Nonlinear Differential Inequalities Research Articles

On a Dini type blow-up condition for solutions of nonlinear higher order differential inequalities

We obtain a Dini type blow-up condition for solutions of the differential inequality∑a=m∂αaα(x,u)≥g(u)inℝn where m,n≥1are integers and aαand gare some functions.

On a Dini Type Blow-Up Condition for Solutions of Higher Order Nonlinear Differential Inequalities

On a Dini Type Blow-Up Condition for Solutions of Higher Order Nonlinear Differential Inequalities

A view on Liouville theorems in PDEs

Abstract Our review of Liouville theorems includes a special focus on nonlinear partial differential equations and inequalities.

Portfolio-consumption choice problem with voluntary retirement and consumption constraints

We study the investment–consumption choice problem with voluntary retirement and upside/downside consumption constraints in infinite horizon, which can be formulated as a two-phase mixed stochastic control problem including stopping time coupled with controls. By verification theorem, we show the strong solution of a fully nonlinear differential variational inequality, satisfying some special properties, is the value function of this problem. Then we eliminate the nonlinearity through Legendre transformation, and obtain the closed-form formulas of the optimal investment–consumption feedback strategy, the optimal retirement strategy and the value function by solving the dual equation. Since the introduction of consumption constraints, these formulas are both segmented and semi-explicit, and bring certain difficulties to the analysis of the properties of the optimal strategies and the value function. The regularity of the feedback strategy and the value function are obtained, as well as their monotonicity or non-monotonicity with respect to various parameters are discussed by some stochastic analysis methods and differential equation techniques.

Nonexistence for fractional differential inequalities and systems in the sense of Erdélyi-Kober

<p>Nonexistence theorems constitute an important part of the theory of differential and partial differential equations. Motivated by the numerous applications of fractional differential equations in diverse fields, in this paper, we studied sufficient conditions for the nonexistence of solutions (or, equivalently, necessary conditions for the existence of solutions) for nonlinear fractional differential inequalities and systems in the sense of Erdélyi-Kober. Our approach is based on nonlinear capacity estimates specifically adapted to the Erdélyi-Kober fractional operators and some integral inequalities.</p>

Stability and Stabilization of Nonlinear Stochastic Systems With Synchronous and Asynchronous Switching Parameters to the States.

This article is concerned with the stability and stabilization of nonlinear hybrid stochastic systems. First, the concepts of synchronous and asynchronous switching parameters to the system states are proposed for a generalization of the asynchronous control. As preliminaries, a hybrid nonlinear differential inequality is established for coping with the synchronous models, and the -continuity of the system parameters driven by Markov chains is established for coping with the asynchronous models by the attribute of Markov chains for less conservativeness. Second, stability criteria with moment exponential estimates are established for two kinds of system models under the nonlinear growth condition. Third, the stabilization problem is illustrated based on the stability criteria and Riccati like matrix equations. A corollary is given to improve some stability theorems obtained in the related literature. New initial condition is formally proposed and explained for the nonlinear stochastic systems. Finally, a numerical example with simulation is proposed to illustrate the method, verify the conclusions of this article, and show the superiority of this work.

Nonexistence results for nonlinear fractional differential inequalities involving weighted fractional derivatives

In this paper, we study the nonexistence of global solutions for certain classes of nonlinear fractional differential inequalities involving weighted fractional derivatives and a singular potential function. Namely, using the test function method with a judicious choice of the test function, we obtain sufficient criteria depending on the parameters of the considered problems, under which we have absence of global solutions. Next, some special cases of the potential function are discussed.

Challenges in Optimization with Complex PDE-Systems

The workshop concentrated on various aspects of optimization problems with systems of nonlinear partial differential equations (PDEs) or variational inequalities (VIs) as constraints. In particular, discussions around several keynote presentations in the areas of optimal control of nonlinear or non-smooth systems, optimization problems with functional and discrete or switching variables leading to mixed integer nonlinear PDE constrained optimization, shape and topology optimization, feedback control and stabilization, multi-criteria problems and multiple optimization problems with equilibrium constraints as well as versions of these problems under uncertainty or stochastic influences, and the respectively associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, aspects of optimal control of data-driven PDE constraints (e.g. related to machine learning) were addressed.

Sharp lifespan estimates for the weakly coupled system of semilinear damped wave equations in the critical case

The open question, which seems to be also the final part, in terms of studying the Cauchy problem for the weakly coupled system of damped wave equations or reaction-diffusion equations, is so far known as the sharp lifespan estimates in the critical case. In this paper, we mainly investigate lifespan estimates for solutions to the weakly coupled system of semilinear damped wave equations in the critical case. By using a suitable test function method associated with nonlinear differential inequalities, we catch upper bound estimates for the lifespan. Moreover, we establish polynomial-logarithmic type time-weighted Sobolev spaces to obtain lower bound estimates for the lifespan in low spatial dimensions. Then, together with the derived lifespan estimates, new and sharp results on estimates for the lifespan in the critical case are claimed. Finally, we give an application of our results to the semilinear reaction-diffusion system in the critical case.

Necessary Conditions for the Existence of Global Solutions to Nonlinear Fractional Differential Inequalities and Systems

In this paper, fractional differential inequalities and systems of fractional differential inequalities involving fractional derivatives in the sense of Caputo are investigated. Namely, necessary conditions for the existence of global solutions are obtained. Our approach is based on the test function method and some integral inequalities.

