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

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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

“…For the last term in Equation , we recall Young's inequality to get

\int_{s}^{T} E^{q} \int_{Ω} u | u_{t} |^{m (.) - 2} u_{t} \leq δ \int_{s}^{T} E^{q} \int_{Ω} | u |^{m (.)} + \int_{s}^{T} E^{q} \int_{Ω} c_{δ} (x) | u_{t} |^{m (.)},

where

C_{δ} (x) = δ^{1 - m (x)} {(m (x))}^{m (x)} {(m (x) - 1)}^{m (x) - 1} .

See Messaoudi et al for more details. By using Equation , Lemma , and Lemma , we have the following:

\begin{matrix} \int_{s}^{T} E^{q} \int_{normalΩ} false| u {false|}^{m false(. false)} & \leq C \int_{s}^{T} E^{q} ((), false| false| u false| {false|}_{m false(. false)}^{m_{1}}) \end{matrix}

…”

Section: Decay Resultsmentioning

confidence: 99%

u_{t t} - d i v (| \nabla u |^{r (.) - 2} \nabla u) + | u_{t} |^{m (.) - 2} u_{t} = 0

was studied in a bounded domain

normalΩ

, with a smooth boundary and for exponents

r false(. false)

and

m false(. false)

satisfying

2 \leq r false(. false) \leq

m false(. false)

and other specific properties. 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

.…”

Section: Introductionmentioning

confidence: 99%

“…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,

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Messaoudi

2020

Self Cite

“…Moreover, when r ( x ) ≡ 2, Sun et al 26 gave lower and upper bounds for the blow‐up time. Recently, Messaoudi et al 27 looked at (), with

b = 0

and 2 ≤ m ( x ) < 2 ∗ , and proved decay estimates for the solution under suitable assumptions on the variable exponents m , r and the initial data. We also refer to Gao and Gao 28 who studied a nonlinear viscoelastic equation with variable exponents and proved the existence of weak solutions.…”

Section: Introductionmentioning

confidence: 99%

Energy estimates to the Cauchy problem of a weakly damped Klein‐Gordon equation with variable‐exponent nonlinearity

Mustafa

2021

“…On the other hand, recently, researchers have much interested in the study of nonlinear models of elliptic, parabolic, and hyperbolic equations with variable exponent nonlinearities 9–15 . Messaoudi et al 13 considered the nonlinear damped equation

w_{t t} - d i v (| \nabla w |^{r (x) - 2} \nabla w) + | w_{t} |^{γ (x) - 2} w_{t} = 0 .

Under appropriate conditions on the initial data and the exponents r (·), γ (·), they showed the decay estimates applying the multiplier method.…”

Section: Introductionmentioning

confidence: 99%

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

Park

2021

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.