A Non-Autonomous Wave Equation With Time-Dependent Energy Damping

In this paper, we establish the existence of a random attractor for a random dynamical system generated by the non-autonomous wave equation with strong damping and multiplicative noise when the nonlinear term satisfies a critical growth condition.

Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain

The problem of the blow-up of solutions to the initial boundary value problem for a nonautonomous semilinear wave equation with damping and accelerating terms under the Robin boundary condition is studied. Sufficient conditions for the blow up in finite time of solutions to semilinear damped wave equations with arbitrary large initial energy are obtained. A result on the blow-up of solutions with negative initial energy to a semilinear second-order wave equation with an accelerating term is also obtained.

We study the global existence of solutions to the Cauchy problem for the wave equation with time-dependent damping and a power nonlinearity in the overdamping case: { u t t − Δ u + b ( t ) u t = N ( u ) , a m p ; t ∈ [ 0 , T ) , x ∈ R d , u ( 0 ) = u 0 , u t ( 0 ) = u 1 , a m p ; x ∈ R d . \begin{align*}\left \{\begin {array}{ll} u_{tt} - \Delta u + b(t) u_t = N(u),&t\in [0,T),\ x\in \mathbb {R}^d,\\ u(0) = u_0,\ u_t(0) = u_1,&x\in \mathbb {R}^d. \end{array}\right . \end{align*} Here, b ( t ) b(t) is a positive C 1 C^1 -function on [ 0 , ∞ ) [0,\infty ) satisfying \[ b ( t ) − 1 ∈ L 1 ( 0 , ∞ ) , b(t)^{-1} \in L^1(0,\infty ), \] whose case is called overdamping. N ( u ) N(u) denotes the p p th order power nonlinearities. It is well known that the problem is locally well-posed in the energy space H 1 ( R d ) × L 2 ( R d ) H^1(\mathbb {R}^d)\times L^2(\mathbb {R}^d) in the energy-subcritical or energy-critical case 1 ≤ p ≤ p 1 1\le p\le p_1 , where p 1 := 1 + 4 d − 2 p_1:=1+\frac {4}{d-2} if d ≥ 3 d\ge 3 or p 1 = ∞ p_1=\infty if d = 1 , 2 d=1,2 . It is known that when N ( u ) := ± | u | p N(u):=\pm |u|^p , small data blow-up in L 1 L^1 -framework occurs in the case b ( t ) − 1 ∉ L 1 ( 0 , ∞ ) b(t)^{-1} \notin L^1(0,\infty ) and 1 > p > p c ( > p 1 ) 1>p>p_c(> p_1) , where p c p_c is a critical exponent, i.e., threshold exponent dividing the small data global existence and the small data blow-up. The main purpose in the present paper is to prove the global well-posedness to the problem for small data ( u 0 , u 1 ) ∈ H 1 ( R d ) × L 2 ( R d ) (u_0,u_1)\in H^1(\mathbb {R}^d)\times L^2(\mathbb {R}^d) in the whole energy-subcritical case, i.e., 1 ≤ p > p 1 1\le p>p_1 . This result implies that the small data blow-up does not occur in the overdamping case, different from the other case b ( t ) − 1 ∉ L 1 ( 0 , ∞ ) b(t)^{-1}\notin L^1(0,\infty ) , i.e., the effective or noneffective damping.

We study blow-up behavior of solutions for the Cauchy problem of the semilinear wave equation with time-dependent damping. When the damping is effective, and the nonlinearity is subcritical, we show the blow-up rates and the sharp lifespan estimates of solutions. Upper estimates are proved by an ODE argument, and lower estimates are given by a method of scaling variables.

We consider the Cauchy problem for the wave equation with time-dependent damping and absorbing semilinear term � utt − Δu + b(t)ut + |u| ρ−1 u = 0, (t, x) ∈ R+ × R N ,

Of concern is the energy decay property of solutions to wave equations with time-dependent damping. A reasonable class of damping coefficients for the framework of weighted energy methods is proposed, which contains not only the model of “effective” damping (1+t)^{-beta }(-1le beta <1), but also non-differentiable functions with a suitable behavior at trightarrow infty . As an application of the weighted energy estimate, global existence for the corresponding semilinear wave equation is discussed.

