Existence and Uniqueness of Solution for Cahn-Hilliard Hyperbolic Phase-Field System with Dirichlet Boundary Condition and Regular Potentials

Our aim in this paper is to study the existence and the uniqueness of the solutions for hyperbolic Cahn-Hilliard phase-field system, with initial conditions, Dirichlet boundary condition and regular potentials.

Boundary Control for Cooperative Elliptic Systems under Conjugation Conditions

The existence and uniqueness of the state for 2 × 2 Dirichlet cooperative elliptic systems under conjugation conditions are proved using Lax-Milgram lemma, then the boundary control for these systems is discussed. The set of equations and inequalities that characterizes this boundary control is found by theory of Lions, Sergienko and Deineka. The problem for cooperative Neumann elliptic systems under conjugation conditions is also considered. Finally, the problem for n × n cooperative elliptic systems under conjugation conditions is established.

Distributed Control for <i>n</i>×<i>n</i>Cooperative Systems Governed by Hyperbolic Operator of Infinite Order

In this study, a distributed optimal control problem for n × n cooperative hyperbolic systems with infinite order operators and Dirichlet conditions are considered. The existence and uniqueness of the state of these systems are proved. The necessary and sufficient conditions for optimality of distributed control with constraints are found, and the set of equations and inequalities that defining the optimal control of these systems is also obtained. Finally, some examples for the control problem without constraints are given.

LOWER BOUNDS ESTIMATE FOR THE BLOW-UP TIME OF A NONLINEAR NONLOCAL POROUS MEDIUM EQUATION

The lower bounds for the blow-up time of blow-up solutions to the nonlinear nolocal porous equation ut=△u^m＋u^p∫Ωu^qdxwith either null Dirichlet boundary condition or homogeneous Neumann boundary condi- tion is given in this article by using a differential inequality technique.

A Smallness Condition Ensuring Boundedness in a Two-dimensional Chemotaxis-Navier-Stokes System involving Dirichlet Boundary Conditions for the Signal

The chemotaxis-Navier-Stokes system{nt+u・∇n=Δn−∇・(n∇c),ct+u・∇c=Δc−nc,ut+(u・∇)u=Δu+∇P+n∇φ,∇・u=0is considered in a smoothly bounded planar domainΩunder the boundary conditions(∇n−n∇c)・ν=0,c=c,u=0,x∈∂Ω,t>0,with a given nonnegative constant c_(*).It is shown that if(n_(0),c_(0),u_(0))is sufficiently regular and such that the product||n_(0)||L^(1)(Ω)||c_(0)||^(2)L^(∞)(Ω)is suitably small,an associated initial value problem possesses a bounded classical solution with(n,c,u)|_(t=0)=(n_(0),c_(0),u_(0)).

EXISTENCE AND NONEXISTENCE OF GLOBAL POSITIVE SOLUTIONS FOR DEGENERATE PARABOLIC EQUATIONS IN EXTERIOR DOMAINS

This article deals with the degenerate parabolic equations in exterior domains and with inhomogeneous Dirichlet boundary conditions. We obtain that pc = （σ ＋ m ）n / （ n-σ- 2 ） is its critical exponent provided max{-1, [ （1- m ）n- 2] / （ n ＋ l ） } 〈 σ 〈 n- 2. This critical exponent is not the same as that for the corresponding equations with the boundary value 0, but is more closely tied to the critical exponent of the elliptic type degenerate equations. Futhermore, we demonstrate that if max（1, σ ＋ m） 〈 p 〈 pc, then every positive solution of the equations blows up in finite time; whereas for p 〉 pc, the equations admit global positive solutions for some boundary values and initial data. Meantime, we also demonstrate that its positive solutions blow up in finite time provided n〈σ＋2.

BLOW-UP AND LIFE SPAN ESTIMATES FOR A CLASS OF NONLINEAR DEGENERATE PARABOLIC SYSTEM WITH TIME-DEPENDENT COEFFICIENTS

This paper deals with the singularity and global regularity tor a class oI nonlinear porous medium system with time-dependent coefficients under homogeneous Dirichlet boundary conditions. First, by comparison principle, some global regularity results are established. Secondly, using some differential inequality technique, we investigate the blow-up solution to the initial-boundary value problem. Furthermore, upper and lower bounds for the maximum blow-up time under some appropriate hypotheses are derived as long as blow-up Occurs.

