Ordinary Differential Inequalities Research Articles

On the critical exponents for a fractional diffusion-wave equation with a nonlinear memory term in a bounded domain

In this paper, we prove sharp blow-up and global existence results for a time fractional diffusion-wave equation with a nonlinear memory term in a bounded domain, where the fractional derivative in time is taken in the sense of the Caputo type. Moreover, we also give a result for nonexistence of global solutions to a wave equation with a nonlinear memory term in a bounded domain. The proof of blow-up results is based on the eigenfunction method and the asymptotic properties of solutions for an ordinary fractional differential inequality.

Rellich inequalities via Riccati pairs on model space forms

We present a simple method for proving Rellich inequalities on Riemannian manifolds with constant, non-positive sectional curvature. The method is built upon simple convexity arguments, integration by parts, and the so-called Riccati pairs, which are based on the solvability of a Riccati-type ordinary differential inequality. These results can be viewed as the higher order counterparts of the recent work by Kajántó, Kristály, Peter, and Zhao, discussing Hardy inequalities using Riccati pairs.

Blow-up time of solutions to a class of pseudo-parabolic equations

In this paper, we study the Dirichlet problem for a semilinear pseudo-parabolic equation. By using the energy estimates and ordinary differential inequalities, we studied the upper and lower bounds of blow-up time of the solutions. The results of this paper extend and complete the results on this model.

Global Existence and Extinction Behaviour for a Doubly Nonlinear Parabolic Equation with Logarithmic Nonlinearity

This paper is mainly focused on the global existence and extinction behaviour of the solutions to a doubly nonlinear parabolic equation with logarithmic nonlinearity. By making use of energy estimates method and a series of ordinary differential inequalities, the global existence of the solution is obtained. Moreover, we give the sufficient conditions on the occurrence (or absence) of the extinction behaviour.

On a semilinear wave equation in anti-de Sitter spacetime: The critical case

In the present paper, we prove the blow-up in finite time for local solutions of a semilinear Cauchy problem associated with a wave equation in anti-de Sitter spacetime in the critical case. According to this purpose, we combine a result for ordinary differential inequalities with an iteration argument by using an explicit integral representation formula for the solution to a linear Cauchy problem associated with the wave equation in anti-de Sitter spacetime in one space dimension.

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.

Global existence and extinction of solutions to a fast diffusion $p$-Laplace equation with special medium void

A fast diffusion p-Laplace equation with special medium void is considered. Global existence of weak solutions is obtained by using comparison principle and ordinary differential inequalities. Moreover, sufficient conditions for the solutions to vanish in finite time or not are derived with the help of Hardy–Littlewood–Sobolev inequality.

Extinction and Non-extinction of Solutions to a Fast Diffusion p-Laplace Equation with Logarithmic Non-linearity

This paper deals with a class of fast diffusion p-Laplace equation with logarithmic non-linearity in a bounded smooth domain with homogeneous Dirichlet boundary condition. By using energy estimates and some ordinary differential inequalities, we study the conditions on extinction and non-extinction of global solutions. The results of this paper extend and complete the previous studies on this equation.

Global Existence and Extinction Singularity for a Fast Diffusive Polytropic Filtration Equation with Variable Coefficient

In this article, we deal with an inhomogeneous fast diffusive polytropic filtration equation. By using the energy estimate approach, Hardy–Littlewood–Sobolev inequality, and a series of ordinary differential inequalities, we prove the global existence result and obtain the conditions on the occurrence of the extinction phenomenon of the weak solution.

Absence of Solutions for a System of Ordinary Differential Inequalities

Absence of Solutions for a System of Ordinary Differential Inequalities

Asymptotic property of singular solutions in some nonstandard parabolic equation

In this paper, we study the asymptotic properties of singular solutions to some nonstandard parabolic equation involving p(x,t)-Laplacian and nonlocal anisotropic sources. First, we show the existence and uniqueness of weak solutions by using the Galerkin’s approximations in the anisotropic Orlicz–Sobolev spaces. Second, in order to determine extinction rates and time of solutions, we prove some ordinary differential inequalities with the Sobolev embedding inequalities. Third, we combine priori estimates and the embedding theorems to show the rates and time estimates of blow-up solutions.

Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source

<p style='text-indent:20px;'>This paper deals with the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. By using some ordinary differential inequalities, the conditions on finite time blow-up of solutions are given with suitable assumptions on initial values. Moreover, the upper and lower bounds of the blow-up time are also investigated.</p>

Extinction behavior for a parabolic $ p $-Laplacian equation with gradient source and singular potential

<abstract><p>We concern with the extinction behavior of the solution for a parabolic $ p $-Laplacian equation with gradient source and singular potential. By energy estimate approach, Hardy-Littlewood-Sobolev inequality, a series of ordinary differential inequalities, and super-solution and sub-solution methods, we obtain the conditions on the occurrence of the extinction phenomenon of the weak solution.</p></abstract>

Existence, Uniqueness and Stabilization of Solutions of a Generalized Telegraph Equation on Star Shaped Networks

The existence, uniqueness, strong and exponential stability of a generalized telegraph equation set on one dimensional star shaped networks are established. It is assumed that a dissipative boundary condition is applied at all the external vertices and an improved Kirchhoff law at the common internal vertex is considered. First, using a general criteria of Arendt-Batty (see Arendt and Batty in Trans. Am. Math. Soc. 306(2):837–852, 1988), combined with a new uniqueness result, we prove that our system is strongly stable. Next, using a frequency domain approach, combined with a multiplier technique and the construction of a new multiplier satisfying some ordinary differential inequalities, we show that the energy of the system decays exponentially to zero.

Adaptive-Gains Enforcing Constraints in Closed-Loop QP Control

In this letter, we revisit an open problem of constraints formulation in the context of task-space control frameworks formulated as quadratic programs. In most inverse dynamics implementations, the decision variables are: robot joints acceleration, interaction forces (mostly physical contacts), and robot torques. Nevertheless, many constraints, like distance and velocity bounds, are not written originally in terms of one of these decision variables. Previous work proposed solutions to formulate and enforce joint limits constraints. Yet, none of them worked properly in closed-loop, specifically when bounds are reached or when they are time-varying. First, we show that constraints like collision avoidance, bounds of center of mass, constraints on field-of-view, Cartesian and velocity bounds on a given link... are written in a generic class. Then, we formulate such a class of constraints with gain-parameterized ordinary differential inequality. An adaptive-gain method enforces systematically such class of constraints, and results on a stable behavior when their bounds (even when they vary with time) are reached in closed-loop. Experimental results performed on a humanoid robot validate our solution on a large panel of constraints.

Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity

<p style='text-indent:20px;'>In this paper, the initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity is invsitgated. First, we establish the local well-posedness of solutions by means of the semigroup theory. Then by using ordinary differential inequalities, potential well theory and energy estimate, we study the conditions on global existence and finite time blow-up. Moreover, the lifespan (i.e., the upper bound of the blow-up time) of the finite time blow-up solution is estimated.</p>

Global Existence, Extinction, and Non-Extinction of Solutions to a Fast Diffusion p-Laplace Evolution Equation with Singular Potential

In this paper, we study a class of fast diffusion p-Laplace equation with singular potential in a bounded smooth domain with homogeneous Dirichlet boundary condition. By using energy estimates, Hardy-Littlewood-Sobolev inequality, and some ordinary differential inequalities, we get the solution of the equation exists globally. Moreover, the conditions of extinction and non-extinction are studied. The results of this paper extend and complete the previous studies on this equation.

Lifespan of Solutions for a Weakly Coupled System of Semilinear Heat Equations

We introduce a direct method to analyze the blow-up of solutions to systems of ordinary differential inequalities, and apply it to study the blow-up of solutions to a weakly coupled system of semilinear heat equations. In particular, we give upper and lower estimates of the lifespan of the solution in the subcritical case.

New general decay results in an infinite memory viscoelastic problem with nonlinear damping

This work is concerned with a viscoelastic equation with a nonlinear frictional damping and a relaxation function satisfying g^{prime }(t) le -xi (t) g^{p}(t), t ge 0, 1le p<frac{3}{2}. We establish general decay rate results using the multiplier method and some properties of non-homogeneous ordinary differential inequalities. These results extend and improve many results in the literature.

Explicit lower bound of blow–up time for an attraction–repulsion chemotaxis system

In this paper we study classical solutions to the zero–flux attraction–repulsion chemotaxis–system(◇){ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)inΩ×(0,t⁎),0=Δv+αu−βvinΩ×(0,t⁎),0=Δw+γu−δwinΩ×(0,t⁎), where Ω is a smooth and bounded domain of R2, t⁎ is the blow–up time and α,β,γ,δ,χ,ξ are positive real numbers. From the literature it is known that under a proper interplay between the above parameters and suitable assumptions on the initial data u(x,0)=u0∈C0(Ω¯), system (◇) has a unique classical solution which becomes unbounded as t↗t⁎. The main result of this investigation is to provide an explicit lower bound for t⁎ estimated in terms of ∫Ωu02dx and attained by means of well–established techniques based on ordinary differential inequalities.