Finite-Time Blow-Up and Local Existence for Chemotaxis System with a General Memory Term

In this paper, we discuss the local existence of weak solutions for a parabolic system modelling chemotaxis with memory term, and we show the finite-time blowup and chemotactic collapse for this system. The main methods we used are the fixed point theorem and the semigroup theory.     

Reflected forward–backward stochastic differential equations and related PDEs

In this article, we study a type of coupled reflected forward–backward stochastic differential equations (reflected FBSDEs, for short) with continuous coefficients, including the existence and the uniqueness of the solution of our reflected FBSDEs as well as the comparison theorem. We prove that the solution of our reflected FBSDEs gives a probabilistic interpretation for the viscosity solution of an obstacle problem for a quasilinear parabolic partial differential equation.     

Classical solvability of 1‐D Cahn–Hilliard equation coupled with elasticity

In this paper, we prove the classical solvability of a nonlinear 1‐D system of hyperbolic–parabolic type arising as a model of phase separation in deformable binary alloys. The system is governed by the nonstationary elasticity equation coupled with the Cahn–Hilliard equation. The existence proof is based on the application of the Leray–Schauder fixed point theorem and standard energy methods. Copyright © 2006 John Wiley & Sons, Ltd.     

A parabolic–hyperbolic system modeling the tumor growth with angiogenesis

In this paper, we study the parabolic–hyperbolic system about the growth of a tumor. The model is a coupled system of PDEs with Robin boundary, which involves nutrient density, extracellular matrix and matrix degrading enzyme. By transforming the free boundary into a fixed boundary and using strict mathematical analysis, we can prove the existence and uniqueness of the radially symmetric stationary solution. By the fixed point theorem, we obtain the existence and uniqueness of the radially symmetric solution globally in time.     

Existence of Solutions for a Parabolic System Modelling Chemotaxis with Memory Term

In this paper, we come up with a parabolic system modelling chemotaxis with memory term, and establish the local existence and uniqueness of weak solutions. The main methods we use are the fixed point theorem and semigroup theory.     

The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications

This paper deals with a class of backward stochastic differential equations with Poisson jumps and with random terminal times. We prove the existence and uniqueness result of adapted solution for such a BSDE under the assumption of non-Lipschitzian coefficient. We also derive two comparison theorems by applying a general Girsanov theorem and the linearized technique on the coefficient. By these we first show the existence and uniqueness of minimal solution for one-dimensional BSDE with jumps when its coefficient is continuous and has a linear growth. Then we give a general Feynman-Kac formula for a class of parabolic types of second-order partial differential and integral equations (PDIEs) by using the solution of corresponding BSDE with jumps. Finally, we exploit above Feynman-Kac formula and related comparison theorem to provide a probabilistic formula for the viscosity solution of a quasi-linear PDIE of parabolic type.     

一类抛物型趋化模型的解的存在性

本文研究了一类抛物型趋化模型的解的存在性问题.利用算子半群理论,Sobolev嵌入定理及不等式技巧对解进行一些重要的先验估计,然后构造压缩映射证明了模型存在局部解.进一步构造迭代估计来说明不可能存在爆破,从而证明全局解的存在.     

Optimal control of the bidomain system (I): The monodomain approximation with the Rogers-McCulloch model

For the monodomain approximation of the bidomain equations, a weak solution concept is proposed. We analyze it for the FitzHugh-Nagumo and the Rogers-McCulloch ionic models, obtaining existence and uniqueness theorems. Subsequently, we investigate optimal control problems subject to the monodomain equations. After proving the existence of global minimizers, the system of the first-order necessary optimality conditions is rigorously characterized. For the adjoint system, we prove an existence and regularity theorem as well.     

Some properties of solutions for a fourth‐order parabolic equation

This paper is concerned with a fourth‐order parabolic equation in one spatial dimension. On the basis of Leray–Schauder's fixed point theorem, we prove the existence and uniqueness of global weak solutions. Moreover, we also consider the regularity of solution and the existence of global attractor. Copyright © 2012 John Wiley & Sons, Ltd.     

A quasistatic contact problem for viscoelastic materials with friction and damage

We consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is quasistatic, the material is assumed to be viscoelastic with long memory and the frictional contact is modelled with subdifferential boundary conditions. The mechanical damage of the material is described by the damage function, which is modelled by a nonlinear partial differential equation. We derive the variational formulation of the problem, which is a coupled system of a hemivariational inequality and a parabolic equation. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of abstract stationary inclusion and a fixed point theorem.     

Complex analysis in subsystems of second order arithmetic

This research is motivated by the program of Reverse Mathematics. We investigate basic part of complex analysis within some weak subsystems of second order arithmetic, in order to determine what kind of set existence axioms are needed to prove theorems of basic analysis. We are especially concerned with Cauchy’s integral theorem. We show that a weak version of Cauchy’s integral theorem is proved in RCAo. Using this, we can prove that holomorphic functions are analytic in RCAo. On the other hand, we show that a full version of Cauchy’s integral theorem cannot be proved in RCAo but is equivalent to weak König’s lemma over RCAo.     

