Theorem For Equations Research Articles

Classification and a priori estimates for the singular prescribing Q-curvature equation on 4-manifold

On (M,g) a compact riemannian 4-manifold we consider the prescribed Q-curvature equation defined on M with finite singular sources. We first prove a classification theorem for singular Liouville equations defined on R4 and perform a concentration compactness analysis. Then we derive a quantization result for bubbling solutions and establish a priori estimate under the assumption that certain conformal invariant does not take some quantized values. Furthermore we prove a spherical Harnack inequality around singular sources provided their strength is not an integer. Such an inequality implies that in this case singular sources are isolated simple blow up points.

Vallée-Poussin theorem for fractional functional differential equations

An analog of the classical Vallée-Poussin theorem about differential inequality in the theory of ordinary differential equations is developed for fractional functional differential equations. The main results are obtained in a form of a theorem about several equivalent assertions. Among them solvability of two-point boundary value problems with fractional functional differential equation, negativity of Green’s function, and its derivatives and existence of a function v(t) satisfying a corresponding differential inequality. Thus the Vallée-Poussin theorem presents one of the possible “entrances” to assertions on nonoscillating properties and assertions about the negativity of Green’s functions and their derivatives for various two-point problems. Choosing the function v(t) in the condition, we obtain explicit tests of sign-constancy of Green’s functions and their derivatives. It can be stressed that a choice of a corresponding function in the Vallée-Poussin theorem leads to explicit criteria in the form of algebraic inequalities, which, as we demonstrate with examples, cannot be improved. Replacing strict inequalities with non-strict ones will already lead to incorrect statements. In some cases, the well-known results obtained in the form of the Lyapunov inequalities for fractional differential equations can be improved based on ours. Another development, we propose, is a concept to consider fractional functional differential equations. The basis of this concept is to reduce boundary value problems for fractional functional differential equations to operator equations in the space of essentially bounded functions. Fractional functional differential equations can appear in various applications and in the process of construction of equations for a corresponding component of a solution-vector of a system of fractional equations.

Attractivity theorems for second order nonlinear differential equations

Attractivity theorems for second order nonlinear differential equations

A Support Theorem for Stochastic Differential Equations Driven by a Fractional Brownian Motion

A Support Theorem for Stochastic Differential Equations Driven by a Fractional Brownian Motion

Wong-Zakai approximation and support theorem for 2D and 3D stochastic convective Brinkman-Forchheimer equations

In this work, we demonstrate the Wong-Zakai approximation results for two- and three-dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations forced by Hilbert space valued Wiener noise on bounded domains. Even though the existence and uniqueness of a pathwise strong solution to SCBF equations is known, the existence of a unique solution to the approximating system is not immediate from the solvability results of SCBF equations, and we prove it by using the Faedo-Galerkin approximation technique and monotonicity arguments. Moreover, as an application of the Wong-Zakai approximation, we obtain the support of the distribution of solutions to SCBF equations.

A rigidity theorem for parabolic 2-Hessian equations

In this paper, we consider the entire solution to the parabolic 2 2 -Hessian equation − u t σ 2 ( D 2 u ) = 1 -u_t\sigma _2(D^2 u)=1 in R n × ( − ∞ , 0 ] \mathbb {R}^n\times (-\infty ,0] . We prove a rigidity theorem for the parabolic 2 2 -Hessian equation in R n × ( − ∞ , 0 ] \mathbb {R}^n\times (-\infty ,0] by establishing Pogorelov type estimates for 2 2 -convex-monotone solutions.

Central limit theorems for parabolic stochastic partial differential equations

Soit {u(t,x)}t≥0,x∈Rd la solution d’une équation de la chaleur stochastique non-linéaire d-dimensionnelle, perturbée par un bruit gaussien, blanc en temps et avec une covariance homogène en espace donnée par une mesure de Borel finie qui satisfait la condition de Dalang. Nous démontrons deux théorèmes de la limite centrale fonctionnels pour des champs d’occupation de la forme N−d∫Rdg(u(t,x))ψ(x/N)dx quand N→∞, où g est une function lipschitzienne sur Rd et ψ∈L2(Rd). La preuve utilise des inegalités de type Poincaré, le calcul de Malliavin, des arguments de compacité et la caractérisation du mouvement brownien comme le seul processus de Lévy continu de moyenne nulle. Notre résultat généralise les théorèmes de la limite centrale de Huang et al (Stochastic Process. Appl. 131 (2020) 7170–7184 ; Stoch. Partial Differ. Equ., Anal. Computat. 8 (2020) 402–421) qui sont valables lorsque g(u)=u et ψ=1[0,1]d.

