First-order Differential Inequality Research Articles

Properties of second order differential equations with advanced and delay argument

In this paper, we study the second-order linear delay differential equation of the form (E)y′′(t)+p(t)y(τ(t))=0.We establish new oscillation criteria for (1.1), which improve a number of related ones in the literature. Our approach essentially involves establishing stronger monotonicity properties for the positive solutions of (1.1) than those presented in known works. Two main approaches for the investigation of (1.1) will be used, namely a comparison principle with first-order delay differential inequalities and generalization of very effective Koplatadze’s technique. We illustrate the improvement over the known results by applying and comparing our method with the other known results for (1.1).

On the distribution of adjacent zeros of solutions to first-order neutral differential equations

The purpose of this paper is to study the distribution of zeros of solutions to a first-order neutral differential equation of the form \begin{equation*} \left[x(t) + p(t) x(t-\tau)\right]' + q(t) x(t-\sigma) = 0, \quad t \geq t_0, \end{equation*} where $p\in C([t_0,\infty),[0,\infty))$, $q \in C([t_0,\infty),(0,\infty))$, $\tau,\sigma>0$, and $\sigma>\tau$. We obtain new upper bound estimates for the distance between consecutive zeros of solutions, which improve upon many of the previously known ones. The results are formulated so that they can be generalized without much effort to equations for which the distribution of zeros problem is related to the study of this property for a first-order delay differential inequality. The strength of our results is demonstrated viatwo illustrative examples.

Phragmén–Lindelöf type alternative results for the solutions to generalized heat conduction equations

This paper investigates the spatial behavior of the solutions of the generalized heat conduction equations on a semi-infinite cylinder by means of a first-order differential inequality. We consider three kinds of semi-infinite cylinders with boundary conditions of Dirichlet type. For each cylinder, we prove the Phragmén–Lindelöf alternative for the solutions. In the case of decay, we also present a method for obtaining explicit bounds for the total energy.

New Oscillation Results of Even-Order Emden–Fowler Neutral Differential Equations

The objective of this study is to establish new sufficient criteria for oscillation of solutions of even-order delay Emden-Fowler differential equations with neutral term rıyı+mıygın−1γ′+∑i=1jqiıyγμiı=0. We use Riccati transformation and the comparison with first-order differential inequalities to obtain theses criteria. Moreover, the presented oscillation conditions essentially simplify and extend known criteria in the literature. To show the importance of our results, we provide some examples. Symmetry plays an essential role in determining the correct methods for solutions to differential equations.

Blow-up results of the positive solution for a class of degenerate parabolic equations

Abstract This paper is devoted to discussing the blow-up problem of the positive solution of the following degenerate parabolic equations: ( r ( u ) ) t = div ( ∣ ∇ u ∣ p ∇ u ) + f ( x , t , u , ∣ ∇ u ∣ 2 ) , ( x , t ) ∈ D × ( 0 , T ∗ ) , ∂ u ∂ ν + σ u = 0 , ( x , t ) ∈ ∂ D × ( 0 , T ∗ ) , u ( x , 0 ) = u 0 ( x ) , x ∈ D ¯ . \left\{\begin{array}{ll}{(r\left(u))}_{t}={\rm{div}}(| \nabla u{| }^{p}\nabla u)+f\left(x,t,u,| \nabla u\hspace{-0.25em}{| }^{2}),& \left(x,t)\in D\times \left(0,{T}^{\ast }),\\ \frac{\partial u}{\partial \nu }+\sigma u=0,& \left(x,t)\in \partial D\times \left(0,{T}^{\ast }),\\ u\left(x,0)={u}_{0}\left(x),& x\in \overline{D}.\end{array}\right. Here p > 0 p\gt 0 , the spatial region D ⊂ R n ( n ≥ 2 ) D\subset {{\mathbb{R}}}^{n}\hspace{0.33em}\left(n\ge 2) is bounded, and its boundary ∂ D \partial D is smooth. We give the conditions that cause the positive solution of this degenerate parabolic problem to blow up. At the same time, for the positive blow-up solution of this problem, we also obtain an upper bound of the blow-up time and an upper estimate of the blow-up rate. We mainly carry out our research by means of maximum principles and first-order differential inequality technique.

Oscillation Results for Second-Order Neutral Functional Differential Equations

In this article, we obtain several comparison theorems for the second-order neutral differential equation , t ⩾ t 0, where γ, λ, β are ratios of positive odd integers. We compare above equation with the first-order differential inequalities in the sense that the absence of the eventually positive solutions of these first-order inequalities implies the oscillation of the studied equation.

