Variable Exponent Of Nonlinearity Research Articles

Blow-Up of Solutions for a Class Quasilinear Wave Equation with Nonlinearity Variable Exponents

This work deals with the blow-up of solutions for a new class of quasilinear wave equation with variable exponent nonlinearities. To clarify more, we prove in the presence of dispersion term − Δ u t t a finite-time blow-up result for the solutions with negative initial energy and also for certain solutions with positive energy. Our results are extension of the recent work (Appl Anal. 2017; 96(9): 1509-1515).

Existence and decay results for a von Karman equation with variable exponent nonlinearities

In this article, we consider a von Karman equation with variable exponent nonlinearities in a bounded domain . We firstly discuss an existence result of solutions by utilizing Faedo‐Galerkin approximation technique. Then, we undertake an investigation of asymptotic stability to the solutions by making use of the multiplier method.

Existence and blow up of solutions for a strongly damped Petrovsky equation with variable-exponent nonlinearities

In this article, we consider a nonlinear plate (or beam) Petrovsky equation with strong damping and source terms with variable exponents. By using the Banach contraction mapping principle we obtain local weak solutions, under suitable assumptions on the variable exponents p(.) and q(.). Then we show that the solution is global if \(p(.) \geq q(.)\). Also, we prove that a solution with negative initial energy and \(p(.)<q(.)\) blows up in finite time. For more information see https://ejde.math.txstate.edu/Volumes/2021/06/abstr.html

On the existence and stability of a nonlinear wave system with variable exponents

Problems with variable exponents have attracted a great deal of attention lately and various existence, nonexistence and stability results have been established. The importance of such problems has manifested due to the recent advancement of science and technology and to the wide application in areas such as electrorheological fluids (smart fluids) which have the property that the viscosity changes drastically when exposed to heat or electrical fields. To tackle and understand these models, new sophisticated mathematical functional spaces have been introduced, such as the Lebesgue and Sobolev spaces with variable exponents. In this work, we are concerned with a system of wave equations with variable-exponent nonlinearities. This system can be regarded as a model for interaction between two fields describing the motion of two “smart” materials. We, first, establish the existence of global solutions then show that solutions of enough regularities stabilize to the rest state ( 0 , 0 ) either exponentially or polynomially depending on the range of the variable exponents. We also present some numerical tests to illustrate our theoretical findings.

Ground state solution for a class of supercritical nonlocal equations with variable exponent

In this paper, we obtain the existence of positive critical point with least energy for a class of functionals involving nonlocal and supercritical variable exponent nonlinearities by applying the variational method and approximation techniques. We apply our results to the supercritical Schrödinger–Poisson type systems and supercritical Kirchhoff type equations with variable exponent, respectively.

Cauchy problem for hyperbolic equations of third order with variable exponent of nonlinearity

We investigate weak solutions of the Cauchy problem for the third order hyperbolic equations with variable exponent of the nonlinearity. The problem is considered in some classes of functions namely in Lebesgue spaces with variable exponents. The sufficient conditions of the existence and uniqueness of the weak solutions to given problem are found.

On the decay of solutions of a viscoelastic wave equation with variable sources

In this paper, we consider the following viscoelastic problem with variable exponent nonlinearities: where m(.) and q(.) are two functions satisfying specific conditions. This type of problems appears in fluid dynamics, the electrorheological fluids (smart fluids), which show changing (often dramatically) in the viscosity when an electrical field is applied. The Lebesgue and Sobolev spaces with variable exponents are efficient tools to analyze such problems. In this work, we prove a global existence result using the well‐depth method and establish explicit and general decay results under a very general assumption on the relaxation function. Our results extend and generalize many results in the literature.

Blow-up of solutions for a Kirchhoff type equation with variable-exponent nonlinearities

Abstract This paper deals with a Kirchhoff type equation with variable exponent nonlinearities, subject to a nonlinear boundary condition. Under appropriate conditions and regarding arbitrary positive initial energy, it is proved that solutions blow up in a finite time. Moreover, we obtain the upper bound estimate of the blow-up time.

Global Nonexistence of Solutions for a Class of Viscoelastic Lamé System

The aim of this work is to study the blow-up result for a Lame system of viscoelastic equation with variable-exponent nonlinearities and strong damping. Under some suitable conditions on the coefficients, memory term and initial data, we proved a global nonexistence result for the weak solution with positive initial energy.

On the viscoelastic equation with Balakrishnan-Taylor damping and nonlinear boundary/interior sources with variable-exponent nonlinearities

This work is devoted to the study of a nonlinear viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping and nonlinear boundary interior sources with variable exponents. Under appropriate assumptions, we establish a uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function.