CONVERGENCE OF FINITE VOLUME SCHEMES FOR HAMILTON-JACOBI EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS

We study numerical methods for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. We first propose a new class of abstract monotone approximation schemes and get a convergence rate of 1/2 . Then, according to the abstract convergence results, by newly constructing monotone finite volume approximations on interior and boundary points, we obtain convergent finite volume schemes for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. Finally give some numerical results.

THE INITIAL BOUNDARY VALUE PROBLEM FOR QUASI-LINEAR SCHRODINGER-POISSON EQUATIONS

In this article, the author studies the iuitial （Dirichlet.） boundary problem for a high field version of the Schroedinger-Poisson equations, which include a nonlinear term in the Poisson equation corresponding to a field-dependent dielectric constant and an effective potential in the Schroedinger equations on the unit cube. h global existence and uniqueness is established for a solution to this problem.

Lower Bound Estimate of Blow Up Time for the Porous Medium Equations under Dirichlet and Neumann Boundary Conditions

In this paper,we establish the lower bounds estimate of the blow up time for solutions to the nonlocal cross-coupled porous medium equations with nonlocal source terms under Dirichlet and Neumann boundary conditions.The results are obtained by using some differential inequality technique.

On the Caginalp for a Conserve Phase-Field with a Polynomial Potentiel of Order 2<i>p</i>- 1

Our aim in this paper is to study on the Caginalp for a conserved phase-field with a polynomial potentiel of order 2p - 1. In this part, one treats the conservative version of the problem of generalized phase field. We consider a regular potential, more precisely a polynomial term of the order 2p - 1 with edge conditions of Dirichlet type. Existence and uniqueness are analyzed. More precisely, we precisely, we prove the existence and uniqueness of solutions.

Two-Relaxation-Time Lattice Boltzmann Scheme:About Parametrization,Velocity,Pressure and Mixed Boundary Conditions

We develop a two-relaxation-time (TRT) Lattice Boltzmann model for hydrodynamicequations with variable source terms based on equivalent equilibriumfunctions. A special parametrization of the free relaxation parameter is derived. Itcontrols, in addition to the non-dimensional hydrodynamic numbers, any TRT macroscopicsteady solution and governs the spatial discretization of transient flows. Inthis framework, the multi-reflection approach [16, 18] is generalized and extended forDirichlet velocity, pressure and mixed (pressure/tangential velocity) boundary conditions.We propose second and third-order accurate boundary schemes and adapt themfor corners. The boundary schemes are analyzed for exactness of the parametrization,uniqueness of their steady solutions, support of staggered invariants and for the effectiveaccuracy in case of time dependent boundary conditions and transient flow.When the boundary scheme obeys the parametrization properly, the derived permeabilityvalues become independent of the selected viscosity for any porous structureand can be computed efficiently. The linear interpolations [5, 46] are improved withrespect to this property.

Existence of Positive Solutions to Semipositone Singular Dirichlet Boundary Value Problems

The paper presents the conditions which guarantee that for some positive value of μ there are positive solutions of the differential equation （Ф（x＇））＇＋μQ（t, x, x＇） = 0 satisfying the Dirichlet boundary conditions x（0） = x（T） = 0. Here Q is a continuous function on the set [0, T] × （0, ∞） ~ （R ／ {0}） of the semipositone type and Q is singular at the value zero of its phase variables.

EXISTENCE AND MULTIPLICITY OF SOLUTIONS TO A CLASS OF DIRICHLET PROBLEMS WITH IMPULSIVE EFFECTS

In this paper, we study the existence and multiplicity of solutions to a class of Dirichlet problems with impulsive effects via variational methods. Under an assumption that the nonlinearity f is superlinear but does not necessarily satisfy the Ambrosetti-Rabinowitz condition, we extend and improve some recent results.

Numerical Reconstruction of Locally Rough Surfaces with a Newton Iterative Algorithm

In this paper,we propose a Newton iterative algorithm to numerically reconstruct a locally rough surface with Dirichlet and impedance boundary conditions by near-field measurements of acoustic waves.The algorithm relies on the Frechet differentiability analysis of the locally rough surface scattering problem,which is established by reducing the original model into an equivalent boundary value problem with compactly supported boundary data.With a slight modification,the algorithm can be also extended to reconstruct the local perturbation of a non-local rough surface.Finally,numerical results are presented to illustrate the effectiveness of the inversion algorithm with the multi-frequency data.

