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1

Stabilization of the wave equation with a nonlinear delay term in the boundary conditions

Ghecham Wassila, Rebiai Salah-Eddine, Sidiali Fatima Zohra

Journal of Applied Analysis

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2022

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Vol. 28, nr 1

35--55

EN

A wave equation in a bounded and smooth domain of ℝn with a delay term in the nonlinear boundary feedback is considered. Under suitable assumptions, global existence and uniform decay rates for the solutions are established. The proof of existence of solutions relies on a construction of suitable approximating problems for which the existence of the unique solution will be established using nonlinear semigroup theory and then passage to the limit gives the existence of solutions to the original problem. The uniform decay rates for the solutions are obtained by proving certain integral inequalities for the energy function and by establishing a comparison theorem which relates the asymptotic behavior of the energy and of the solutions to an appropriate dissipative ordinary differential equation.

2

General decay for a nonlinear pseudo-parabolic equation with viscoelastic term

Vu Ngo Tran, Dung Dao Bao, Dung Huynh Thi Hoang

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

737--751

EN

This work is concerned with a multi-dimensional viscoelastic pseudo-parabolic equation with critical Sobolev exponent. First, with some suitable conditions, we prove that the weak solution exists globally. Next, we show that the stability of the system holds for a much larger class of kernels than the ones considered in previous literature. More precisely, we consider the kernel g:[0,∞)⟶(0,∞) satisfying g′(t)⩽−ξ(t)G(g(t)) , where ξ and G are functions satisfying some specific properties.

3

Energy decay result for a nonlinear wave p-Laplace equation with a delay term

Zennir K., Laouar L. K.

Mathematica Applicanda

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2017

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Vol. 45, No. 1

65--80

EN

We consider the nonlinear (in space and time) wave equation with delay term in the internal feedback. Under conditions on the delay term and the term without delay, we study the asymptotic behavior of solutions using the multiplier method and general weighted integral inequalities.

PL

Rozważamy nieliniowe równanie falowe (w czasie i przestrzeni) z członem wewnętrznego sprzężenia zwrotnego. Przy pewnych warunkach na poszczególne człony równania badane jest asymptotyczne zachowanie rozwiązań.

4

Decay rates of the solution to the Cauchy problem of the type III Timoshenko model without any mechanical damping

Said-Houari B.

Demonstratio Mathematica

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2015

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Vol. 48, nr 3

379--390

EN

In this paper, we study the asymptotic behavior of the solutions of the one-dimensional Cauchy problem in Timoshenko system with thermal effect. The heat conduction is given by the type III theory of Green and Naghdi. We prove that the dissipation induced by the heat conduction alone is strong enough to stabilize the system, but with slow decay rate. To show our result, we transform our system into a first order system and, applying the energy method in the Fourier space, we establish some pointwise estimates of the Fourier image of the solution. Using those pointwise estimates, we prove the decay estimates of the solution and show that those decay estimates are very slow and, in the case of nonequal wave speeds, are of regularity–loss type. This paper solves the open problem stated in [10] and shows that the stability of the solution holds without any additional mechanical damping term.

5

Photolytic degradation of 4-tert-octylphenol in aqueous solution

Błędzka D., Gryglik D., Miller J. S.

Environment Protection Engineering

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2009

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Vol. 35, nr 3

235-247

EN

4-tert-octylphenol (OP), the endocrine disrupting compound, in aqueous solution was degraded by polychromatic UV lamp. It was found that the rate of OP depletion depends on light intensity, pH of the reaction mixture, initial compound concentration and the presence of natural water constituents. Nitrate addition seemed to give the best results in terms of OP decay rate. The time of 50% OP degradation in the presence of NO3- (2 mg l -1) was about 8 times shorter in comparison to degradation without any additives.