Nonlocal Source Term Research Articles

Liouville type theorems for a quasilinear elliptic differential inequality with weighted nonlocal source and gradient absorption terms

Liouville type theorems for a quasilinear elliptic differential inequality with weighted nonlocal source and gradient absorption terms

Existence and multiplicity of weak solutions for a system of fourth‐order elliptic equations with combined nonlocal and indefinite source terms

Existence and multiplicity of weak solutions for a system of fourth‐order elliptic equations with combined nonlocal and indefinite source terms

Scattering threshold for the focusing energy-critical generalized Hartree equation

Abstract This work investigates the asymptotic behavior of energy solutions to the focusing nonlinear Schrödinger equation of Choquard type i ∂ t u + Δ u + ∣ u ∣ p − 2 ( I α * ∣ u ∣ p ) u = 0 , p = 1 + 2 + α N − 2 , N ≥ 3 . i{\partial }_{t}u+\Delta u+{| u| }^{p-2}\left({I}_{\alpha }* {| u| }^{p})u=0,\hspace{1.0em}p=1+\frac{2+\alpha }{N-2},\hspace{1.0em}N\ge 3. Indeed, in the energy-critical spherically symmetric regime, one proves a global existence and scattering versus finite time blow-up dichotomy. Precisely, if the data have an energy less than the ground state one, two cases are possible. If the kinetic energy of the radial data is less than the ground state one, then the solution is global and scatters. Otherwise, if the data have a finite variance or is spherically symmetric and have a finite mass, then the solution is nonglobal. The main difficulty is to deal with the nonlocal source term. The argument is the concentration-compactness-rigidity method introduced by Kenig and Merle (Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), no. 3, 645–675). This note naturally complements the work by Saanouni (Scattering theory for a class of defocusing energy-critical Choquard equations, J. Evol. Equ. 21 (2021), 1551–1571), where the scattering of the defocusing energy-critical generalized Hartree equation was obtained.

Mean field limit for one dimensional opinion dynamics with Coulomb interaction and time dependent weights

The mean field limit with time dependent weights for a 1D singular case, given by the attractive Coulomb interactions, is considered. This extends recent results (Ayi and Duteil, 2021; Duteil, 2022) for the case of regular interactions. The approach taken here is based on transferring the kinetic target equation to a Burgers-type equation through the distribution function of the measures. The analysis leading to the stability estimates of the latter equation makes use of Kruzkov entropy type estimates adapted to deal with nonlocal source terms.

Lower Bounds of Blow Up Time for a Class of Slow Reaction Diffusion Equations with Inner Absorption Terms

In this paper, a class of slow reaction-diffusion equations with nonlocal source and inner absorption terms are studied. By using the technique of improved differential inequality, the lower bounds of blow up time for the system under either homogeneous Dirichlet or nonhomogeneous Neumann boundary conditions are obtained.

Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term

Abstract This work is concerned with the nonexistence of nontrivial nonnegative weak solutions for a quasilinear parabolic differential inequality with weighted nonlocal source term in the whole space, which involves weighted polytropic filtration operator or generalized mean curvature operator. We establish the new critical Fujita exponents containing the first and second types. The key ingredient of the technique in proof is the test function method developed by Mitidieri and Pohozaev. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required.

Exponential Scattering for a Damped Hartree Equation

This note studies the linearly damped generalized Hartree equation iu˙−(−Δ)su+iau=±|u|p−2(Jγ∗|u|p)u,0<s<1,a>0,p≥2. Indeed, one proves an exponential scattering of the energy global solutions, with spherically symmetric datum. This means that, for large time, the solution goes exponentially to the solution of the associated free problem iu˙−(−Δ)su+iau=0, in Hs norm. The radial assumption avoids a loss of regularity in Strichartz estimates. The exponential scattering, which means that v:=eatu scatters in Hs, is proved in the energy sub-critical defocusing regime and in the mass-sub-critical focusing regime. This result is presented because of the gap due to the lack of scattering in the mass sub-critical regime, which seems not to be well understood. In this manuscript, one needs to overcome three technical difficulties which are mixed together: the first one is a fractional Laplace operator, the second one is a Choquard (non-local) source term, including the Hartree-type term when p=2 and the last one is a damping term iau. In a work in progress, the authors investigate the exponential scattering of global solutions to the above Schrödinger problem, with different kind of damping terms.

Boundary Control and Estimation for Underbalanced Drilling With Uncertain Reservoir Parameters

In underbalanced drilling (UBD), the bottom hole pressure is kept below pore pressure, causing pressure-dependent influx of reservoir gas into the wellbore that makes the system unstable at low drawdowns. In this article, we propose a feedback controller which stabilizes the system, represented by the “reduced drift flux model (DFM),” around an arbitrary pressure setpoint, while using only topside measurement and assuming unknown reservoir parameters. A particular challenge with this problem is the distributed and highly nonlinear nature of the system dynamics, where the “reduced DFM” models gas–liquid flow as a nonlinear transport equation with a nonlocal integral source term. An observer estimates the distributed gas concentration, downhole pressure, and reservoir parameters by solving the system dynamics backward relative to how the gas rises in the well. The control inputs are then constructed by designing target states over the next sampling period and again solving reversed dynamics to obtain the required topside pressures. The resulting controller is implemented with a 2-min zero-order hold to accommodate the actuation limitation situation on an actual drilling rig. Finally, the results are illustrated in simulations with an industry standard drift flux formulation as the plant model.

