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

In this article, we consider a von Karman equation with variable exponent nonlinearities wtt(x,t)+Δ2w(x,t)+|wt(x,t)|γ(x)−2wt(x,t)=[w(x,t),F(w(x,t))]+|w(x,t)|p(x)−2w(x,t) in a bounded domain normalΩ⊂ℝ2. 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.

Section: Stabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Park

2021

“…They proved a blowup in finite time with negative initial energy under suitable conditions on g, f and the variable exponent of the ⃗ p(x, t)-Laplace operator. For more results regarding this matter, we refer the reader to previous studies [13][14][15][16][17][18] and the review paper. 19 Motivated by the aforementioned works, in the present paper, we study a class of elastic inverse source problem with variable-exponent nonlinearities.…”

Section: Introductionmentioning

confidence: 99%

General decay and blow up of solutions for a class of inverse problem with elasticity term and variable‐exponent nonlinearities

Shahrouzi

2021

In this paper, we study a class of Lamé inverse source problem with variable‐exponent nonlinearities. Under some suitable conditions on the coefficients and initial data, we proved general decay of solutions when the integral overdetermination tends to zero as time goes to infinity in appropriate range of variable exponents. Furthermore, in the absence of damping term, we show that there are solutions under some conditions on initial data and variable exponents which blow up in finite time.

“…It has been shown that the solution energy decays exponentially if

m \equiv 2

and polynomially at the rate of

t^{2 false/ false(m_{2} - 2 false)}

, if

m_{2} = e s s \sup m false(. false) > 2

. Also, Ghegal et al considered, in a bounded domain, the following equation

u_{t t} - Δ u + | u_{t} |^{m (.) - 2} u_{t} = | u |^{p (.) - 2} u,

together with Dirichlet‐boundary conditions, and proved under suitable conditions on the initial data, a global existence and a stability result similar to that of Messaoudi et al We refer the reader to the recent review paper for more results concerning stability and blow up in wave problems.…”

Section: Introductionmentioning

confidence: 99%

On the decay of solutions of a damped quasilinear wave equation with variable‐exponent nonlinearities

Messaoudi

2020

Self Cite

In this work, we consider the following nonlinear wave equation with variable exponents:where Ω is a bounded domain, T > 0, and m(.) and r(.) are continuous functions. We will establish several decay results depending on the range of the variable exponents m and r.KEYWORDS exponential decay, polynomial decay, strong damping, variable exponent, wave MSC CLASSIFICATION 35B35; 35L20; 35L70in a bounded domain Ω, with a smooth boundary, was considered at a large scale. For instance, Nakao 6 looked into Equation (1.2), in the case when (u) = |u| p−2 u, and g(u t ) = |u t | m−2 u t , m, p > 2 and proved a unique global weak solution if 2 ≤ p ≤ 2(n − 1)∕(n − 2), n ≥ 3 and a global unique strong solution if p > 2(n − 1)∕(n − 2), n ≥ 3. Moreover, he addressed Math Meth Appl Sci. 2020;43:5114-5126. wileyonlinelibrary.com/journal/mma