American Institute of Mathematical Sciences

We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier work [1], we constructed a blow-up solution for that equation, and showed that it blows up (at least) at the origin. We also derived the so called "intermediate blow-up profile". In this paper, we prove the single point blow-up property and determine the final blow-up profile.

This paper aims to solve numerically the linearized Korteweg-de Vries equation. We begin by deriving suitable boundary conditions then approximate them using finite difference method. The methodology of derivation, used in this paper, yields to Non-Standard Boundary Conditions (NSBC) that perfectly absorb wave reflections at the boundary. In addition, these NSBC are exact and local in time and space for non necessarily supported initial data and source terms. We finish with numerical examples that show the absorbing quality of these boundary conditions. Further comparisons are made using standard boundary conditions like, Dirichlet, Neumann and a variant of absorbing boundary conditions called discrete artificial ones.

We discuss the asymptotic behavior of the solutions for the fractional nonlinear Schrödinger equation that reads

\begin{document}$ u_t-iD u+ig(|u|^2)u+\gamma u = f\, . $\end{document}

We prove that this behavior is characterized by the existence of a compact global attractor in the appropriate energy space.

In this paper, we consider the inverse problem of determining two spatially varying coefficients appearing in the two-dimensional Boussinesq system from observed data of velocity vector and the temperature in a given arbitrarily subboundary. Based on Carleman estimates, we prove a Lipschitz stability result.

We present here different situations in which the filtering of high or low modes is used either for stabilizing semi-implicit numerical schemes when solving nonlinear parabolic equations, or for building adapted damping operators in the case of dispersive equation. We consider numerical filtering provided by mutigrid-like techniques as well as the filtering resulting from operator with monotone symbols. Our approach applies to several discretization techniques and we focus on finite elements and finite differences. Numerical illustrations are given on Cahn-Hilliard, Korteweig-de Vries and Kuramoto-Sivashinsky equations.

Our aim in this paper is to study a mathematical model for tumor growth and lactate exchanges in a glioma. We prove the existence of nonnegative (i.e. biologically relevant) solutions and, under proper assumptions, the uniqueness of the solution. We also state the permanence of the tumor when necrosis is not taken into account in the model and obtain linear stability results. We end the paper with numerical simulations.

We study a kinetic toy model for a spray of particles immersed in an ambient fluid, subject to some additional random forcing given by a mixing, space-dependent Markov process. Using the perturbed test function method, we derive the hydrodynamic limit of the kinetic system. The law of the limiting density satisfies a stochastic conservation equation in Stratonovich form, whose drift and diffusion coefficients are completely determined by the law of the stationary process associated with the Markovian perturbation.

In this paper, we study the solution behavior of two coupled non–linear Schrödinger equations (CNLS) in the critical case, where one equation includes gain, while the other includes losses. Next, we present two numerical methods for solving the CNLS equations, for which we have made a comparison. These numerical experiments permit to illustrate other theoretical results proven by the authors [11]. We also obtain several numerical results for different non–linearities and investigate on the value of the blow up time relatively to some parameters.

In this paper, we derive a simple model for the description of an ecological system including several subgroups with distinct ages, in order to analyze the influence of various phenomena on temporal evolution of the considered species. Our aim is to address the question of resilience of the global system, defined as its ability to stabilize itself to equilibrium, when being perturbed by exterior fluctuations. It is shown that a under a critical condition involving growth rate and mortality rate of each subgroup, extinction of all species may occur.

Based on recent works of Dodson-Murphy [12] and Arora-Dodson-Murphy [3], we give a unified approach for the energy scattering with radially symmetric initial data for nonlinear Schrödinger equations and nonlinear Choquard equations in any dimensions \begin{document}$ N\geq 2 $\end{document}. We also discuss its applications for other Schrödinger-type equations.

In this article, the asymptotic behavior of the solution to the following one dimensional Schrödinger equations with white noise dispersion

\begin{document}$ idu + u_{xx}\circ dW+ |u|^{p-1}udt = 0 $\end{document}

is studied. Here the equation is written in the { Stratonovich} formulation, and \begin{document}$ W(t) $\end{document} is a standard real valued Brownian motion. After establishing the global well-posedness, theoretical proof and numerical investigations are provided showing that, for a deterministic small enough initial data in \begin{document}$ L^1_x\cap H^1_x $\end{document}, the expectation of the \begin{document}$ L^\infty_x $\end{document} norm of the solutions decay to zero at \begin{document}$ O(t^{-\frac14}) $\end{document} as \begin{document}$ t $\end{document} goes to \begin{document}$ +\infty $\end{document}, as soon as \begin{document}$ p>7 $\end{document}.

