Variable Exponent Of Nonlinearity Research Articles

On a biharmonic coupled system with non-standard nonlinearity: Existence, blow up and numerics

In this paper, we consider a coupled system of two biharmonic equations with damping and source terms of variable-exponent nonlinearities, supplemented with initial and mixed boundary conditions. We establish an existence and uniqueness result of a weak solution, under suitable assumptions on the variable exponents. Then, we show that solutions with negative-initial energy blow up in finite time. To illustrate our theoritical findings, we present two numerical examples.

Nonlocal hyperbolic Stokes system with variable exponent of nonlinearity

In this paper, we study the problem for a nonlinear hyperbolic Stokes system of the second order with an integral term.Sufficient conditions for the uniqueness of the weak solution of this problem are found in a bounded domain. The nonlinear term of the system contains a variable exponent of nonlinearity, which is a function of spatial variables.The problem is studied in ordinary Sobolev spaces and generalized Lebesgue spaces, which is quite natural in this case.

Existence and blow-up results for a weak-viscoelastic plate equation involving $$p(x)-$$Laplacian operator and variable-exponent nonlinearities

Existence and blow-up results for a weak-viscoelastic plate equation involving $$p(x)-$$Laplacian operator and variable-exponent nonlinearities

General decay and blow up of solutions for a plate viscoelastic p(x)-Kirchhoff type equation with variable exponent nonlinearities and boundary feedback

In this paper, we consider a plate viscoelastic p(x)–Kirchhoff type equation with variable-exponent nonlinearities of the form associated with initial and boundary feedback. Under appropriate conditions on p(·), m(·) and q(·), general decay result along the solution energy is proved. By introducing a suitable auxiliary function, it is also shown that regarding negative initial energy and a suitable range of variable exponents, solutions blow up in a finite time.

Hyperbolic Stokes system of the third order with variable exponent of nonlinearity

Hyperbolic Stokes system of the third order with variable exponent of nonlinearity

Global existence, asymptotic stability and blow up of solutions for a nonlinear viscoelastic plate equation involving (p(x), q(x))-Laplacian operator

This study aims at investigating the global existence, asymptotic stability and blow up of solutions for a nonlinear viscoelastic fourth-order (p(x); q(x))-Laplacian equation with variable-exponent nonlinearities. First, we prove the global existence of solutions, and next, we show that the solutions are asymptotically stable if initial data p(x) and q(x) are in the appropriate range. Moreover, under suitable conditions on initial data, we prove that there exists a finite time in which some solutions blow up with positive as well as negative initial energies.

The Exponential Growth of Solution, Upper and Lower Bounds for the Blow-Up Time for a Viscoelastic Wave Equation with Variable- Exponent Nonlinearities

This paper aims to study the model of a nonlinear viscoelastic wave equation with damping and source terms involving variable-exponent nonlinearities. First, we prove that the energy grows exponentially, and thus in 𝐿p2 and 𝐿p1 norms. For the case 2 ≤ 𝑘(. ) < 𝑝(. ), we reach the exponential growth result of a blowup in finite time with positive initial energy and get the upper bound for the blow-up time. For the case 𝑘(. ) = 2, we use the concavity method to show a finite time blow-up result and get the upper bound for the blow-up time. Furthermore, for the case 𝑘(. ) ≥ 2, under some conditions on the data, we give a lower bound for the blow-up time when the blow-up occurs.

Initial-boundary value problem for higher-orders nonlinear elliptic-parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinity

Initial-boundary value problems for parabolic and elliptic-parabolic (that is degenerated parabolic) equations in unbounded domains with respect to the spatial variables were studied by many authors. It is well known that in order to guarantee the uniqueness of the solution of the initial-boundary value problems for linear and some nonlinear parabolic and elliptic-parabolic equations in unbounded domains we need some restrictions on behavior of solution as $|x|\to +\infty$ (for example, growth restriction of solution as $|x|\to +\infty$, or the solution to belong to some functional spaces).Note, that we need some restrictions on the data-in behavior as$|x|\to +\infty$ for the initial-boundary value problemsfor equations considered above to be solvable.
 However, there are nonlinear parabolic equations for whichthe corresponding initial-boundary value problems are uniquely solvable withoutany conditions at infinity.
 We prove the unique solvability of the initial-boundary value problemwithout conditions at infinity for some of the higher-orders anisotropic parabolic equationswith variable exponents of the nonlinearity. A priori estimate of the weak solutionsof this problem was also obtained. As far as we know, the initial-boundary value problem for the higher-orders anisotropic elliptic-parabolic equations with variable exponents of nonlinearity in unbounded domains were not considered before.

The Dirichlet problem in an unbounded cone-like domain for second order elliptic quasilinear equations with variable nonlinearity exponent

In this paper we consider the Dirichlet problem for quasi-linear second-order elliptic equation with the m ( x ) -Laplacian and the strong nonlinearity on the right side in an unbounded cone-like domain. We study the behavior of weak solutions to the problem at infinity and we find the sharp exponent of the solution decreasing rate. We show that the exponent is related to the least eigenvalue of the eigenvalue problem for the Laplace–Beltrami operator on the unit sphere.

