Liu Qiang

Shenzhen University · college of mathematics

The Functionalized Cahn–Hilliard equation has been proposed as a model for the interfacial energy of phase-separated mixtures of amphiphilic molecules. We study the existence of a nonnegative weak solutions of a gradient flow of the Functionalized Cahn–Hilliard equation subject to a degenerate mobility M(u) that is zero for u≤0. Assuming the initia...

In this paper, we mainly show a novel fast fractional order anisotropic diffusion algorithm for noise removal based on the recent numerical scheme called the Fast Explicit Diffusion. To balance the efficiency and accuracy of the algorithm, the truncated matrix method is used to deal with the iterative matrix in the model and its error is also estim...

In this paper, we propose a new image denosing model to remove the multiplicative noise by a maximum a posteriori estimation and an inhomogeneous fractional \begin{document}$ 1 $\end{document}-Laplace evolution equation. The main difficulty of the problem is the equation will become very singular when \begin{document}$ u(x) = u(y) $\end{document}....

The Functionalized Cahn-Hilliard equation has been proposed as a model for the interfacial energy of phase-separated mixtures of amphiphilic molecules. We study the existence of a nonnegative weak solutions of a gradient flow of the Functionalized Cahn-Hilliard equation subject to a degenerate mobility M(u) that is zero for u<=0. Assuming the initi...

Recently, some fractional Cahn–Hilliard equations are proposed for phase transition and image process, etc., which have attracted a lot of attention. In this paper, we concern the L∞ bound of the solutions to a class of fractional Cahn–Hilliard equations, which extends the results of integer order. By an invariant derivative technique, the crucial...

In this paper, we study the fractional p -Laplacian evolution equation with arbitrary initial energy, $$\begin{array}{} \displaystyle u_t(x,t) + (-{\it\Delta})_p^s u(x,t) = f(u(x,t)), \quad x\in {\it\Omega}, \,t \gt 0, \end{array} $$ where $\begin{array}{} (-{\it\Delta})_p^s \end{array} $ is the fractional p -Laplacian with $\begin{array}{} p \gt \...

In this paper, we propose a new model using a fractional reaction–diffusion system for image restoration and image decomposition which is motivated by work of S. Osher, A. Solé and L. Vese’s. Our model decomposes the degraded image into cartoon component belonging to a fractional Sobolev space and textured component belonging to a negative Hilbert...

The functionalized Cahn–Hilliard free energy describes phase separation in mixtures of amphiphilic molecules in solvent. Applications to highly amphiphilic molecules such as lipids requires degenerate diffusion that eliminates bulk diffusion, resulting in surface driven diffusion. We study the existence of weak solutions of a gradient flow of the f...

In this paper, we construct fixed-point algorithms for the second-order total variation models through discretization models and the subdifferential and proximity operators. Particularly, we focus on the convergence conditions of our algorithms by analyzing the eigenvalues of the difference matrix. The algorithms are tested on various images to ver...

Abstract This paper concerns the asymptotic behavior of the solution to a class of coupled semilinear parabolic systems with gradient terms. The Fujita-type blow-up theorems are established and the critical Fujita curve is determined not only by the behavior of the coefficients of the gradient term and the source terms at infinity, but also by the...

This paper concerns a control system governed by a convection-diffusion equation, which is weakly degenerate at the boundary. In the governing equation, the convection is independent of the degeneracy of the equation and cannot be controlled by the diffusion. The Carleman estimate is established by means of a suitable transformation, by which the d...

In this paper, we consider control systems governed by a class of semilinear parabolic equations, which are singular at the boundary and possess singular convection and reaction terms. The systems are shown to be null controllable by establishing Carleman estimates, observability inequalities and energy estimates for solutions to linearized equatio...

In this paper, we prove the existence and uniqueness of weak solutions for a singular evolutionary system, which is deduced from a model for image decomposition combining staircase reduction and texture extraction. The main method we used is p-Laplace regularization. The numerical experimental result shows the efficiency of this kind of model.