In this paper, we investigate numerical methods for a backward problem of the time-fractional wave equation in bounded domains. We propose two fractional filter regularization methods, which can be regarded as an extension of the classical Landweber iteration for the time-fractional wave backward problem. The idea is first to transform the ill-posed backward problem into a weighted normal operator equation, then construct the regularization methods for the operator equation by introducing suitable fractional filters. Both a priori and a posteriori regularization parameter choice rules are investigated, together with an estimate for the smallest regularization parameter according to a discrepancy principle. Furthermore, an error analysis is carried out to derive the convergence rates of the regularized solutions generated by the proposed methods. The theoretical estimate shows that the proposed fractional regularizations efficiently overcome the well-known over-smoothing drawback caused by the classical regularizations. Some numerical examples are provided to confirm the theoretical results. In particular, our numerical tests demonstrate that the fractional regularization is actually more efficient than the classical methods for problems having low regularity.

We characterise the pressure term in the incompressible 2D and 3D Navier–Stokes equations for solutions defined on the whole space.

We consider here a damped forced nonlinear logarithmic Schrödinger equation in \begin{document}$ \mathbb{R}^N $\end{document}. We prove the existence of a global attractor in a suitable energy space. We complete this article with some open issues for nonlinear logarithmic Schrödinger equations in the framework of infinite-dimensional dynamical systems.

This essay is concerned with the one-dimensional Green-Naghdi equations in the presence of a non-zero vorticity according to the derivation in [5], and with the addition of a small surface tension. The Green-Naghdi system is first rewritten as an equivalent system by using an adequate change of unknowns. We show that solutions to this model may be obtained by a standard Picard iterative scheme. No loss of regularity is involved with respect to the initial data. Moreover solutions exist at the same level of regularity as for first order hyperbolic symmetric systems, i.e. with a regularity in Sobolev spaces of order \begin{document}$ s>3/2 $\end{document}.

We consider in this article a nonlinear vibrating Timoshenko system with thermoelasticity with second sound. We first recall the results obtained in [2] concerning the well-posedness, the regularity of the solutions and the asymptotic behavior of the associated energy. Then, we use a fourth-order finite difference scheme to compute the numerical solutions and we prove its convergence. The energy decay in several cases, depending on the stability number \begin{document}$ \mu $\end{document}, are numerically and theoretically studied.

We study a modified version of an initial-boundary value problem describing the formation of colony patterns of bacteria Escherichia Coli. The original system of three parabolic equations was studied numerically and analytically and gave insights into the underlying mechanisms of chemotaxis. We focus here on the parabolic-elliptic-parabolic approximation and the hyperbolic-elliptic-parabolic limiting system which describes the case of pure chemotactic movement without random diffusion. We first construct local-in-time solutions for the parabolic-elliptic-parabolic system. Then we prove uniform a priori estimates and we use them along with a compactness argument in order to construct local-in-time solutions for the hyperbolic-elliptic-parabolic limiting system. Finally, we prove that some initial conditions give rise to solutions which blow-up in finite time in the \begin{document}$ L^\infty $\end{document} norm in all space dimensions. This last result is true even in space dimension 1, which is not the case for the full parabolic or parabolic-elliptic Keller-Segel systems.

We present numerical implicit schemes based on a geometric approach of the study of the convergence of solutions of gradient-like systems given in [3]. Depending on the globality of the induced metric, we can prove the convergence of these algorithms.

We consider the nonlinear Schrödinger equation in dimension one with a nonlinearity concentrated in one point. We prove that this equation provides an infinite dimensional dynamical system. We also study the asymptotic behavior of the dynamics. We prove the existence of a global attractor for the system.

Let \begin{document}$ H $\end{document} be a complex Hilbert space and let \begin{document}$ \mathcal{B}(H) $\end{document} be the algebra of all bounded linear operators on \begin{document}$ H $\end{document}. The polar decomposition theorem asserts that every operator \begin{document}$ T \in \mathcal{B}(H) $\end{document} can be written as the product \begin{document}$ T = V P $\end{document} of a partial isometry \begin{document}$ V\in \mathcal{B}(H) $\end{document} and a positive operator \begin{document}$ P \in \mathcal{B}(H) $\end{document} such that the kernels of \begin{document}$ V $\end{document} and \begin{document}$ P $\end{document} coincide. Then this decomposition is unique. \begin{document}$ V $\end{document} is called the polar factor of \begin{document}$ T $\end{document}. Moreover, we have automatically \begin{document}$ P = \vert T\vert = (T^*T)^{\frac{1}{2}} $\end{document}. Unlike \begin{document}$ P $\end{document}, we have no representation formula that is required for \begin{document}$ V $\end{document}.

In this paper, we introduce, for \begin{document}$ T\in \mathcal{B}(H) $\end{document}, a family of functions called a "polar function" for \begin{document}$ T $\end{document}, such that the polar factor of \begin{document}$ T $\end{document} is obtained as a limit of a net built via continuous functional calculus from this family of functions. We derive several explicit formulas representing different polar factors. These formulas allow new for methods of approximations of the polar factor of \begin{document}$ T $\end{document}.