Avian–human influenza epidemic model with diffusion, nonlocal delay and spatial homogeneous environment

In this paper, an avian–human influenza epidemic model with diffusion, nonlocal delay and spatial homogeneous environment is investigated. This model describes the transmission of avian influenza among poultry, humans and environment. The behavior of positive solutions to a reaction–diffusion system with homogeneous Neumann boundary conditions is investigated. By means of linearization method and spectral analysis the local asymptotical stability is established. The global asymptotical stability for the poultry sub-system is studied by spectral analysis and by using a Lyapunov functional. For the full system, the global stability of the disease-free equilibrium is studied using the comparison Theorem for parabolic equations. Our result shows that the disease-free equilibrium is globally asymptotically stable, whenever the contact rate for the susceptible poultry is small. This suggests that the best policy to prevent the occurrence of an epidemic is not only to exterminate the asymptomatic poultry but also to reduce the contact rate between susceptible humans and the poultry environment. Numerical simulations are presented to illustrate the main results.

Classification and Liouville-type theorems for semilinear elliptic equations in unbounded domains

We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.

Population growth and competition models with decay and competition consistent delay.

We derive an alternative expression for a delayed logistic equation in which the rate of change in the population involves a growth rate that depends on the population density during an earlier time period. In our formulation, the delay in the growth term is consistent with the rate of instantaneous decline in the population given by the model. Our formulation is a modification of Arino et al. (J Theor Biol 241(1):109-119, 2006) by taking the intraspecific competition between the adults and juveniles into account. We provide a complete global analysis showing that no sustained oscillations are possible. A threshold giving the interface between extinction and survival is determined in terms of the parameters in the model. The theory of chain transitive sets and the comparison theorem for cooperative delay differential equations are used to determine the global dynamics of the model. We extend our delayed logistic equation to a system modeling the competition between two species. For the competition model, we provide results on local stability, bifurcation diagrams, and adaptive dynamics. Assuming that the species with shorter delay produces fewer offspring at a time than the species with longer delay, we show that there is a critical value, [Formula: see text], such that the evolutionary trend is for the delay to approach [Formula: see text].

Editorial note to: Existence theorem for the Einsteinian gravitational field equations in the non-analytic case, by Yvonne Fourès-Bruhat

Editorial note to: Existence theorem for the Einsteinian gravitational field equations in the non-analytic case, by Yvonne Fourès-Bruhat

Global Limit Theorem for Parabolic Equations with a Potential

We obtain the asymptotics, as $t + |x| \rightarrow \infty$, of the fundamental solution to the heat equation with a compactly supported potential. It is assumed that the corresponding stationary operator has at least one positive eigenvalue. Two regions with different types of behavior are distinguished: inside a certain conical surface in the $(t,x)$ space, the asymptotics is determined by the principal eigenvalue and the corresponding eigenfunction; outside of the conical surface, the main term of the asymptotics is a product of a bounded function and the fundamental solution of the unperturbed operator, with the contribution from the potential becoming negligible if $|x|/t \rightarrow \infty$. A formula for the global asymptotics, as $t + |x| \rightarrow \infty$, of the solution in the entire half-space $t > 0$ is provided. In probabilistic terms, the result describes the asymptotics of the density of particles in a branching diffusion with compactly supported branching and killing potentials.

Republication of: Existence theorem for the Einsteinian gravitational field equations in the non-analytic case

Republication of: Existence theorem for the Einsteinian gravitational field equations in the non-analytic case

Oscillation theorems for second-order nonlinear difference equations with advanced superlinear neutral term

Oscillation theorems for second-order nonlinear difference equations with advanced superlinear neutral term

Liouville theorems for parabolic systems with homogeneous nonlinearities and gradient structure

Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess nontrivial entire solutions) guarantee optimal universal estimates of solutions of related initial and initial-boundary value problems. Assume that \(p>1\) is subcritical in the Sobolev sense. In the case of nonnegative solutions and the system $$\begin{aligned} U_t-\varDelta U=F(U)\quad \hbox {in}\quad \mathbb {R}^n\times \mathbb {R}\end{aligned}$$where \(U=(u_1,\dots ,u_N)\), \(F=\nabla G\) is p-homogeneous and satisfies the positivity assumptions \(G(U)>0\) for \(U\ne 0\) and \(\xi \cdot F(U)>0\) for some \(\xi \in \mathbb {R}^N\) and all \(U\ge 0\), \(U\ne 0\), it has recently been shown in [P. Quittner, Duke Math. J. 170 (2021), 1113–1136] that the parabolic Liouville theorem is true whenever the corresponding elliptic Liouville theorem for the system \(-\varDelta U=F(U)\) is true. By modifying the arguments in that proof we show that the same result remains true without the positivity assumptions on G and F, and that the class of solutions can also be enlarged to contain (some or all) sign-changing solutions. In particular, in the scalar case \(N=1\) and \(F(u)=|u|^{p-1}u\), our results cover the main result in [T. Bartsch, P. Poláčik and P. Quittner, J. European Math. Soc. 13 (2011), 219–247]. We also prove a parabolic Liouville theorem for solutions in \(\mathbb {R}^n_+\times \mathbb {R}\) satisfying homogeneous Dirichlet boundary conditions on \(\partial \mathbb {R}^n_+\times \mathbb {R}\) since such theorem is also needed if one wants to prove universal estimates of solutions of related systems in \(\varOmega \times (0,T)\), where \(\varOmega \subset \mathbb {R}^n\) is a smooth domain. Finally, we use our Liouville theorems to prove universal estimates for particular parabolic systems.

On a comparison theorem for parabolic equations with nonlinear boundary conditions

Abstract In this article, a new type of comparison theorem for some second-order nonlinear parabolic systems with nonlinear boundary conditions is given, which can cover classical linear boundary conditions, such as the homogeneous Dirichlet or Neumann boundary condition. The advantage of our comparison theorem over the classical ones lies in the fact that it enables us to compare two solutions satisfying different types of boundary conditions. As an application of our comparison theorem, we can give some new results on the existence of blow-up solutions of some parabolic equations and systems with nonlinear boundary conditions.

Oscillation Theorems for Conformable Differential Equations

Oscillation Theorems for Conformable Differential Equations

Transformations, symmetries and Noether theorems for differential-difference equations

The first part of this paper develops a geometric setting for differential-difference equations that resolves an open question about the extent to which continuous symmetries can depend on discrete independent variables. For general mappings, differentiation and differencing fail to commute. We prove that there is no such failure for structure-preserving mappings, and identify a class of equations that allow greater freedom than is typical. For variational symmetries, the above results lead to a simple proof of the differential-difference version of Noether’s theorem. We state and prove the differential-difference version of Noether’s second theorem, together with a Noether-type theorem that spans the gap between the analogues of Noether’s two theorems. These results are applied to various equations from physics.

Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics

In this paper, we present a center manifold reduction theorem for quasilinear elliptic equations posed on infinite cylinders that is done without a phase space in the sense that we avoid explicitly reformulating the PDE as an evolution problem. Under suitable hypotheses, the resulting center manifold is finite dimensional and captures all sufficiently small bounded solutions. Compared with classical methods, the reduced ODE on the manifold is more directly related to the original physical problem and also easier to compute. The analysis is conducted directly in Hölder spaces, which is often desirable for elliptic equations. We then use this machinery to construct small bounded solutions to a variety of systems. These include heteroclinic and homoclinic solutions of the anti-plane shear problem from nonlinear elasticity; exact slow moving invasion fronts in a two-dimensional Fisher–KPP equation; and hydrodynamic bores with vorticity in a channel. The last example is particularly interesting in that we find solutions with critical layers and distinctive ‘half cat’s eye’ streamline patterns.

Invariant sample measures and random Liouville type theorem for the two-dimensional stochastic Navier-Stokes equations

In this article, we first prove some sufficient conditions guaranteeing the existence of invariant sample measures for random dynamical systems via the approach of global random attractors. Then we consider the two-dimensional incompressible Navier-Stokes equations with additive white noise as an example to show how to check the sufficient conditions for concrete stochastic partial differential equations. Our results generalize the Liouville type theorem to the random case and reveal that the invariance of the sample measures is a particular situation of the random Liouville type theorem.