot concentrate for a short time and no such a singular solutions exists. The results of previous work [11] and this paper give a perfect answer to such topical researches.
2016, 21(5): 1587-1601
doi: 10.3934/dcdsb.2016012
+[Abstract](2021)
+[PDF](497.8KB)
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In this paper we will introduce for a convex domain $K$ in the Euclidean plane a function $\Omega_{n}(K, \theta)$ which is called by us the biwidth of $K$, and then try to find out the least area convex domain with constant biwidth $\Lambda$ among all convex domains with the same constant biwidth. When $n$ is an odd integer, it is proved that our problem is just that of Blaschke-Lebesgue, and when $n$ is an even number, we give a lower bound of the area of such constant biwidth domains.
In this paper we will introduce for a convex domain $K$ in the Euclidean plane a function $\Omega_{n}(K, \theta)$ which is called by us the biwidth of $K$, and then try to find out the least area convex domain with constant biwidth $\Lambda$ among all convex domains with the same constant biwidth. When $n$ is an odd integer, it is proved that our problem is just that of Blaschke-Lebesgue, and when $n$ is an even number, we give a lower bound of the area of such constant biwidth domains.
2016, 21(5): 1603-1615
doi: 10.3934/dcdsb.2016013
+[Abstract](3526)
+[PDF](432.8KB)
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We apply the general theory of pricing in incomplete markets, due to the author, on the problem of pricing bonds for the Hull-White stochastic interest rate model. As pricing in incomplete markets involves more market parameters than the classical theory, and as the derived risk premium is time-dependent, the proposed methodology might offer a better way for replicating different shapes of the empirically observed yield curves. For example, the so-called humped yield curve can be obtained from a normal yield curve by only increasing the investors risk aversion.
We apply the general theory of pricing in incomplete markets, due to the author, on the problem of pricing bonds for the Hull-White stochastic interest rate model. As pricing in incomplete markets involves more market parameters than the classical theory, and as the derived risk premium is time-dependent, the proposed methodology might offer a better way for replicating different shapes of the empirically observed yield curves. For example, the so-called humped yield curve can be obtained from a normal yield curve by only increasing the investors risk aversion.
2016, 21(5): 1617-1633
doi: 10.3934/dcdsb.2016014
+[Abstract](2166)
+[PDF](466.6KB)
Abstract:
In this paper, we consider the compressible magnetohydrodynamic equations with nonnegative thermal conductivity and electric conductivity. The coefficients of the viscosity, heat conductivity and magnetic diffusivity depend on density and temperature. Inspired by the framework of [11], [13] and [15], we use the maximal regularity and contraction mapping argument to prove the existence and uniqueness of local strong solutions with positive initial density in the bounded domain for any dimension.
In this paper, we consider the compressible magnetohydrodynamic equations with nonnegative thermal conductivity and electric conductivity. The coefficients of the viscosity, heat conductivity and magnetic diffusivity depend on density and temperature. Inspired by the framework of [11], [13] and [15], we use the maximal regularity and contraction mapping argument to prove the existence and uniqueness of local strong solutions with positive initial density in the bounded domain for any dimension.
2016, 21(5): 1635-1649
doi: 10.3934/dcdsb.2016015
+[Abstract](1993)
+[PDF](385.6KB)
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In this paper we present a new proof for the interior $C^{1,\alpha}$ regularity of weak solutions for a class of quasilinear elliptic equations, whose prototype is the $p$-Laplace equation.
In this paper we present a new proof for the interior $C^{1,\alpha}$ regularity of weak solutions for a class of quasilinear elliptic equations, whose prototype is the $p$-Laplace equation.
2016, 21(5): 1651-1669
doi: 10.3934/dcdsb.2016016
+[Abstract](1980)
+[PDF](1204.9KB)
Abstract:
An efficient parallelization method for numerically solving Lagrangian radiation hydrodynamic problems with three-temperature modeling on structural quadrilateral grids is presented. The three-temperature heat conduction equations are discretized by implicit scheme, and their computational cost are very expensive. Thus a parallel iterative method for three-temperature system of equations is constructed, which is based on domain decomposition for physical space, and combined with fixed point (Picard) nonlinear iteration to solve sub-domain problems. It can avoid global communication and can be naturally implemented on massive parallel computers. The space discretization of heat conduction equations uses the well-known local support operator method (LSOM). Numerical experiments show that the parallel iterative method preserves the same accuracy as the fully implicit scheme, and has high parallel efficiency and good stability, so it provides an effective solution procedure for numerical simulation of the radiation hydrodynamic problems on parallel computers.
An efficient parallelization method for numerically solving Lagrangian radiation hydrodynamic problems with three-temperature modeling on structural quadrilateral grids is presented. The three-temperature heat conduction equations are discretized by implicit scheme, and their computational cost are very expensive. Thus a parallel iterative method for three-temperature system of equations is constructed, which is based on domain decomposition for physical space, and combined with fixed point (Picard) nonlinear iteration to solve sub-domain problems. It can avoid global communication and can be naturally implemented on massive parallel computers. The space discretization of heat conduction equations uses the well-known local support operator method (LSOM). Numerical experiments show that the parallel iterative method preserves the same accuracy as the fully implicit scheme, and has high parallel efficiency and good stability, so it provides an effective solution procedure for numerical simulation of the radiation hydrodynamic problems on parallel computers.
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