Finite time blow up of solutions of the Kirchhoff-type equation with variable exponents

In this work, we investigate the following Kirchhoff-type equation with variable exponent nonlinearities u_{tt}-M(‖∇u‖²)△u+|u_{t}|^{p(x)-2}u_{t}=|u|^{q(x)-2}u. We proved the blow up of solutions in finite time by using modified energy functional method.

Существование решений квазилинейных эллиптических уравнений в пространствах Музилака-Орлича-Соболева для неограниченных областей

Рассматривается вопрос существования решений задачи Дирихле для нелинейных эллиптических уравнений второго порядка в неограниченных областях. Ограничения на структуру квазилинейных уравнений формулируются в терминах специального класса выпуклых функций - обобщенных $N$-функций. А именно, нелинейности определяются функциями Музилака-Орлича такими, что дополнительные к ним функции подчиняются $\Delta_2$-условию. Соответствующее пространство Музилака-Орлича-Соболева не обязано быть рефлексивным. Именно этот факт является существенной проблемой, поскольку теорема для псевдомонотонных операторов здесь не применима. Для рассматриваемого класса уравнений доказательство теоремы существования проводится на основе абстрактной теоремы для дополнительных систем. Важным инструментом, который позволил обобщить имеющиеся результаты существования решений рассматриваемых уравнений для ограниченных областей на неограниченные области, является теорема вложения пространств Музилака-Орлича-Соболева. Таким образом, в работе найдены условия на структуру квазилинейных уравнений в терминах функций Музилака-Орлича, достаточные для разрешимости задачи Дирихле в неограниченных областях. Кроме того, приведены примеры уравнений, показывающие, что класс нелинейностей, рассматриваемый в работе, шире, чем нестепенные и степенные нелинейности.

BLOW UP OF SOLUTIONS FOR A NONLINEAR VISCOELASTIC WAVE EQUATIONS WITH VARIABLE EXPONENTS

The aim of this work is to study the blow up of solutions for the viscoelastic wave equation with variable exponents in a bounded domain. Our result extends the one in Messaoudi1 to problems with variable exponent nonlinearities.

Initial–boundary value problems for nonlinear elliptic–parabolic equations with variable exponents of nonlinearity in unbounded domains without conditions at infinity

The paper is devoted to results of the investigation of the initial–boundary value problems for the nonlinear second order elliptic–parabolic equations with variable exponents of the nonlinearity in unbounded domains. We consider weak solutions which belong to the Lebesgue and Sobolev spaces with variable exponents. Under certain conditions on data-in the existence and uniqueness of the weak solutions are proved without conditions at infinity. Also some estimates of the weak solutions are obtained.

Existence and asymptotic stability for the semilinear wave equation with variable-exponent nonlinearities

In this paper, we review a class of semilinear wave equation with variable-exponent nonlinearities and boundary conditions. This work is devoted to prove the existence of solutions and uniform decay rates of the wave equation with boundary damping and source terms.

On solutions of anisotropic elliptic equations with variable exponent and measure data

ABSTRACTThe Dirichlet problem in arbitrary domains for a wide class of anisotropic elliptic equations of the second order with variable exponent nonlinearities and the right-hand side as a measure is considered. The existence of an entropy solution in anisotropic Sobolev spaces with variable exponents is established. It is proved that the obtained entropy solution is a renormalized solution of the considered problem.

Quasilinear parabolic equations with p(x)-Laplacian diffusion terms and nonlocal boundary conditions

In this study, we prove the existence of local solution for a quasi linear generalized parabolic equation with nonlocal boundary conditions for an elliptic operator involving the variable-exponent nonlinearities, using Faedo-Galerkin arguments and compactness method.

Global existence and stability of a nonlinear wave equation with variable-exponent nonlinearities

ABSTRACTIn this paper, we consider a nonlinear wave equation with damping and source terms of variable-exponent types. First, we use the stable-set method to prove a global result. Then, by applying an integral inequality due to Komornik, we obtain the stability result.

Decay for solutions of a nonlinear damped wave equation with variable-exponent nonlinearities

In this paper, we consider the following nonlinear waveequation with variable exponents: utt−div(|∇u|r(⋅)−2∇u)+|ut|m(⋅)−2ut=0.By using a lemma by Komornik, we prove the decay estimates for the solution under suitable assumptions on the variable exponents m,r and the initial data. We also give two numerical applications to illustrate our theoretical results.

On behaviour of solutions for a nonlinear viscoelastic equation with variable-exponent nonlinearities

This paper deals with a viscoelastic wave equation with variable-exponent nonlinearities, subject to nonlinear boundary feedback. Under appropriate conditions, a general decay result associated to solution energy is proved. It is also shown that regarding arbitrary positive initial energy, solutions blow-up in a finite time.