In this paper we consider the wave equation with structural damp ing 1 y tt. Dabbicco and ebert 4 proved the global in time existence of small data solutions for any pa p1pn. Massatt, limiting behavior for strongly damped nonlinear wave equations, j. The approximation solutions show that the methodology of local fractional variational iteration method is an efficient and simple tool for solving mathematical problems arising in fractal wave motions. In order to treat space dimensions more than one, we work in the setting of lqspaces with possibly. Comparison of two nite di erence methods for solving the damped wave equation h. At present, there do not seem to be so many related manuscripts. Second order linear partial differential equations part iv. Global solutions to the cauchy problem for the weakly.
A qualitative study of the damped dung equation and. On the strongly damped wave equation 5 and the ehrling lemma, that is, given srq, for every. Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent. Exponential attractor for the wave equation with structural. We obtain spacetime estimates on the solution to the cauchy problem of damped fractional wave equation. Illustrate the nature of the solution by sketching the uxpro. Damped wave equation and dissipative wave equation in fractal. Many other nonlinear wave theories mentioned above have also achieved similar success. Recently, the semilinear structurally damped wave models. If we have more than one spatial dimension a membrane for example, the wave equation will look a bit. Together with the heat conduction equation, they are sometimes referred to as the evolution equations because their solutions evolve, or change, with passing time. Optimal blowup rates for the minimal energy null control for. Exponential energy decay estimate for the solutions of. The wave equation is an important secondorder linear partial differential equation for the description of wavesas they occur in classical physicssuch as mechanical waves e.
Null controllability of the structurally damped wave equation with moving point control philippe martin, lionel rosier, and pierre rouchon abstract. It arises in fields like acoustics, electromagnetics, and fluid dynamics historically, the problem of a vibrating string such as that of a musical. When the nonlinearity is subcritical, we prove the existence of an exponential attractor of optimal regularity, having a basin of attraction coinciding with the whole phasespace. Regarding the structurally damped wave equation 1 with the power nonlinearity u, the critical exponent has been investigated by dabbicco and reissig 7, where they proposed to distinguish between parabolic like models in the case. Nov 11, 2004 we prove the existence of the universal attractor for the strongly damped semilinear wave equation, in the presence of a quite general nonlinearity of critical growth. Numericalanalysislecturenotes university of minnesota. Asymptotic profiles of solutions for structural damped wave equations article pdf available in journal of dynamics and differential equations july 2016 with 163 reads how we measure reads.
Interior controllability of a strongly damped wave equation. Weakly coupled systems of semilinear elastic waves with different. Indeed, the fact that the trajectory of any wave intersects the support of the damping should imply that any solution relaxes to an equilibrium. Dabbicco and ebert 4 proved the global in time existence of small data solutions for. Structurally damped plate and wave equations with random point force in arbitrary space dimensions roland schnaubelt and mark veraar abstract. This lack of regularity prevents a more detailed asymptotic analysis. Chapter 4 the wave equation another classical example of a hyperbolic pde is a wave equation. This article is to study the nonexistence of global solutions to semilinear structurally damped wave equation with.
Existence of an attractor and determining modes for. Stabilization for the wave equation with variable coefficients and balakrishnantaylor damping ha, tae gab, taiwanese journal of mathematics, 2017. Pdf we investigate the internal controllability of the wave equation with structural damping on the onedimensional torus. Null controllability of the structurally damped wave. In such equations the damping term has half of the order of the leading elastic term, as it has been proposed in the seminal paper 38. However, if there is some from of friction, then the amplitude will decrease as a function of time g t a0 a0 x if the damping is sliding friction, fsf constant, then the work done by the.
For the damped wave equation, we consider the extension of the energetic weak. Null controllability of the structurally damped wave equation with moving control article pdf available in siam journal on control and optimization 511. Recently, hayashi et al 4 studied the existence of small global solutions to the initialvalue problem for sy. However, the universal attractor is not shown to have the best possible regularity, even in the subcritical case. Semantic scholar extracted view of on the diffusive structure for the damped wave equation with variable coefficients by yuta wakasugi. We assume that the control is acting on a moving point or on a moving small interval with a constant velocity. Pdf asymptotic profiles of solutions for structural. Decay rates for a viscoelastic wave equation with balakrishnantaylor and frictional dampings feng, baowei and kang, yong han, topological methods in nonlinear analysis, 2019. Asymptotic profiles of solutions for structural damped. By constructing a bounded absorbing set with higher global regularity rather than the longstanding partial regularity and by using the weak quasistability estimates rather than the strong ones as usual. Sobolev spaces for the structurally damped plate equation with dirichletneumann clamped boundary conditions. The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. Null controllability of the structurally damped wave equation with moving point control.