This paper concerns the Neumann problem of a reaction-diffusion system, which has a variable exponent Laplacian term and could be applied to image denoising. It is shown that the problem admits a unique renormalized solution for each integrable initial datum.

This paper concerns the Neumann problem of a reaction-diffusion system, which has a variable exponent Laplacian term and could be applied to image denoising. It is shown that the problem admits a unique renormalized solution for each integrable initial datum.

In this paper, we prove the existence and uniqueness of weak solutions for some singular evolutionary system, which is deduced from one of the Chambolle–Lions denoising models. The main method we used is p-Laplace regularization. Copyright © 2013 John Wiley & Sons, Ltd.

This paper is concerned with the Cauchy problem of a nonlocal equation that takes into account convective and p-Laplacian diffusive effects ∂u ∂t(x,t)=∫ ℝ N J(x-y)|u(y,t)-u(x,t)| p-2 (u(y,t)-u(x,t))dy+(G*f(u)-f(u))(x,t) with J radially symmetric and G not necessarily symmetric. First, we prove the existence and uniqueness of solutions, and if the c...

In this paper, a nondivergence pp-Laplace equation is proposed to remove the multiplicative noise by using a maximum a posteriori estimator. The existences of the weak solution of the Neumann problem for this equation are proved. Numerical examples demonstrate the better capability of this model on some heavy multiplicative noised images.

In this paper, we study the initial–boundary value problem for a class of singular parabolic equations. Under some additional conditions, we prove that there exists a weak solution for the problem by parabolic regularization. Moreover, we show that the solution obtained by parabolic regularization is the maximal solution among all weak solutions.

This paper is concerned with the electromagnetic scattering by a non-perfectly conductor obstacle in a chiral environment. A two-dimensional mathematical model is established. The existence and the uniqueness of the problem are discussed by potential theory.

A new anisotropic diffusion model is proposed for image denoising, which is based on reaction–diffusion systems with p(x)-growth. By Galerkin’s method, we establish the existence and uniqueness of weak solutions of the system for Neumann boundary conditions. Experimental results illustrate the effectiveness of the model in image restoration.

In this paper, we study a reaction–diffusion system applied to image restoration and image decomposition into cartoon and texture, based on Osher, Solé and Vese’s model in [S. Osher, A. Solé, L. Vese, Image decomposition and restoration using total variation minimization and the H−1 norm, SIAM Journal on Multiscale Modeling and Simulation 1 (3) (20...

In this paper, we discuss a class of nonlinear parabolic-hyperbolic equations which could be applied to image restoration. After theoretical analysis, we give an experimental approach to show the efficiency of this kind of model.

The paper concerns the existence theorem of weak solutions for the initial-boundary value problem of a nonlinear diffusion equation with convection. The equation may be regarded as a generalized non-Newtonian polytropic filtration equation. By doing the necessary BVBV estimate and other estimates for approximating solutions, we establish the existe...

This paper concerns the uniqueness of the bounded solution to a strongly degenerate parabolic problem. The equation considered may have two kinds of strong degeneracies and there is no restriction on the relation between the two degeneracies. By using Holmgren’s approach, we prove that the bounded solution of the associated initial–boundary value p...

In this paper, we discuss the existence and uniqueness of weak solutions for a fourth-order partial differential equation stemmed from image processing for noise removal. We also present some numerical tests for high order filters.

In this paper, we establish the existence and uniqueness of entropy solutions for a fourth-order nonlinear degenerate parabolic problem for noise removal in dimension 1≤d<4.

This paper deals with the initial and boundary value problem for a singular nonlinear parabolic equation. The existence of solutions is established by parabolic regularization. Some properties of solutions, for instance localization and large time behaviour are also discussed. Copyright © 2007 John Wiley & Sons, Ltd.

We show the existence of the renormalized solutions and the entropy solutions of a class of strongly degenerate quasilinear parabolic equations.