Nonlinear Dissipative Wave Equations Research Articles

Semigroup theory and finite element method applied to a non-linear dissipative wave equation

We study the wave equation with a non-linear dissipative term associated to a bidimensional membrane with fixed boundary. We use the semigroup theory to consider the existence and uniqueness of solutions to the problem and we implement the finite element method to analyse the vibrating evolutionary equation. In particular we use Comsol Multiphysics software with a rectangular mesh to analyze the corresponding evolutionary system. We mention that this system can appear, for example, in the diaphragm of a centrifugal pump in mining processes.

Destruction of solutions for class of wave $ p(x)- $bi-Laplace equation with nonlinear dissipation

<abstract><p>An initial value problem is considered for the nonlinear dissipative wave equation containing the $ p(x) $-bi-Laplacian operator. For this problem, sufficient conditions for the blow-up with nonpositive initial energy of a generalized solution are obtained in finite time where a wide variety of techniques are used.</p></abstract>

Comparing Solutions to the Nonlinear Dissipative Wave Equation

In previous decades, many of the practical problems arising in scientific fields such as physics, engineering, and mathematics have been related to nonlinear fractional partial differential equations. One of these nonlinear partial differential equations, the dissipative wave equation, has been found to have a plethora of useful applications in different fields. A special class of solutions has been studied for the dissipative wave equation including exact solutions and approximate solutions. The aim of this article is to compare the non-polynomial spline method and the cubic B-spline method with the solution of a nonlinear dissipative wave equation. We will conduct a comparison of the stability of the two methods using the Von Neumann stability analysis. In addition, a numerical example will be presented to illustrate the accuracy of these methods.

A numerical solution of the dissipative wave equation by means of the cubic B-spline method

In the past few decades, partial differential equations have drawn considerable attention, owing to their ability to model certain physical phenomena. The aim of this paper is to investigate a cubic B-spline polynomial to obtain a numerical solution of a nonlinear dissipative wave equation. For the numerical procedure, the time derivative is obtained by the usual finite difference scheme. The approximate solution and its principal derivatives over the subinterval is approximated by the combination of the cubic B-spline and unknown element parameters. The accuracy of the proposed method will be shown by computing L∞ error norms for different time levels. By applying Von Neumann stability analysis, the developed method is shown to be conditionally stable for given values of specified parameters. A numerical example is given to illustrate the accuracy of the cubic B-spline polynomial method. The obtained numerical results show that our proposed method maintains good accuracy.

A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives

In this manuscript, we consider an initial-boundary-value problem governed by a (1+1)-dimensional hyperbolic partial differential equation with constant damping that generalizes many nonlinear wave equations from mathematical physics. The model considers the presence of a spatial Laplacian of fractional order which is defined in terms of Riesz fractional derivatives, as well as the inclusion of a generic continuously differentiable potential. It is known that the undamped regime has an associated positive energy functional, and we show here that it is preserved throughout time under suitable boundary conditions. To approximate the solutions of this model, we propose a finite-difference discretization based on fractional centered differences. Some discrete quantities are proposed in this work to estimate the energy functional, and we show that the numerical method is capable of conserving the discrete energy under the same boundary conditions for which the continuous model is conservative. Moreover, we establish suitable computational constraints under which the discrete energy of the system is positive. The method is consistent of second order, and is both stable and convergent. The numerical simulations shown here illustrate the most important features of our numerical methodology.

On the energy decay for dissipative nonlinear wave equations in one space dimension

The energy decay problem is studied for the nonlinear dissipative wave equation in one space dimension. It is shown by Mochizuki and Motai [6] that the decay rate is at least logarithmic when the exponent of the nonlinearity is greater than one and less than three. In this paper, an improvement is found which implies a polynomial decay rate for the same range of exponents.

Global regularity for supercritical nonlinear dissipative wave equations in 3D

The nonlinear wave equation utt−Δu+|ut|p−1ut=0 is shown to be globally well-posed in the Sobolev spaces of radially symmetric functions Hradk(R3)×Hradk−1(R3) for all p≥3 and k≥3. Moreover, global C∞ solutions are obtained when the initial data are C0∞ and exponent p is an odd integer.The radial symmetry allows a reduction to the one-dimensional case where an important observation of Haraux (2009) can be applied, i.e., dissipative nonlinear wave equations contract initial data in Wk,q(R)×Wk−1,q(R) for all k∈[1,2] and q∈[1,∞].

ON DECAY ESTIMATES OF SOLUTIONS FOR SOME DEGENERATE NONLINEAR KIRCHHOFF STRINGS WITH DISSIPATION

We consider the initial-boundary value problem for the degenerate nonlinear dissipative wave equation of Kirchhoff type:

Existence of Global Decaying Solutions to the Cauchy Problem for a Nonlinear Dissipative Wave Equation of Klein-Gordon Type with a Derivative Nonlinearity

We prove the existence of global decaying solutions to the Cauchy problem for the wave equation of Klein-Gordon type with a nonlinear dissipation and a derivative nonlinearity. To derive required estimates of solutions we employ a delicate 'loan' method.

Global existence and decay for nonlinear dissipative wave equations with a derivative nonlinearity

We prove the global existence of the so-called H 2 solutions for a nonlinear wave equation with a nonlinear dissipative term and a derivative type nonlinear perturbation. To show the boundedness of the second order derivatives we need a precise energy decay estimate and for this we employ a ‘loan’ method.

