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Introduction
In this paper, we study the asymptotic behavior of solutions of a damped semilinear wave equation with nonlinearity of a critical growth exponent over the Euclidean space ℝn of arbitrary dimension n ≥ 3,
$$\begin{array}{}
\displaystyle
u_{tt} -\Delta u + \beta u_t +f(x, u)+\alpha u = g(x)
\end{array}$$
where α and β are arbitrary positive constants, g is a given functions defined on ℝn and f (x, u) is a nonlinear function satisfying some typical dissipative conditions to be specified.
The asymptotic dynamics of global weak solutions for deterministic nonlinear wave equations and for more general nonlinear hyperbolic evolutionary equations with linear or nonlinear damping have been studied in last three decades by many authors, e.g. [1]- [4], [6]- [8], [12]- [14], [16], [19]- [22], [26]. The obtained results focused on the existence of global attractors under certain assumptions.
In the arena of stochastic wave equations driven by additive or multiplicative noise, the solution mapping defines a random dynamical system or called a cocycle on a state space with a parametric base space. The existence of random attractors for stochastic damped wave equations has been studied in [9], [11], [15], [17],[18], [23]- [25].
However, the existence problem of global attractors remains open for damped nonlinear wave equations with nonlinearity of a critical growth exponent and on the unbounded domain Rn with arbitrary dimension. This is the topic of this work.
In case of nonlinearity with higher or critical growth exponents and on the unbounded domain, the issue of asymptotic compactness for the weak or mild solutions of nonlinear damped wave equations becomes difficult to handle due to not only the lack of compactness of the Sobolev embeddings but also the necessarily involved high-order integrable function spaces, in addition to the local existence and regularity of solutions in such spaces. In this work we shall tackle this challenging problem and prove the existence of a global attractor by means of
1) the uniform estimates for absorbing property and norm-smallness of solutions outside a large ball,
2) the esimates of the extended energy functional for the compactness in the space H1(ℝn) × L2(ℝn) and
3) the Vitali-type convergence criterion (Theorem 8) for the function space Lp(ℝn) shown in the paper. This new approach has potential applications to many other nonlinear and stochastic PDEs and to longtime and asymptotic dynamics of various problems with complex and nonlinear interactions.
In Section 2, we briefly recall basic concepts and results related to semiflow and global attractors. In Section 3, we shall conduct uniform estimates of the weak solutions for absorbing sets and for tail parts. In Section 4, we shall establish the intricate asymptotic compactness of the solution semiflow with respect to the Hilbert energy space H1(ℝn) × L2(ℝn). In Section 5, we prove the crucial asymptotic compactness of the first component of solutions in Lp(ℝn). Then the existence of a global attractor for this nonlinear damped wave equation is finally proved.
In this paper, we shall use || · || and 〈·,·〉 to denote the norm and inner product of L2(ℝn), respectively. The norm of Lr(ℝn) with r ≠ 2 or a Banach space X will be denoted by || · ||r or || · ||X . We use c, C or Ci to denote generic or specific positive constants.
Preliminaries and Assumptions
Let (X,|| · ||X ) be a real Banach space. The following are the basic concepts and result on the topic of global attractor for infinite dimensional dynamical systems, cf. [2], [8], [20] and [22].
Definition 1
A mapping Φ : ℝ+ × X → X is called a semiflow on X, if the following conditions are satisfied:
(i) Φ(0, ·) is the identity on X.
(ii) Φ(t + s, ·) = Φ(t, Φ(s, ·)), for any t, s ≥ 0.
(iii) Φ : ℝ+ × X → X is a continuous mapping.
Definition 2
Let Φ be a semiflow on X. A bounded set K ⊂ X is called an absorbing set for Φ if for any bounded subset B ⊂ X there exists a finite time TB > 0 such that
Now we formulate the original initial value problem of the nonlinear damped wave equation (1)-(2). Let ξ = ut + δ u, where δ is a positive number to be specified later. Then (1)-(2) can be rewritten as
where the linear operator A = −Δ : H2(ℝn) → L2(ℝn).