Finite Time Blowup in a Fourth-Order Dispersive Wave Equation with Nonlinear Damping and a Non-Local Source

In this work, we consider a class of initial boundary value problems for fourth-order dispersive wave equations with superlinear damping and non-local source terms as well as time-dependent coefficients in Ω×(t>0), where Ω is a bounded domain in RN and N≥2. We prove that there exists a safe time interval of existence in the solution [0,T], with T being a lower bound of the blowup time t*. Moreover, we find an explicit lower bound of t*, assuming the coefficients are positive constants.

Nonexistence of solutions to higher order evolution inequalities with nonlocal source term on Riemannian manifolds

We establish sufficient conditions for the nonexistence of nontrivial solutions to higher order evolution inequalities, with respect to the time variable. We consider a nonlocal source term, and work on complete noncompact Riemannian manifolds. The obtained conditions depend on the parameters of the problem and the geometry of the manifold. Our main result recovers some nonexistence theorems from the literature, established in the whole Euclidean space.

Blow-up of solutions for a wave equation with nonlinear averaged damping and nonlocal nonlinear source terms

The initial-boundary value problem for a wave equation with nonlinear averaged damping and nonlocal nonlinear source term is considered. We provide the sufficient conditions of finite time blow-up for weak solutions with suitable conditions on initial data by an ordinary differential inequality for an appropriately chosen functional.

Mean-field limit of collective dynamics with time-varying weights

<p style='text-indent:20px;'>In this paper, we derive the mean-field limit of a collective dynamics model with time-varying weights, for weight dynamics that preserve the total mass of the system as well as indistinguishability of the agents. The limit equation is a transport equation with source, where the (non-local) transport term corresponds to the position dynamics, and the (non-local) source term comes from the weight redistribution among the agents. We show existence and uniqueness of the solution for both microscopic and macroscopic models and introduce a new empirical measure taking into account the weights. We obtain the convergence of the microscopic model to the macroscopic one by showing continuity of the macroscopic solution with respect to the initial data, in the Wasserstein and Bounded Lipschitz topologies.</p>

Nonexistence of solutions for quasilinear hyperbolic inequalities

In this paper, we study the Cauchy problems for quasilinear hyperbolic inequalities with nonlocal singular source term and prove the nonexistence of global weak solutions in the homogeneous and nonhomogeneous cases by the test function method.

Energy scattering for radial focusing inhomogeneous bi-harmonic Schrödinger equations

This note studies the asymptotic behavior of global solutions to the fourth-order Schrödinger equation $$\begin{aligned} i\dot{u}+\Delta ^2 u+F(x,u)=0 . \end{aligned}$$Indeed, for both cases, local and non-local source term, the scattering is obtained in the focusing mass super-critical and energy sub-critical regimes, with radial setting. This work uses a new approach due to Dodson and Murphy (Proc Am Math Soc 145(11):4859–4867, 2017).

Solvability for a New Class of Moore-Gibson-Thompson Equation with Viscoelastic Memory, Source Terms, and Integral Condition

This paper deals with the existence and uniqueness of solutions of a new class of Moore-Gibson-Thompson equation with respect to the nonlocal mixed boundary value problem, source term, and nonnegative memory kernel. Galerkin’s method was the main used tool for proving our result. This work is a generalization of recent homogenous work.

Fractional elliptic problems with nonlinear gradient sources and measures

In this manuscript we deal with existence/uniqueness and regularity issues of suitable weak solutions to nonlocal problems driven by fractional Laplace type operators. Different from previous researches, in our approach we consider gradient non-linearity sources with subcritical growth, as well as appropriated measures as sources and boundary datum. We provide an in-depth discussion on the notions of solutions involved together with existence/uniqueness results in different regimes and for different boundary value problems. Finally, this work extends previous ones by dealing with more general nonlocal operators, source terms and boundary data.

Blow-Up Phenomena for a Class of Parabolic or Pseudo-parabolic Equation with Nonlocal Source

In this paper, we consider a class of parabolic or pseudo-parabolic equation with nonlocal source term: $$\begin{aligned} u_t-\nu \triangle u_t-\hbox {div}(\rho (|\nabla u|)^2\nabla u)=u^p(x,t)\int _{\Omega }k(x,y)u^{p+1}(y,t)dy, \end{aligned}$$ where $$\nu \ge 0$$ and $$p>0$$ . Using some differential inequality techniques, we prove that blow-up cannot occur provided that $$q>p$$ , also, we obtain some finite-time blow-up results and the lifespan of the blow-up solution under some different suitable assumptions on the initial energy. In particular, we prove finite-time blow-up of the solution for the initial data at arbitrary energy level. Furthermore, the lower bound for the blow-up time is determined if blow-up does occur.

Scattering for Radial Defocusing Inhomogeneous Bi-Harmonic SchrÖDinger Equations

This note studies the asymptotic behavior of global solutions to the fourth-order Schrodinger equation $$ i\dot u+{\Delta}^{2} u-F(x,u)=0 . $$ Indeed, for both cases, local and non-local source term, the scattering is obtained in the defocusing mass super-critical and energy sub-critical regimes, with radial setting.

Nonlocal approaches for multilane traffic models

We present a multilane traffic model based on balance laws, where the nonlocal source term is used to describe the lane changing rate. The modelling framework includes the consideration of local and nonlocal flux functions. Based on a Godunov type numerical scheme, we provide BV estimates and a discrete entropy inequality. Together with the $L^1$-contractivity property, we prove existence and uniqueness of weak solutions. Numerical examples show the nonlocal impact compared to local flux functions and local sources.

Asymptotic stability and blow-up for the wave equation with degenerate nonlocal nonlinear damping and source terms

This work is devoted to studying a wave equation with degenerate nonlocal nonlinear damping and source terms. By potential well theory, we show the asymptotic stability of energy in the presence of a degenerate damping of polynomial type when the initial energy is small. Also, we firstly derive some sufficient conditions on initial data which lead to finite time blow-up.