Solutions Of Damped Wave Equation Research Articles

Global existence and dynamic structure of solutions for damped wave equation involving the fractional Laplacian

Abstract We consider strong damped wave equation involving the fractional Laplacian with nonlinear source. The results of global solution under necessary conditions on the critical exponent are established. The existence is proved by using the Galerkin approximations combined with the potential well theory. Moreover, we showed new decay estimates of global solution.

Distributional solutions for damped wave equations

This work presents results on solutions to the one-dimensional damped wave equation, also called telegrapher's equation, when the initial conditions are general distributions. We make a complete deduction of its fundamental solutions, both for positive and negative times. To obtain them we only use self-similarity arguments and distributional calculus, making no use of Fourier or Laplace transforms. We next use these fundamental solutions to prove both the existence and the uniqueness of solutions to the distributional initial value problem. As applications we recover the semi-group property for initial data in classical function spaces, and we find the probability distribution function for a recent financial model of evolution of prices. For more information see https://ejde.math.txstate.edu/Volumes/2020/131/abstr.html

Weighted L2-estimates of solutions for damped wave equations with variable coefficients

The authors establish weighted L2-estimates of solutions for the damped wave equations with variable coefficients utt− divA(x)∇u+aut = 0 in IRn under the assumption a(x) ≥ a0[1+ρ(x)]−l, where a0 > 0, l < 1, ρ(x) is the distance function of the metric g = A−1(x) on IRn. The authors show that these weighted L2-estimates are closely related to the geometrical properties of the metric g = A−1(x).

General decay of solutions for damped wave equation of Kirchhoff type with density in $$\mathbb {R}^{n}$$ R n

In order to compensate the lack of Poincare’s inequality in \(\mathbb {R}^{n}\) and for wider class of relaxation functions, we are going to use weighted spaces to establish a very general decay rate of solutions of viscoelastic wave equations in Kirchhoff-type.

Global existence of solutions for damped wave equations with nonlinear memory

This article studies the Cauchy problem for the damped wave equation with nonlinear memory. For a noncompactly supported initial data with small energy, global existence and asymptotic behaviour of solutions are obtained when 1 ≤ n ≤ 3. This result generalized the previous result by Fino [Critical exponent for damped wave equations with nonlinear memory, Nonlinear Anal. 74 (2011), pp. 5495–5505], which dealt with the solution with compactly supported initial data.

The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space

In this paper, we study thetime-asymptotic behavior of the solution for the Cauchy problem ofthe damped wave equation with a nonlinear convection term in themulti-dimensional space. When the initial data is a small perturbation around aconstant state $u^*$, we obtain the point-wise decay estimates of thesolution under theso-called dissipative condition $|b| < 1$, where $b$ depends on $u^*$ and the nonlinear term.

Global existence and Lp estimates for solutions of damped wave equation with nonlinear convection in multi-dimensional space

In this article, the author studies the Cauchy problem of the damped wave equation with a nonlinear convection term in multi-dimensions. The author shows that a classical solution to the Cauchy problem exists globally in time under smallness condition on the initial perturbation. Furthermore, the author obtains the Lp (2 ≤p ≤ ∞) decay estimates of the solution.

Asymptotic behavior of solutions for damped wave equations with non-convex convection term on the half line

We study the asymptotic stability of nonlinear waves for damped wave equations with a convection term on the half line. In the case where the convection term satisfies the convex and sub-characteristic conditions, it is known by the work of Ueda [7] and Ueda--Nakamura--Kawashima [10] that the solution tends toward a stationary solution. In this paper, we prove that even for a quite wide class of the convection term, such a linear superposition of the stationary solution and the rarefaction wave is asymptotically stable. Moreover, in the case where the solution tends to the non-degenerate stationary wave, we derive that the time convergence rate is polynomially (resp. exponentially) fast if the initial perturbation decays polynomially (resp. exponentially) as $x \to \infty$. Our proofs are based on a technical $L^{2}$ weighted energy method.