On the absence of global solutions to two-times-fractional differential inequalities involving Hadamard-Caputo and Caputo fractional derivatives

<abstract><p>In this paper, we consider a two-times nonlinear fractional differential inequality involving both Hadamard-Caputo and Caputo fractional derivatives of different orders, with a singular potential term. We obtain sufficient criteria depending on the parameters of the problem, for which a global solution does not exist. Some examples are provided to support our main results.</p></abstract>

Stochastic Robust H∞ Decentralized Network Formation Tracking Control of Large-Scale Team Satellites via Event-Triggered Mechanism

The team formation of large-scale satellites for communication services of future smart cities in the 5G and 6G era is an important research topic currently. In this study, a robust H∞ with an event-triggered mechanism is proposed for the formation of large-scale satellites for communication tasks. At first, a satellite in the team formation is described by a nonlinear stochastic system with intrinsic random fluctuation, external disturbance, and coupling from other satellites. The proposed team formation tracking control for each satellite needs to consider the effect of network-induced delay and packet-dropout as well as save communication resources via an event-triggered mechanism. Further, the desired team formation of large-scale satellites can be prescribed by a set of reference models with the desired formation shape embedded in their reference inputs. Therefore, the large-scale team formation design problem can be simplified as an independentH∞ network model reference tracking control design problem for each satellite to efficiently attenuate the worst-effect of external disturbance, coupling of other satellites, intrinsic random fluctuation, network-induced delay, and packet-dropout on the model reference tracking performance. In order to avoid solving a nonlinear partial differential Hamillion-Jacobin inequality(HJI) for the H∞ decentralized network formation control for each satellite in the team the formation, the T-S fuzzy model is employed to interpolate several local linearized systems to approximate nonlinear satellite systems. Then the HJI in the H∞ decentralized network team formation design problem can be transformed into a set of linear matrix inequalities (LMIs) which can be easily solved by LMI TOOLBOX in MATLAB. Finally, a simulation example of team formation composed of ten satellites is given to illustrate the design procedure and to validate the proposed method in comparison with other methods.

Nonlinear dynamics of a new seasonal epidemiological model with age-structure and nonlinear incidence rate

In this article, we study the dynamics of a new proposed age-structured population mathematical model driven by a seasonal forcing function that takes into account the variability of the climate. We introduce a generalized force of infection function to study different potential disease outcomes. Using nonlinear analysis tools and differential inequalities theorems, we obtain sufficient conditions that guarantee the existence of a positive periodic solution. Moreover, we provide sufficient conditions that assure the global attractivity of the positive periodic solution. Numerical results corroborate the theoretical results in the sense that the solutions are positive and the periodic solution is a global attractor. This type of models are important, since they take into account the variability of the weather and the impact on some epidemics such as the current COVID-19 pandemic.

Nonexistence Results for Some Classes of Nonlinear Fractional Differential Inequalities

We study the nonexistence of global solutions for new classes of nonlinear fractional differential inequalities. Namely, sufficient conditions are provided so that the considered problems admit no global solutions. The proofs of our results are based on the test function method and some integral estimates.

Blow-Up Case for Some Nonlinear Differential Inequalities

Blow-Up Case for Some Nonlinear Differential Inequalities

Boundary estimates for superharmonic functions and solutions of semilinear elliptic equations with source

In a certain Lipschitz domain $$\Omega \subset {\mathbb {R}}^n$$ , we establish the boundary Harnack principle for positive superharmonic functions satisfying the nonlinear differential inequality $$-\Delta u\le cu^p$$ , where $$c>0$$ and $$1<p<(n+\alpha )/(n+\alpha -2)$$ with constant $$\alpha $$ regarding the lower bound estimate of the Green function on $$\Omega $$ . An argument combined estimates for certain Green potentials and iteration methods enables us to prove it. Results are applicable to positive solutions of semilinear elliptic equations like $$-\Delta u=a(x)u^p$$ with a(x) being nonnegative and bounded on $$\Omega $$ . Also, we present an a priori estimate and a removability theorem for positive solutions having isolated singularities at a boundary point. The former extends one given by Bidaut-Veron and Vivier (Rev Mat Iberoam 16:477–513, 2000) in the case where $$\Omega $$ has a smooth boundary and $$a(x)\equiv 1$$ .

Absence of Solutions to Certain Nonlinear Inequalities with Transformed Arguments

We obtain sufficient conditions for the absence of solutions to certain nonlinear partial differential inequalities with transformed arguments.

Nonexistence of Solutions for Nonlinear Differential Inequalities with Singularities on Unbounded Sets

We study some new Liouville theorems for nonlinear differential inequalities with gradient terms and singular variable coefficients that have singularities on unbounded sets. The proofs are based on the test function method developed by Mitidieri and Pohozaev.

Krylov's Boundary Gradient Type Estimates for Solutions to Fully Nonlinear Differential Inequalities with Quadratic Growth on the Gradient

In this paper we prove Krylov's boundary gradient type estimates (and regularity) for solutions to fully nonlinear differential inequalities with unbounded coefficients and quadratic growth on the ...

A comparison theorem for super- and subsolutions of $\mathbf {\nabla ^2 u + f (u) = 0}$ and its application to water waves with vorticity

A comparison theorem is proved for a pair of solutions that satisfy opposite nonlinear differential inequalities in a weak sense. The nonlinearity is of the form f (u) with f belonging to the class ...