Blow-up of solutions for wave equation with multiple ?(x)-Laplacian and variable exponent nonlinearities

We consider an initial value problem related to the equation \begin{equation*} u_{tt}-{div}\left( \left\vert \nabla u\right\vert ^{m\left( x\right) -2}\nabla u\right) -{div}\left( \left\vert \nabla u_{t}\right\vert ^{r\left( x\right) -2}\nabla u_{t}\right) -\gamma \Delta u_{t}=\left\vert u\right\vert ^{p\left( x\right) -2}u, \end{equation*} with homogeneous Dirichlet boundary condition in a bounded domain $\Omega $. Under suitable conditions on variable-exponent $m\left( .\right) ,$ $r\left( .\right), $ and $p\left( .\right) ,$ we prove a blow-up of solutions with negative initial energy.

On the existence and decay of a viscoelastic system with variable-exponent nonlinearity

In the present paper, we discuss the existence and the long-time behavior of a system of two viscoelastic equations with variable-exponent nonlinearities acting in both equations. First, we establish in details the existence of solutions, then prove explicit and general decay results, under a very general assumption on the relaxation function and for suitable given data, to solutions with enough regularities. Our results extend and generalize many earlier results in the literature.

Blow-up results for a Boussinesq-type plate equation with a logarithmic damping term and variable-exponent nonlinearities

Blow-up results for a Boussinesq-type plate equation with a logarithmic damping term and variable-exponent nonlinearities

Logarithmic Wave Equation Involving Variable-exponent Nonlinearities:well-posedness and Blow-up

Logarithmic Wave Equation Involving Variable-exponent Nonlinearities:well-posedness and Blow-up

Analytical and computational results for the decay of solutions of a damped wave equation with variable-exponent nonlinearities

Analytical and computational results for the decay of solutions of a damped wave equation with variable-exponent nonlinearities

General decay for a viscoelastic von Karman equation with delay and variable exponent nonlinearities

In this paper, we consider a viscoelastic von Karman equation with damping, delay, and source effects of variable exponent type. Firstly, we show the global existence of solution applying the potential well method. Then, by making use of the perturbed energy method and properties of convex functions, we derive general decay results for the solution under more general conditions of a relaxation function. General decay results of solutions for viscoelastic von Karman equations with variable exponent nonlinearities have not been discussed before. Our results extend and complement many results for von Karman equations in the literature.

Blow up of negative initial-energy solutions of a system of nonlinear wave equations with variable-exponent nonlinearities

<p style='text-indent:20px;'>This work is concerned with a system of wave equations with variable-exponent nonlinearities acting in both equations. We, first, discuss the well-posedness then prove a blow up result for solutions with negative initial energy.</p>

Blow up of solutions for a r(x)-Laplacian Lam'{e} equation with variable-exponent nonlinearities and arbitrary initial energy level

In this paper, we consider the nonlinear $r(x)-$Laplacian Lam'{e} equation $$ u_{tt}-Delta_{e}u-divbig(|nabla u|^{r(x)-2}nabla ubig)+|u_{t}|^{m(x)-2}u_{t}=|u|^{p(x)-2}u $$ in a smoothly bounded domain $Omegasubseteq R^{n}, ngeq1$, where $r(.), m(.)$ and $p(.)$ are continuous and measurable functions. Under suitable conditions on variable exponents and initial data, the blow-up of solutions is proved with negative initial energy as well as positive.

Global nonexistence and blow-up results for a quasi-linear evolution equation with variable-exponent nonlinearities

"In this paper, we consider a class of quasi-linear parabolic equations with variable exponents, $$a\left( x,t\right) u_{t}-\Delta _{m\left( .\right) }u=f_{p\left( .\right)}\left( u\right)$$ in which $f_{p\left( .\right)}\left( u\right)$ the source term, $a(x,t)>0$ is a nonnegative function, and the exponents of nonlinearity $m(x)$, $p(x)$ are given measurable functions. Under suitable conditions on the given data, a finite-time blow-up result of the solution is shown if the initial datum possesses suitable positive energy, and in this case, we precise estimate for the lifespan $T^{\ast }$ of the solution. A blow-up of the solution with negative initial energy is also established."

Liouville-Type Theorems for Sign-Changing Solutions to Nonlocal Elliptic Inequalities and Systems with Variable-Exponent Nonlinearities

We consider the fractional elliptic inequality with variable-exponent nonlinearity $$\begin{aligned} (-\Delta )^{\frac{\alpha }{2}} u+\lambda \, \Delta u \ge |u|^{p(x)}, \quad x\in {\mathbb {R}}^N, \end{aligned}$$where \(N\ge 1\), \(\alpha \in (0,2)\), \(\lambda \in {\mathbb {R}}\) is a constant, \(p: {\mathbb {R}}^N\rightarrow (1,\infty )\) is a measurable function, and \((-\Delta )^{\frac{\alpha }{2}}\) is the fractional Laplacian operator of order \(\frac{\alpha }{2}\). A Liouville-type theorem is established for the considered problem. Namely, we obtain sufficient conditions under which the only weak solution is the trivial one. Next, we extend our study to systems of fractional elliptic inequalities with variable-exponent nonlinearities. Besides the consideration of variable-exponent nonlinearities, the novelty of this work consists in investigating sign-changing solutions to the considered problems. Namely, to the best of our knowledge, only nonexistence results of positive solutions to fractional elliptic problems were investigated previously. Our approach is based on the nonlinear capacity method combined with a pointwise estimate of the fractional Laplacian of some test functions, which was derived by Fujiwara (Math Methods Appl Sci 41:4955–4966, 2018) [see also Dao and Reissig (A blow-up result for semi-linear structurally damped \(\sigma \)-evolution equations, arXiv:1909.01181v1, 2019)]. Note that the standard nonlinear capacity method cannot be applied to the considered problems due to the change of sign of solutions.

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).