We consider the Cauchy problem for the semilinear wave equation with time-dependent damping $$ \left\{ \begin{array}{ll} u_{tt} - \Delta u + b(t)u_t=|u|^{\rho}, & (t,x) \in \mathbb{R}^+ \times \mathbb{R}^N \\ (u,u_t)(0,x) = (u_0,u_1)(x), & x \in \mathbb{R}^N. \end{array}\right. (*) $$ When $b(t)=b_0(t+1)^{-\beta}$ with $b_0>0$ and $-1 < \beta <1$ and $\int_{{\bf R}^N} u_i(x)\,dx >0\,(i=0,1)$, we show that the time-global solution of ($*$) does not exist provided that $1<\rho \leq \rho_F(N):= 1+2/N$ (Fujita exponent). On the other hand, when $\rho_F(N)<\rho<\frac{N+2}{[N-2]_+}:= \left\{ \begin{array}{ll} \infty & (N=1,2), (N+2)/(N-2) & (N \ge 3), \end{array} \right.$ the small data global existence of solution has been recently proved in [K. Nishihara, Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math. 34 (2011), 327-343] provided that $0 \le \beta<1$. We can prove the small data global existence even if $-1<\beta<0$. Thus, we conclude that the Fujita exponent $\rho_F(N)$ is still critical even in the time-dependent damping case. For the proofs we apply the weighted energy method and the method of test functions by [Qi S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001), 109--114].

Uniform attractors of non-autonomous dissipative semilinear wave equations*

We study the problem of the well-posedness for the abstract Cauchy problem associated to the non-autonomous one-dimensional wave equation utt = A(t)u with general Wentzell boundary conditions Here A(t)u := (a(x, t)ux)x, a(x, t) ≥ ε &gt; 0 in [0, 1] × [0, + ∞) and βj(t) &gt; 0, γj(t) ≥ 0, (γ0(t), γ1(t)) ≠ (0,0). Under suitable regularity conditions on a, βj, γj we prove the well-posedness in a suitable (energy) Hilbert space

Exact boundary observability for nonautonomous quasilinear wave equations

This article deals with the geometric control of a one-dimensional non-autonomous linear wave equation. The idea consists in reducing the wave equation to a set of first-order linear hyperbolic equations. Then, based on geometric control concepts, a distributed control law that enforces the exponential stability and output tracking in the closed-loop system is designed. The presented control approach is applied to obtain a distributed control law that brings a stretched uniform string, modelled by a wave equation with Dirichlet boundary conditions, to rest in infinite time by considering the displacement of the middle point of the string as the controlled output. The controller performances have been evaluated in simulation by considering both tracking and disturbance rejection problems. The robustness of the controller has also been studied when the string tension is subjected to sudden variations.

In this paper we analyze the asymptotic behavior of the pullback attractors for non-autonomous dynamical systems generated by a family of non-autonomous damped wave equations when some reaction terms are concentrated in a neighbourhood of the boundary and this neighbourhood shrinks to boundary as a parameter $\varepsilon$ goes to zero. We show the gradient-like structure of the limit pullback attractor, the existence and continuity of global hyperbolic solutions and the lower semicontinuity of the pullback attractors at $\varepsilon=0$. Finally, we obtain the continuity of the pullback attractors at $\varepsilon=0$.

We study the isentropic Euler equations with time-dependent damping, given by . Here, λ and μ are two non-negative constants to describe the decay rate of damping with respect to time. We will investigate the global existence and asymptotic behavior of small data solutions to the Euler equations when in multi-dimensions . Our strategy of proving the global existence is to convert the Euler system to a time-dependent damped wave equation and use a kind of weighted energy estimate. Investigation to the asymptotic behavior of the solution is based on the detailed analysis to the fundamental solutions of the corresponding linear damped wave equation and it coincides with that of standard results if λ deduces to zero.

In this paper, by means of ourZp index theory developed recently we study the existence of multiple periodic solutions for asymptotically linear nonautonomous wave equations. All previous known results rely either on the oddness of nonlinear terms, or on autonomous systems, and the best result for the general case is the existence of two nontrivial periodic solutions (Amann, Zehnder, K. C. Chang, S. P. Wu, S. J. Li, Z. Q. Wang). In this paper, under the assumption that the nonlinear term isT/p periodic we discuss multiple periodic solutions of nonautonomous systems and generalize a series of previous results.