Also, several examples of partial differential equations with. Pdf optimal blowup rates for the minimal energy null. For the single semilinear classical damped wave equation. It is worth noting that nondissipative \energybuilding sources, especially those on the boundary, arise when one considers a wave equation being coupled. Wave equations, examples and qualitative properties. The paper studies the existence of an exponential attractor for the wave equation with structural damping and supercritical nonlinearity u t t. Depending on the medium and type of wave, the velocity v v v can mean many different things, e. Zelik, finitedimensional attractors for the quasilinear stronglydamped wave equation, j. Critical exponent for semilinear structurally damped wave. Wave equation, boundary element method, finite element method. Another works concerning with structurally damped elastic systems are the papers 6 and 7 by a. We investigate the internal controllability of the wave equation with structural damping on the one dimensional torus. The almost everywhere convergence of the solution to linear equations as is also studied, with the initial data satisfying certain regularity conditions 1.
In such equations the damping term has half of the order of the leading elastic term, as it has been proposed in the seminal paper 29. For example, in a transverse wave traveling along a string, each point in the string oscillates back and forth in the transverse direction not along the direction of the string. Pdf critical exponent for semilinear structurally damped. Izadiy 1department of applied mathematics, lahijan branch, islamic azad university 2department of applied mathematics, lahijan branch, islamic azad university october 23, 2014 communicated by prof. We investigate the internal controllability of the wave equation with structural damping on the onedimensional torus. Pdf structurally damped plate and wave equations with.
Pdf on the diffusive structure for the damped wave equation with. In this paper, using a different approach, we prove the existence of a universal attrac. Damped wave equation and dissipative wave equation in. The wave equation is a linear secondorder partial differential equation which describes the propagation of oscillations at a fixed speed in some quantity y y y a solution to the wave equation in two dimensions propagating over a fixed region 1. It turns out that the rstorder system related to the scalar equation on rn is.
Comparison of two nite di erence methods for solving the. Point controls and feedbacks in elastic systems lead naturally to perturbations of damped. Structurally damped plate and wave equations dimensions arxiv. The dispersion relation can then also be written more compactly as. Energy decay rates for solutions of the kirchhoff type wave equation. Structurally damped plate and wave equations have been studied intensively in the deterministic case. Optimal blowup rates for the minimal energy null control. The wave equation governs a wide range of phenomena, including gravitational waves, light waves, sound waves, and even the oscillations of strings in string theory. When the elasticity k is constant, this reduces to usual two term wave equation u tt c2u xx where the velocity c p k. P p1, 2s, the socalled noneffective damping according to expected decay estimates see more 3. The second type of second order linear partial differential equations in 2 independent variables is the onedimensional wave equation. Cholewa, where the authors studied the existence of an attractor for the following strongly damped wave equation with critical nonlinearities. International audiencewe investigate the internal controllability of the wave equation with structural damping on the onedimensional torus.
To be concrete, we impose timedependent dirichlet boundary conditions. Let us come back our interest to consider the structurally damped wave equation with the power nonlinearity of derivative type 1. Structurally damped plate and wave equations with random point force in arbitrary space dimensions article pdf available in differential and integral equations 239 october 2008 with 28 reads. We shall discuss the basic properties of solutions to the wave equation 1.
A qualitative study of the damped dung equation and applications. In figure 1, the explicit solution for a special case of the di. Blow up results for semilinear structural damped wave model with. It arises in fields like acoustics, electromagnetics, and fluid dynamics. The constant c2 comes from mass density and elasticity, as expected in newtons and hookes laws. Pdf null controllability of the structurally damped wave. Wave equation in 1d part 1 derivation of the 1d wave equation vibrations of an elastic string solution by separation of variables three steps to a solution several worked examples travelling waves more on this in a later lecture dalemberts insightful solution to the 1d wave equation. Motivated by potential applications in physics, engineering, biology and communication theory, the damped dung equation x. Spacetime estimates on damped fractional wave equation. Nov 20, 2011 null controllability of the structurally damped wave equation with moving point control article pdf available november 2011 with 51 reads how we measure reads. We obtain unique solutions with optimal regularity for the inhomogeneous problem in the whole space, in the halfspace, and in bounded domains of class c4.
We assume that the control is acting on a moving point. Herein we shall be concerned with the internally damped wave equation y. In this paper, the local fractional variational iteration method is given to handle the damped wave equation and dissipative wave equation in fractal strings. Study of semilinear \sigma evolution equations with.
To gain an insight to the physical background about the damped wave equation we refer to 2 where it is stated that when the neural. P p0, 1s, the socalled effective damping, and hyperbolic like modelsin the remaining case. Null controllability of the structurally damped wave equation with. Using these estimates, we can solve the problem of finding the critical exponents for the two nonlinear problems above. Asymptotic profiles of solutions for structural damped wave. In this paper we consider structurally damped plate and wave equations with point and distributed random forces. Global solutions to the cauchy problem for the weakly coupled system of damped wave equations. New examples of damped wave equations with gradientlike. Pdf null controllability of the structurally damped wave equation. Null controllability of the structurally damped wave equation.
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