Long-time Dynamics Of Solutions Research Articles

On a generalized Cahn--Hilliard model with $p$-Laplacian

A generalized Cahn--Hilliard model in a bounded interval of the real line with no-flux boundary conditions is considered. The label ``generalized'' refers to the fact that we consider a concentration dependent mobility, the $p$-Laplace operator with $p > 1$ and a double well potential of the form $F(u)=\frac{1}{2\theta}|1-u^2|^\theta$, with $\theta > 1$; these terms replace, respectively, the constant mobility, the linear Laplace operator and the $C^2$ potential satisfying $F''(\pm1) > 0$, which are typical of the standard Cahn--Hilliard model. After investigating the associated stationary problem and highlighting the differences with the standard results, we focus the attention on the long time dynamics of solutions when $\theta\geq p > 1$. In the critical case $\theta= p > 1$, we prove exponentially slow motion of profiles with a transition layer structure, thus extending the well know results of the standard model, where $\theta=p=2$; conversely, in the supercritical case $\theta > p > 1$, we prove algebraic slow motion of layered profiles.

Dimension Estimate of Uniform Attractor for a Model of High Intensity Focussed Ultrasound-Induced Thermotherapy.

High intensity focussed ultrasound (HIFU) has emerged as a novel therapeutic modality, for the treatment of various cancers, that is gaining significant traction in clinical oncology. It is a cancer therapy that avoids many of the associated negative side effects of other more well-established therapies (such as surgery, chemotherapy and radiotherapy) and does not lead to the longer recuperation times necessary in these cases. The increasing interest in HIFU from biomedical researchers and clinicians has led to the development of a number of mathematical models to capture the effects of HIFU energy deposition in biological tissue. In this paper, we study the simplest such model that has been utilized by researchers to study temperature evolution under HIFU therapy. Although the model poses significant theoretical challenges, in earlier work, we were able to establish existence and uniqueness of solutions to this system of PDEs (see Efendiev et al. Adv Appl Math Sci 29(1):231-246, 2020). In the current work, we take the next natural step of studying the long-time dynamics of solutions to this model, in the case where the external forcing is quasi-periodic. In this case, we are able to prove the existence of uniform attractors to the corresponding evolutionary processes generated by our model and to estimate the Hausdorff dimension of the attractors, in terms of the physical parameters of the system.

Long time finite dimensionality in charged fluids

We consider long time dynamics of solutions of 2D periodic Nernst–Planck–Navier–Stokes systems forced by body charges and body forces. We show that, in the absence of body charges, but in the presence of fluid body forces, the charge density of the ions converges exponentially in time to zero, and the ion concentrations converge exponentially in time to equal time independent constants. This happens while the fluid continues to be dynamically active for all time. In the general case of body charges and body forces, the solutions converge in time to an invariant finite dimensional compact set in phase space.

Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms

<p style='text-indent:20px;'>This paper establishes the emergence of slowly moving transition layer solutions for the <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian (nonlinear) evolution equation, <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t = \varepsilon^p(|u_x|^{p-2}u_x)_x - F'(u), \qquad x \in (a,b), \; t &gt; 0, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula> are constants, driven by the action of a family of double-well potentials of the form <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ F(u) = \frac{1}{2\theta}|1-u^2|^\theta, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>indexed by <inline-formula><tex-math id="M5">\begin{document}$ \theta&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \theta\in \mathbb{R} $\end{document}</tex-math></inline-formula> with minima at two pure phases <inline-formula><tex-math id="M7">\begin{document}$ u = \pm1 $\end{document}</tex-math></inline-formula>. The equation is endowed with initial conditions and boundary conditions of Neumann type. It is shown that interface layers, or solutions which initially are equal to <inline-formula><tex-math id="M8">\begin{document}$ \pm 1 $\end{document}</tex-math></inline-formula> except at a finite number of thin transitions of width <inline-formula><tex-math id="M9">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula>, persist for an exponentially long time in the critical case with <inline-formula><tex-math id="M10">\begin{document}$ \theta = p $\end{document}</tex-math></inline-formula>, and for an algebraically long time in the supercritical (or degenerate) case with <inline-formula><tex-math id="M11">\begin{document}$ \theta&gt;p $\end{document}</tex-math></inline-formula>. For that purpose, energy bounds for a renormalized effective energy potential of Ginzburg–Landau type are established. In contrast, in the subcritical case with <inline-formula><tex-math id="M12">\begin{document}$ \theta&lt;p $\end{document}</tex-math></inline-formula>, the transition layer solutions are stationary.

Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation

Author(s): Ifrim, Mihaela; Tataru, Daniel | Abstract: This article represents a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to be known concerning its long time dynamics. Here, we prove that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^2$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach.

The global attractors and their Hausdorff and fractal dimensions estimation for the higher-order nonlinear Kirchhoff-type equation*

We investigate the global well-posedness and the longtime dynamics of solutions for the higher-order Kirchhoff-typeequation with nonlinear strongly dissipation:2( ) ( )m mt t tu    u    D u  ( ) ( ) ( )m  u  g u  f x . Under of the properassume, the main results are that existence and uniqueness of the solution is proved by using priori estimate and Galerkinmethod, the existence of the global attractor with finite-dimension, and estimation Hausdorff and fractal dimensions of theglobal attractor.

Numerical study of fractional nonlinear Schrödinger equations.

Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation.

Dynamics of a bio-reactor model with chemotaxis

Long time dynamics of solutions to a strongly coupled system of parabolic equations modeling the competition in bio-reactors with chemotaxis will be studied. In particular, we show that the dynamical system possesses a global attractor and that it is strongly uniformly persistent if the trivial steady state is unstable. Using a result of Smith and Waltman on perturbation of global attractors, we also show that the positive steady state is unique and globally attracting.

On a time-dependent bio-reactor model with chemotaxis

Long time dynamics of solutions to a strongly coupled system of parabolic equations modeling the competition in bio-reactors with chemotaxis will be studied. If the parameters of the system are per...

On a time-dependent bio-reactor model with chemotaxis

Long time dynamics of solutions to a strongly coupled system of parabolic equations modeling the competition in bio-reactors with chemotaxis will be studied. If the parameters of the system are periodic, sufficient conditions for positive periodic solutions will be derived in terms of parabolic eigenvalue problems.

Long time behavior of solutions of nonlinear classical field equations: the example of NLS defocusing

Hamiltonian classical systems are the farthest from variational systems. Formally they cannot decay because the equation of motion is reversible. Thus the long time behavior of their solutions is naturally described by statistical methods. For instance, a large assembly of interacting particles in a box follow the Gibbs-Boltzmann statistics, as far as average properties are concerned. As noticed already by Jeans, this statistical approach does not hold for the long time behavior of classical nonlinear field equations, because of the infinite heat capacity of a classical field. Thus nonlinear and nonintegrable classical fields cannot relax for long times toward an ill defined thermal equilibrium. The specific example considered here is the long time dynamics of solutions of the nonlinear Schrödinger equation, in the defocusing case. I show, under some assumptions, that for long times a cascade of energy toward smaller and smaller scales introduces a kind of dissipation in a system that is formally reversible, and I give the scaling laws for this. Besides this relaxation by weak interaction between phonons, there is also a relaxation of localized vortical structures emitting phonons.