System Of Linear Parabolic Equations Research Articles

State-constrained controllability of linear reaction-diffusion systems

We study the controllability of a coupled system of linear parabolic equations, with nonnegativity constraint on the state. We establish two results of controllability to trajectories in large time: one for diagonal diffusion matrices with an “approximate” nonnegativity constraint, and a another stronger one, with “exact” nonnegativity constraint, when all the diffusion coefficients are equal and the eigenvalues of the coupling matrix have nonnegative real part. The proofs are based on a “staircase” method. Finally, we show that state-constrained controllability admits a positive minimal time, even with weaker unilateral constraint on the state.

Asset pricing with heterogeneous beliefs and illiquidity

AbstractThis paper studies the equilibrium price of an asset that is traded in continuous time between N agents who have heterogeneous beliefs about the state process underlying the asset's payoff. We propose a tractable model where agents maximize expected returns under quadratic costs on inventories and trading rates. The unique equilibrium price is characterized by a weakly coupled system of linear parabolic equations which shows that holding and liquidity costs play dual roles. We derive the leading‐order asymptotics for small transaction and holding costs which give further insight into the equilibrium and the consequences of illiquidity.

Asset Pricing with Heterogeneous Beliefs and Illiquidity

This paper studies the equilibrium price of an asset that is traded in continuous time between N agents who have heterogeneous beliefs about the state process underlying the asset's payoff. We propose a tractable model where agents maximize expected returns under quadratic costs on inventories and trading rates. The unique equilibrium price is characterized by a weakly coupled system of linear parabolic equations which shows that holding and liquidity costs play dual roles. We derive the leading-order asymptotics for small transaction and holding costs which give further insight into the equilibrium and the consequences of illiquidity.

Asymptotics of the Solution of Bisingular Problem for a System of Linear Parabolic Equations. I

Given a bisingular parabolic problem for a system of linear parabolic equations, we construct an asymptotics for the solution of any order with respect to a small parameter, without using the joining procedure for asymptotic expansions.

Hodograph Transformations and Cauchy Problem to Systems of Nonlinear Parabolic Equations

In this paper, we provide a method to solve the Cauchy problem of systems of quasi‐linear parabolic equations, such systems can be transformed to the systems of linear parabolic equations with variable coefficients via the hodograph transformations. Our approach to solve the linear systems with variable coefficients is to use their fundamental solutions, which are constructed by using the Lie's symmetry method. In consequence, we can derive explicit solutions to the Cauchy problem of the quasi‐linear systems in terms of the solutions of the linear systems and the hodograph transformations relating to the quasi‐linear and the linear systems.

Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model

We consider the classical parabolic–parabolic Keller–Segel system { u t = Δ u − ∇ ⋅ ( u ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n . It is proved that in space dimension n ⩾ 3 , for each q > n 2 and p > n one can find ε 0 > 0 such that if the initial data ( u 0 , v 0 ) satisfy ‖ u 0 ‖ L q ( Ω ) < ε and ‖ ∇ v 0 ‖ L p ( Ω ) < ε then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic equations. In particular, ( u , v ) approaches the steady state ( m , m ) as t → ∞ , where m is the total mass m : = ∫ Ω u 0 of the population. Moreover, we shall show that if Ω is a ball then for arbitrary prescribed m > 0 there exist unbounded solutions emanating from initial data ( u 0 , v 0 ) having total mass ∫ Ω u 0 = m .

A PROBABILISTIC REPRESENTATION FOR THE VORTICITY OF A THREE-DIMENSIONAL VISCOUS FLUID AND FOR GENERAL SYSTEMS OF PARABOLIC EQUATIONS

AbstractA probabilistic representation formula for general systems of linear parabolic equations, coupled only through the zero-order term, is given. On this basis, an implicit probabilistic representation for the vorticity in a three-dimensional viscous fluid (described by the Navier–Stokes equations) is carefully analysed, and a theorem of local existence and uniqueness is proved. The aim of the probabilistic representation is to provide an extension of the Lagrangian formalism from the non-viscous (Euler equations) to the viscous case. As an application, a continuation principle, similar to the Beale–Kato–Majda blow-up criterion, is proved.

On Exponential Splitting for Systems of Linear Parabolic Equations

This article concerns the character of decrease of solutions of linear parabolic systems that are close to splitting ones for time tending to infinity.