Blow-up problem of quasilinear weakly coupled reaction-diffusion systems with Neumann boundary conditions

In this paper, the blow-up solutions of the following reaction-diffusion systems are studied:{(h(u))t=∇⋅(b(u)∇u)+f(x,u,v,t),(k(v))t=∇⋅(d(v)∇v)+g(x,u,v,t),(x,t)∈D×(0,T),∂u∂n=0,∂v∂n=0,(x,t)∈∂D×(0,T),u(x,0)=u0(x),v(x,0)=v0(x),x∈D‾, where D⊂Rn(n≥2) is a bounded region and the boundary ∂D of D is smooth. Sufficient conditions to ensure the existence of the blow-up solution of this problem are obtained. For the blow-up solution, we also get an upper bound on the blow-up time and an upper estimate of the blow-up rate. Our research mainly relies on the use of the maximum principles of weakly coupled parabolic systems and the first-order differential inequality technique.

Attenuation of an external signal in a thermoelastic material with triple porosity in local thermal non-equilibrium

This work aims to highlight, by a series of spatial estimates, the effect of the presence of pore system and local thermal non-equilibrium properties in a triple porosity material matrix on the transmission depth of a thermo-mechanical signal. Two kind of spatial behaviors are established, for which the spatial response obeys a linear first-order differential inequality with two kind different coefficients: one is time independent and the other is time dependent. The first differential inequality furnishes some exponential decaying spatial estimates with uniform in time exponent and such estimates are recommended to be used for appropriately large values of the time variable. While the second first-order differential inequality furnishes some exponential decaying spatial estimates with time dependent exponent, thus showing a rapid spatial decrease of the solution for appropriate short values of time. These results are compared with their counterparts without mechanical effects, that is for purely pore system or local thermal non-equilibrium effects theories: this is done for appropriate short and large values of the time variable.

Blow-up analysis of solutions for weakly coupled degenerate parabolic systems with nonlinear boundary conditions

This paper is devoted to the study of the blow-up solutions of the following weakly coupled degenerate parabolic systems with nonlinear boundary conditions: ut=div|∇u|p∇u+f(x,u,v,t),vt=div|∇v|q∇v+g(x,u,v,t),(x,t)∈D×(0,T∗),∂u∂n=b(u),∂v∂n=d(v),(x,t)∈∂D×(0,T∗),u(x,0)=u0(x),v(x,0)=v0(x),x∈D¯.Here p>0,q>0, D is a bounded spatial region in Rn(n≥2), and the boundary ∂D is smooth. We mainly combine the maximum principles of the weakly coupled parabolic systems with the first-order differential inequality technique to discuss the blow-up phenomenon of the above problem. Sufficient conditions for the blow-up of the nonnegative solution (u,v) of this problem are given. In addition, for the nonnegative blow-up solution (u,v), we also obtain an upper bound on the blow-up time and an upper estimate of the blow-up rate.

Structural stability for the Boussinesq equations interfacing with Darcy equations in a bounded domain

A priori bounds were derived for the flow in a bounded domain for the viscous-porous interfacing fluids. We assumed that the viscous fluid was slow in Omega _{1}, which was governed by the Boussinesq equations. For a porous medium in Omega _{2}, we supposed that the flow satisfied the Darcy equations. With the aid of these a priori bounds we were able to demonstrate the result of the continuous dependence type for the Boussinesq coefficient λ. Following the method of a first-order differential inequality, we can further obtain the result that the solution depends continuously on the interface boundary coefficient α. These results showed that the structural stability is valid for the interfacing problem.

Third-order neutral differential equations of the mixed type: Oscillatory and asymptotic behavior.

In this work, by using both the comparison technique with first-order differential inequalities and the Riccati transformation, we extend this development to a class of third-order neutral differential equations of the mixed type. We present new criteria for oscillation of all solutions, which improve and extend some existing ones in the literature. In addition, we provide an example to illustrate our results.