Tensor Fields on Self-Dual Warped AdS_3 Background

Considering rank s fields obey first order equation of motion, we study the dynamics of such fields in a 3-dimensionaJ self-duaJ space-like warped AdS3 black hole background. We obtain a Klein-Gordon-like equation for tensor fields. By using gauge constraint and traceless condition, we will find the exact solutions of the equations of motion. Then, we will Compute the quasi-normal modes by imposing appropriate boundary conditions at horizon and infinity.

Solving forward and inverse problems of the nonlinear Schrodinger equation with the generalized PT-symmetric Scarf-Ⅱpotential via PINN deep learning

In this paper,based on physics-informed neural networks(PINNs),a good deep learning neural network framework that can be used to effectively solve the nonlinear evolution partial differential equations(PDEs)and other types of nonlinear physical models,we study the nonlinear Schrodinger equation(NLSE)with the generalized PT-symmetric Scarf-Ⅱpotential,which is an important physical model in many fields of nonlinear physics.Firstly,we choose three different initial values and the same Dinchlet boundaiy conditions to solve the NLSE with the generalized PT-symmetric Scarf-Ⅱpotential via the PINN deep learning method,and the obtained results are compared with ttose denved by the toditional numencal methods.Then,we mvestigate effect of two factors(optimization steps and activation functions)on the performance of the PINN deep learning method in the NLSE with the generalized PT-symmetric Scarf-Ⅱpotential.Ultimately,the data-driven coefficient discovery of the generalized PT-symmetric Scarf-Ⅱpotential or the dispersion and nonlinear items of the NLSE with the generalized PT-symmetric Scarf-Ⅱpotential can be approximately ascertained by using the PINN deep learning method.Our results may be meaningful for further investigation of the nonlinear Schrodmger equation with the generalized PT-symmetric Scarf-Ⅱpotential in the deep learning.

Study of Simple Hydrodynamic Solutions with the Two-Relaxation-Times Lattice Boltzmann Scheme

For simple hydrodynamic solutions, where the pressure and the velocity arepolynomial functions of the coordinates, exact microscopic solutions are constructedfor the two-relaxation-time (TRT) Lattice Boltzmann model with variable forcing andsupported by exact boundary schemes. We show how simple numerical and analyticalsolutions can be interrelated for Dirichlet velocity, pressure and mixed (pressure/tangential velocity) multi-reflection (MR) type schemes. Special care is taken toadapt themfor corners, to examine the uniqueness of the obtained steady solutions andstaggered invariants, to validate their exact parametrization by the non-dimensionalhydrodynamic and a “kinetic” (collision) number. We also present an inlet/outlet“constant mass flux” condition. We show, both analytically and numerically, that thekinetic boundary schemes may result in the appearance of Knudsen layers which arebeyond the methodology of the Chapman-Enskog analysis. Time dependent Dirichletboundary conditions are investigated for pulsatile flow driven by an oscillating pressuredrop or forcing. Analytical approximations are constructed in order to extend thepulsatile solution for compressible regimes.

Numerical Study of Quantized Vortex Interaction in theGinzburg-Landau Equation on Bounded Domains

In this paper,we study numerically quantized vortex dynamics and their interaction in the two-dimensional(2D)Ginzburg-Landau equation(GLE)with a dimensionless parameter#>0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition.We begin with a reviewof the reduced dynamical laws for time evolution of quantized vortex centers in GLE and show how to solve these nonlinear ordinary differential equations numerically.Then we present efficient and accurate numerical methods for discretizing the GLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition.Based on these efficient and accurate numerical methods for GLE and the reduced dynamical laws,we simulate quantized vortex interaction of GLE with different#and under different initial setups including single vortex,vortex pair,vortex dipole and vortex lattice,compare them with those obtained from the corresponding reduced dynamical laws,and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE on vortex interaction.Finally,we also obtain numerically different patterns of the steady states for quantized vortex lattices under the GLE dynamics on bounded domains.

On the Existence of Feller Semigroups with Discontinuous Coefficients

This paper is devoted to the functional analytic approach to the problem of the existence of Markov processes in probability theory. More precisely, we construct Feller semigroups with Dirichlet conditions for second-order, uniformly elliptic integro-differential operators with discontinuous coefficients. In other words, we prove that there exists a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it dies at the time when it reaches the boundary.