动力边界条件的阻尼波动方程解的爆破性和渐近性

本文讨论动力边界条件的阻尼波动方程解的爆破性和渐近性。利用凸性分析和不稳定集，分别给出了初始能量为负和正时，解爆破的充分条件；借助Nakao不等式和位势井理论得到了解的衰减估计。 In this paper, the blow-up and asymptotic behavior of global solution of damped wave equation with dynamic boundary condition are discussed. By the convexity lemma and unstable set, the sufficient con- dition of the solution of the wave equation with negative and positive initial energy respectively are obtained. With the help of Nakao and stable set, the energy decay of the solution is given.

Existence of asymptotically almost periodic solutions for damped wave equations

In this paper, a class of nonlinear damped wave equations of the form α u ‴ ( t ) + u ″ ( t ) = β A u ( t ) + γ A u ′ ( t ) + f ( t , u ( t ) ) , t ⩾ 0 , satisfying α β < γ with prescribed initial conditions are studied. Some sufficient conditions are established for the existence and uniqueness of an asymptotically almost periodic solution. These results have significance in the study of vibrations of flexible structures possessing internal material damping. Finally, an example is presented to illustrate the feasibility and effectiveness of the results.

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 .

Decay property of solutions for damped wave equations with space–time dependent damping term

We consider the Cauchy problem for the damped wave equation with space–time dependent potential b ( t , x ) and absorbing semilinear term | u | ρ − 1 u . Here, b ( t , x ) = b 0 ( 1 + | x | 2 ) − α 2 ( 1 + t ) − β with b 0 > 0 , α , β ⩾ 0 and α + β ∈ [ 0 , 1 ) . Using the weighted energy method, we can obtain the L 2 decay rate of the solution, which is almost optimal in the case ρ > ρ c ( N , α , β ) : = 1 + 2 / ( N − α ) . Combining this decay rate with the result that we got in the paper [J. Lin, K. Nishihara, J. Zhai, L 2 -estimates of solutions for damped wave equations with space–time dependent damping term, J. Differential Equations 248 (2010) 403–422], we believe that ρ c ( N , α , β ) is a critical exponent. Note that when α = β = 0 , ρ c ( N , α , β ) coincides to the Fujita exponent ρ F ( N ) : = 1 + 2 / N . The new points include the estimate in the supercritical exponent and for not necessarily compactly supported data.

Formula omitted]-estimates of solutions for damped wave equations with space–time dependent damping term

In this paper, we consider the damped wave equation with space–time dependent potential b ( t , x ) and absorbing semilinear term | u | ρ − 1 u . Here, b ( t , x ) = b 0 ( 1 + | x | 2 ) − α 2 ( 1 + t ) − β with b 0 > 0 , α , β ⩾ 0 and α + β ∈ [ 0 , 1 ) . Based on the local existence theorem, we obtain the global existence and the L 2 decay rate of the solution by using the weighted energy method. The decay rate coincides with the result of Nishihara [K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, preprint] in the case of β = 0 and coincides with the result of Nishihara and Zhai [K. Nishihara, J. Zhai, Asymptotic behaviors of time dependent damped wave equations, preprint] in the case of α = 0 .

Bifurcation and multiplicity results for periodic solutions of a damped wave equation in a thin domain

We study the bifurcation problem for periodic solutions of a nonautonomous damped wave equation defined in a thin domain. Here the bifurcation parameter is represented by the thinness ε>0 of the considered domain. This study has as starting point the existence result of periodic solutions already stated by the authors for this equation and it makes use of the condensivity properties of the associated Poincaré map and its linearization around these solutions. We establish sufficient conditions to guarantee that ε=0 is or not a bifurcation point and a related multiplicity result. These results are in the spirit of those given by Krasnosel'skii and they are obtained by using the topological degree theory for k-condensing operators.