20 research outputs found
Periodic solutions for a porous medium equation
In this paper, we study with a periodic porous medium equation with nonlinear convection terms and weakly nonlinear sources under Dirichlet boundary conditions. Based on the theory of Leray-Shauder fixed point theorem, we establish the existence of periodic solutions
Re-initialization-free Level Set Method via Molecular Beam Epitaxy Equation Regularization for Image Segmentation
Variational level set method has become a powerful tool in image segmentation
due to its ability to handle complex topological changes and maintain
continuity and smoothness in the process of evolution. However its evolution
process can be unstable, which results in over flatted or over sharpened
contours and segmentation failure. To improve the accuracy and stability of
evolution, we propose a high-order level set variational segmentation method
integrated with molecular beam epitaxy (MBE) equation regularization. This
method uses the crystal growth in the MBE process to limit the evolution of the
level set function, and thus can avoid the re-initialization in the evolution
process and regulate the smoothness of the segmented curve. It also works for
noisy images with intensity inhomogeneity, which is a challenge in image
segmentation. To solve the variational model, we derive the gradient flow and
design scalar auxiliary variable (SAV) scheme coupled with fast Fourier
transform (FFT), which can significantly improve the computational efficiency
compared with the traditional semi-implicit and semi-explicit scheme. Numerical
experiments show that the proposed method can generate smooth segmentation
curves, retain fine segmentation targets and obtain robust segmentation results
of small objects. Compared to existing level set methods, this model is
state-of-the-art in both accuracy and efficiency
Periodic boundary value problems for two classes of nonlinear fractional differential equations
Abstract By using the coincidence degree theorem, we obtain a new result on the existence of solutions for a class of fractional differential equations with periodic boundary value conditions, where a certain nonlinear growth condition of the nonlinearity needs to be satisfied. Furthermore, we study another class of differential equations of fractional order with periodic boundary conditions at resonance. A new result on the existence of positive solutions is presented by use of a Leggett–Williams norm-type theorem for coincidences. Two examples are given to illustrate the main result at the end of this paper
Asymptotic Behavior of Solutions of a Periodic Diffusion Equation
<p/> <p>We consider a degenerate parabolic equation with logistic periodic sources. First, we establish the existence of nontrivial nonnegative periodic solutions by monotonicity method. Then by using Moser iterative technique and the method of contradiction, we establish the boundedness estimate of nonnegative periodic solutions, by which we show that the attraction of nontrivial nonnegative periodic solutions, that is, all non-trivial nonnegative solutions of the initial boundary value problem, will lie between a minimal and a maximal nonnegative nontrivial periodic solutions, as time tends to infinity.</p
A Two-Species Cooperative Lotka-Volterra System of Degenerate Parabolic Equations
We consider a cooperating two-species Lotka-Volterra model of degenerate
parabolic equations. We are interested in the coexistence of the species in a bounded domain. We establish the existence of global generalized solutions of the initial boundary value problem by means of parabolic regularization and also consider the existence of the nontrivial time-periodic solution for this system