一类非散度型非线性退化抛物方程解的存在性(英文)

In this paper, we discuss the existence of weak solutions to the initial and boundary value problem of a class of nonlinear degenerate parabolic equations in non-divergence form. Applying the method of parabolic regularization, we prove the existence of weak solutions to the problem. By carefully analyzing the approximate solutions to the problem, we make a series of estimates to the solutions and prove the weak convergence of the approximation solution sequence. Finally we testify the existence of weak solutions to the problem.     

Global solution of the Cauchy problem in nonlinear thermodi.usion in solid body

We prove a theorem about global existence (in time) of the solution to the initial‐value problem for a nonlinear hyperbolic parabolic system of coupled partial differential equation of second order describing the process of thermodiffusion in solid body. The corresponding global existence theorems has been proved using the L^p ‐ L^q time decay estimates for the solution of the associated linearized problem. Next, we proved the energy estimate in the Sobolev space with constant independent of time. Such an energy estimate allows us to apply the standard (continuation argument and to continue the local solution to one de.ned for all t ∈ 〈0, ∞)).     

Maximum Principle and Existence of Weak Solutions for Nonlinear System Involving Singular p-Laplacian Operators

In this work we give necessary and sufficient conditions to have a maximum principle for nonlinear system involving singular (p,q)-Laplacian operators on bounded domain Ω of Rⁿ and then we prove the existence of positive weak solutions for the same system by using the Browder theorem method.     

Analysis of a parabolic cross-diffusion population model without self-diffusion

The global existence of non-negative weak solutions to a strongly coupled parabolic system arising in population dynamics is shown. The cross-diffusion terms are allowed to be arbitrarily large, whereas the self-diffusion terms are assumed to disappear. The last assumption complicates the analysis since these terms usually provide H¹ estimates of the solutions. The existence proof is based on a positivity-preserving backward Euler-Galerkin approximation, discrete entropy estimates, and L¹ weak compactness arguments. Furthermore, employing the entropy-entropy production method, we show for special stationary solutions that the transient solution converges exponentially fast to its steady state. As a by-product, we prove that only constant steady states exist if the inter-specific competition parameters disappear no matter how strong the cross-diffusion constants are.     

On a parabolic free boundary problem arising from a Bingham-like flow model with a visco-elastic core

In this paper we study a two-phase one-dimensional free boundary problem for parabolic equation, arising from a mathematical model for Bingham-like fluids with visco-elastic core presented in [L. Fusi, A. Farina, A mathematical model for Bingham-like fluids with visco-elastic core, Z. Angew. Math. Phys. 55 (2004) 826-847]. The main feature of this problem consists in the very peculiar structure of the free boundary condition, not allowing to use classical tools to prove well posedness. Local existence is proved using a fixed point argument based on Schauder's theorem. Uniqueness is proved using a non-standard technique based on a weak formulation of the problem.     

Maximum and minimum solutions for nonlinear parabolic problems with discontinuities

In this paper we examine nonlinear parabolic problems with a discontinuous right hand side. Assuming the existence of an upper solution φ and a lower solution ψ such that ψ ≤ φ, we establish the existence of a maximum and a minimum solution in the order interval [ψ, φ]. Our approach does not consider the multivalued interpretation of the problem, but a weak one side “Lipschitz” condition on the discontinuous term. By employing a fixed point theorem for nondecreasing maps, we prove the existence of extremal solutions in [ψ, φ for the original single valued version of the problem.     

Study of weak solutions for a fourth-order parabolic equation with variable exponent of nonlinearity

The aim of this paper is to study the existence and uniqueness of weak solutions for an initial boundary problem of a fourth-order parabolic equation with variable exponent of nonlinearity. First, the authors of this paper apply Leray-Schauder’s fixed point theorem to prove the existence of solutions of the corresponding nonlinear elliptic problem and then obtain the existence of weak solutions of nonlinear parabolic problem by combining the results of the elliptic problem with Rothe’s method. In addition, the authors also discuss the regularity of weak solutions in the case of space dimension one.     

几类具Pucci算子的椭圆方程组解的存在性

研究完全非线性椭圆方程组解的存在性问题,其中ΩR~n,n≥2是有界光滑区域,—Μ_(λ,Λ)~+为具参数0<λ≤Λ的Pucci算子.首先,对f_i,i=1,2为一致有界函数的情形,证明了此方程组存在有界非负解.其次,当{f_1,f_2}是拟增的,且方程组存在有序上、下解时,利用上、下解方法,并结合增算子的不动点定理证明了此方程组存在最大非负解和最小非负解.当{f_1,f_2}是拟减或混拟单调时,使用Schauder不动点定理证明了此方程组至少存在一个非负解.针对此方程组中f_i,i=1,2的某些特殊形式,证明了相应方程组正解的存在性.最后给出了应用实例.     

On a local in time solvability of the Neumann problem of quasilinear hyperbolic parabolic coupled systems

We prove a local in time existence theorem of classical solutions to some coupled system of quasilinear hyperbolic equations and quasilinear parabolic equations with Neumann boundary condition. This coupled system contains a non-linear thermoelastic equation as an important physical example.