Explicit traveling wave solutions to nonlinear evolution equations

First of all, some technical tools are developed. Then the author studies explicit traveling wave solutions to nonlinear dispersive wave equations, nonlinear dissipative dispersive wave equations, nonlinear convection equations, nonlinear reaction diffusion equations and nonlinear hyperbolic equations, respectively.

Large time behavior of solutions for the nonlinear wave equation with viscosity in odd dimensions

We study the dissipation of solutions of the Cauchy problem for the nonlinear dissipative wave equation in odd multi-spatial dimensions. Pointwise estimates of the time-asymptotic shape of the solutions are obtained and shown to exhibit the generalized Huygens principle. Our approach is based on the detailed analysis of the Green function of the linearized system. This is used to study the coupling of nonlinear diffusion waves.

SMALL GLOBAL SOLUTIONS FOR NONLINEAR COMPLEX GINZBURG–LANDAU EQUATIONS AND NONLINEAR DISSIPATIVE WAVE EQUATIONS IN SOBOLEV SPACES

The Cauchy problems for nonlinear complex Ginzburg–Landau equations and nonlinear dissipative wave equations are considered in Sobolev spaces. The relation between the order of the nonlinear terms and the regularity of solutions is considered in terms of the scaling arguments, and the existence of local solutions and small global solutions is shown in Sobolev and Besov spaces.

Nonlinear dissipative wave equations with space–time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space–time dependent friction a ( t , x ) u t and nonlinear absorption | u | p − 1 u (Ikawa (2000) [17]). We consider 1 < p < ( n + 2 ) / ( n − 2 ) and separable a ( t , x ) = λ ( x ) η ( t ) with λ ( x ) ∼ ( 1 + | x | ) − α and η ( t ) ∼ ( 1 + t ) − β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L 2 and L p + 1 norms of solutions. We also observe the following behavior: if α ∈ [ 0 , 1 ) , β ∈ ( − 1 , 1 ) and 0 < α + β < 1 , there are three different regions for the decay of solutions depending on p ; if α ∈ ( − ∞ , 0 ) and β ∈ ( − 1 , 1 ) , there are only two different regions for the decay of the solutions depending on p .

Local asymptotic stability for polyharmonic Kirchhoff systems†

In this article we treat the question of local asymptotic stability for polyharmonic Kirchhoff systems, governed by time-dependent source forces and nonlinear damping terms. Even the simplest case (𝒫3) studied here includes several systems of great interest, as the physical models for vibrating beams of the Woinowsky–Krieger type. This article extends and generalizes some local stability results of Autuori et al. [Asymptotic stability for anisotropic Kirchhoff systems, J. Math. Anal. Appl. 352 (2009), pp. 149–165] to higher order systems, and also of Nakao and Zhu [Decay rate of solutions of a wave equation with damping and external force, Nonlinear Anal. 46 (2001), pp. 335–345; An attractor for a nonlinear dissipative wave equation of Kirchho type, J. Math. Anal. Appl. 353 (2009), pp. 652–659].

Explicit traveling wave solutions of five kinds of nonlinear evolution equations

First of all, by using Bernoulli equations, we develop some technical lemmas. Then, we establish the explicit traveling wave solutions of five kinds of nonlinear evolution equations: nonlinear convection diffusion equations (including Burgers equations), nonlinear dispersive wave equations (including Korteweg–de Vries equations), nonlinear dissipative dispersive wave equations (including Ginzburg–Landau equation, Korteweg–de Vries–Burgers equation and Benjamin–Bona–Mahony–Burgers equation), nonlinear hyperbolic equations (including Sine–Gordon equation) and nonlinear reaction diffusion equations (including Belousov–Zhabotinskii system of reaction diffusion equations).

Global existence and energy decay for mildly degenerate Kirchhoff’s equations on ℝ N

We study the initial-boundary value problem for the following degenerate non-lineardissipative wave equations of Kirchhoff type: with initial conditions u(x, 0) = u 0(x) and ut (x, 0) = u 1(x) , in the case where N ≥ 3 , δ > 0 , γ ≥ 1 , f (u) is a nonlinear C 1 function and (ϕ(x))-1 = g(x) is a positive function lying in L N/2( ℝ N ) ∩ L ∞( ℝ N ) . It is proved that if the initial data {u 0, u 1} are small and ║∇u 0║ > 0 , then the unique solution exists globally in time and has certain decay properties.

An attractor for a nonlinear dissipative wave equation of Kirchhoff type

In this paper we prove the existence and some absorbing properties of an attractor in a local sense for the initial–boundary value problem of a quasilinear wave equation of Kirchhoff type with a standard dissipation u t .

A singular initial value problem and self-similar solutions of a nonlinear dissipative wave equation

We present a systematic study of local solutions of the ODE of the form x ″ = 1 t f ( t , x , x ′ ) near t = 0 . Such ODEs occur in the study of self-similar radial solutions of some second order PDEs. A general theorem of existence and uniqueness is established. It is shown that there is a dichotomy between the cases γ > 0 and γ < 0 , where γ = ∂ f / ∂ x ′ at t = 0 . As an application, we study the singular behavior of self-similar radial solutions of a nonlinear wave equation with superlinear damping near an incoming light cone. A smoothing effect is observed as the incoming waves are focused at the tip of the cone.

Application of variational iteration method to non-homogeneous non-linear dissipative wave equations

In this paper, the solution of the non-linear non-homogeneous dissipative wave equation is obtained by means of the variational iteration method. This equation describes the propagation of a wave, and it arises in a wide variety of physical problems. The results reveal that the variational iteration method is very effective, convenient and quite accurate to systems of nonlinear partial differential equations.