scholarly journals Asymptotic Periodicity for Strongly Damped Wave Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Claudio Cuevas ◽  
Carlos Lizama ◽  
Herme Soto
Keyword(s):  
Existence And Uniqueness ◽  
Wave Equations ◽  
Mild Solutions ◽  
Almost Periodic ◽  
Semilinear Wave Equations ◽  
Asymptotic Periodicity ◽  
Asymptotically Almost Periodic ◽  
Strongly Damped

This work deals with the existence and uniqueness of asymptotically almost-periodic mild solutions for a class of strongly damped semilinear wave equations.

scholarly journals On the existence of solutions of strongly damped nonlinear wave equations

2000 ◽  
Vol 23 (6) ◽  
pp. 369-382 ◽  
Author(s):  
Jong Yeoul Park ◽  
Jeong Ja Bae
Keyword(s):  
Nonlinear Wave ◽  
Existence And Uniqueness ◽  
Wave Equations ◽  
Existence Of Solutions ◽  
Hyperbolic Type ◽  
Nonlinear Wave Equations ◽  
Gamma Delta ◽  
Uniqueness Of Solutions ◽  
Alpha Beta ◽  
Strongly Damped

We investigate the existence and uniqueness of solutions of the following equation of hyperbolic type with a strong dissipation:utt(t,x)−(α+β(∫Ω|∇u(t,y)|2dy)γ)Δu(t,x)                                −λΔut(t,x)+μ|u(t,x)|q−1u(t,x)=0,     x∈Ω,t≥0            u(0,x)=u0(x),          ut(0,x)=u1(x),      x∈Ω,  u|∂Ω=0, whereq>1,λ>0,μ∈ℝ,α,β≥0,α+β>0, andΔis the Laplacian inℝN.

scholarly journals Ergodicity and asymptotically almost periodic solutions of some differential equations

2001 ◽  
Vol 25 (12) ◽  
pp. 787-801 ◽  
Author(s):  
Chuanyi Zhang
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Existence And Uniqueness ◽  
Nonlinear Differential Equations ◽  
Almost Periodic Solution ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Ergodic Condition ◽  
Asymptotically Almost Periodic

Using ergodicity of functions, we prove the existence and uniqueness of (asymptotically) almost periodic solution for some nonlinear differential equations. As a consequence, we generalize a Massera’s result. A counterexample is given to show that the ergodic condition cannot be dropped.

On a backward problem for nonlinear time fractional wave equations

2021 ◽  
pp. 1-24
Author(s):  
Jia Wei He ◽  
Yong Zhou
Keyword(s):  
Wave Equation ◽  
Regularization Method ◽  
Existence And Uniqueness ◽  
Wave Equations ◽  
Mild Solutions ◽  
Fractional Wave Equation ◽  
Backward Problem ◽  
Fractional Wave Equations ◽  
Time Fractional Wave Equation ◽  
Eigenvalue Expansion

In this paper, we concern with a backward problem for a nonlinear time fractional wave equation in a bounded domain. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, we establish some results about the existence and uniqueness of the mild solutions of the proposed problem based on the compact technique. Due to the ill-posedness of backward problem in the sense of Hadamard, a general filter regularization method is utilized to approximate the solution and further we prove the convergence rate for the regularized solutions.

scholarly journals Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Aimin Liu ◽  
Yongjian Liu ◽  
Qun Liu
Keyword(s):  
Differential Equations ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Stochastic Functional Differential Equations ◽  
Almost Periodic ◽  
Quadratic Mean ◽  
Exponential Trichotomy ◽  
Functional Differential ◽  
Asymptotically Almost Periodic ◽  
Stochastic Functional

This work is concerned with the quadratic-mean asymptotically almost periodic mild solutions for a class of stochastic functional differential equationsdxt=Atxt+Ft,xt,xtdt+H(t,xt,xt)∘dW(t). A new criterion ensuring the existence and uniqueness of the quadratic-mean asymptotically almost periodic mild solutions for the system is presented. The condition of being uniformly exponentially stable of the strongly continuous semigroup{Tt}t≥0is essentially removed, which is generated by the linear densely defined operatorA∶D(A)⊂L2(ℙ,ℍ)→L2(ℙ,ℍ), only using the exponential trichotomy of the system, which reflects a deeper analysis of the behavior of solutions of the system. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable, and central subspaces at each point from the flow’s domain. An example is also given to illustrate our results.

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