Spatial decay estimates for the Fochheimer equations interfacing with a Darcy equations

<abstract><p>Spatial decay estimates for the Fochheimer fluid interfacing with a Darcy flow in a semi-infinite pipe was studied. The exponential decay result can be obtained by integrating a first-order differential inequality. The result can be seen as the usage of Saint-Venant's principle for the interfacing fluids.</p></abstract>

Some New Oscillation Results for Fourth-Order Neutral Differential Equations with a Canonical Operator

By this work, our aim is to study oscillatory behaviour of solutions to 4th-order differential equation of neutral typeLy′+∑j=1kqjyzβgjy=0whereLy=ξyw‴yα,wy:=zy+ryzg˜y. By using the comparison method with first-order differential inequality, we find new oscillation conditions for this equation.

Lower Bound for the Blow-Up Time for the Nonlinear Reaction-Diffusion System in High Dimensions

In this paper, we study the blow-up phenomenon for a nonlinear reaction-diffusion system with time-dependent coefficients under nonlinear boundary conditions. Using the technique of a first-order differential inequality and the Sobolev inequalities, we can get the energy expression which satisfies the differential inequality. The lower bound for the blow-up time could be obtained if blow-up does really occur in high dimensions.

Some oscillation theorems for nonlinear second-order differential equations with an advanced argument

The objective in this work is to study oscillation criteria for second-order quasi-linear differential equations with an advanced argument. We establish new oscillation criteria using both the comparison technique with first-order advanced differential inequalities and the Riccati transformation. The established criteria improve, simplify and complement results that have been published recently in the literature. We illustrate the results by an example.

Global and blow-up solutions for a nonlinear reaction diffusion equation with Robin boundary conditions

In the paper, we investigate global and blow-up solutions for a class of nonlinear reaction diffusion equations with Robin boundary conditions. By using auxiliary functions and a first-order differential inequality technique, we establish conditions on the data to prove the existence of global solution. Moreover, based on maximum principles, we obtain a criterion that guarantees the occurrence of the blow-up. When blow-up occurs, we discuss an upper bound and a lower bound on blow-up time. At last, we apply two examples to illustrate our main results.

Blow-up phenomena for a class of nonlinear reaction-diffusion equations under nonlinear boundary conditions

ABSTRACTThe paper considers the blow-up solution for the following nonlinear reaction-diffusion equations under nonlinear boundary conditions: where p>2 and is a bounded convex domain with smooth boundary . By means of constructing auxiliary functions and a first-order differential inequality technique, we obtain the upper and lower bounds on blow-up time under some appropriate assumptions. Moreover, some examples are also presented as application of our main results.

Improved oscillation results for second-order half-linear delay differential equations

In this paper, we study the second-order half-linear delay differential equation of the form \[(r(t)(y'(t))^\alpha)'+q(t)y^\alpha(\tau(t))= 0.\:\:\:(E)\] We establish new oscillation criteria for (E), which improve a number of related ones in the literature. Our approach essentially involves establishing sharper estimates for the positive solutions of (E) than those presented in known works and a comparison principle with first-order delay differential inequalities. We illustrate the improvement over the known results by applying and comparing our method with the other known methods on the particular example of Euler-type equations.

General problems of explicit integration of differential inequalities

We consider the problem of the explicit search for all solutions of a first-order nonstrict differential inequality. We use the formula of the general solution of the corresponding differential equation. Using an analog of the method of arbitrary constant variation or, in other words, a straightening diffeomorphism, we reduce the original inequality to the simplest form ẋ ≤ 0 or ẋ ≥ 0. Even if the equation is considered in the existence and uniqueness region, theoretical and practical problems arise. The first problem is related to the extension of solutions (i.e., to the interval of determination). The second problem is that the general solution may consist of several functions given on different intervals of the equation domain. As a result, the resulting inequality also may have a solution that is composed of different functions. The situation becomes more complicated when the equation has points of branching. In this case, the method of comparison of theorems cannot be used. In this paper, we describe a method for solving differential inequalities and estimating their solutions for this case as well. The result obtained in this study provides a unified approach to many theorems on differential inequalities available in the literature.

An inverse Hölder inequality and its application in lower bound estimates for blow-up time

This paper deals with the lower bound for blow-up solutions to a nonlinear viscoelastic hyperbolic equation. An inverse Hölder inequality with the correction constant is employed to overcome the difficulty caused by the failure of the embedding inequality W01,r(Ω)↪L2α−2(Ω)(r⁎+22<α<r⁎=NrN−r) and the lack of a version of the Gagliardo–Nirenberg inequality. Moreover, a lower bound for blow-up time is obtained by establishing a first-order differential inequality. This result is a continuation of an earlier work [1].