Standing Assumption. Throughout the paper, assume that the nonlinear term f ∈ C1(ℝn × ℝ, ℝ), n ≥ 3, and its antiderivative $\begin{array}{}
\displaystyle
F(x, u)=\int^u_0 f(x, s)ds
\end{array}$ satisfy the following conditions:
where C1, C2 and C3 are positive constants and $\begin{array}{}
\displaystyle
1 \leq p \leq \frac{n+2}{n-2}
\end{array}$ is arbitrarily given. Assume that g ∈ H1(ℝn).
weakly in E, provided that g0, m = (u0,m, v0, m) ⇀ g0 = (u0, v0) weakly in E.
Proof. The local existence and uniqueness of a weak solution for this problem (4) in the phase space E = (H1(ℝn) ⋂ Lp(ℝn)) × L2(ℝn) and its weakly continuous dependence on the initial data can be established by the Galerkin approximation method as in [8, Chapter XV] and [3]. Also see [20], [22] and [25]. Here the detail is omitted. The proof of the global existence of weak solutions will be included in the proof of Lemma 3 below.
Uniform Estimates of Solution Trajectories
In this section, we shall derive uniform estimates on the solutions of the nonlinear damped wave equation(4) defined on ℝn in a long run. These a priori estimates pave the way to proving the existence of absorbing set and the asymptotic compactness of the semiflow Φ. In particular, we will show that tails of the solutions for large spatial variables are uniformly small when time is sufficiently large.
Obviously the norm || · ||E in (9) and the Sobolev norm || · ||(H1⋂Lp)×L2 in (8) are equivalent.
Absorbing Set
The next lemma shows that there exists an absorbing set in the Banach space E for the semiflow Φ generated by the weak solutions (u(t, u0), (v(t, v0)) to the problem (4),
Therefore, this set K = BE(0, R) is an absorbing set in the phase space E for the solution semiflow Φ. The proof is completed.
Tail Estimate
Next we conduct uniform estimates on the tail parts of the weak solutions for large spatial and time variables. These estimates play key roles in proving the asymptotic compactness in the space E of the dynamical systems F generated by the nonlinear wave equation (4) on the unbounded domain ℝn.
Lemma 4
For every bounded set B ⊂ E and 0 < η ≤ 1, there exists T = T (B, η) > 0 and V = V (η) ≥ 1 such that the semiflow Φ generated by the nonlinear damped wave equation (4) satisfies
for all t > T and every r > V , where$\begin{array}{}
\displaystyle
\zeta_{B_r^c } (x)
\end{array}$is the characteristic function of the set {x ∈ ℝn : |x| > r}.
Proof. Choose a smooth and nondecreasing function ρ such that 0 ≤ ρ(s) ≤ 1 for all s ∈ [0, ∞) and
$$\begin{array}{}
\displaystyle
\rho(s)=\left\{
\begin{array}{l}
0, \qquad if \;\; 0 \leq s <1,\\ \\
1, \qquad if \;\; s >2,
\end{array} \right.
\end{array}$$
with 0 ≤ ρ′(s) ≤ 2 for s ≥ 0. Taking the inner product of the second equation of (4) with ρ(|x|2/r2)v in L2(ℝn), we get
Next we conduct estimates of the terms on the right-hand side of in (30). For the first term, there exists T1 = T1(B, η) > 0 and a constant C4 > 0 such that
for all t > T1. For the second integral term on the right-hand side of (30), applying the Gronwall inequality to(14) while taking the spatial integral over the region $\begin{array}{}
\displaystyle
r \leq |x| \leq
\sqrt{2} r
\end{array}$, with (13) in mind, we get
Now assemble all these estimates and substitute (31) and (33) into (30). It shows that for any bounded set B and any 0 < η ≤ 1, as long as r > V = max {K0, K1} and t > max {T1, T2}, one has
where $\begin{array}{}
\displaystyle
R = \sqrt{2}r
\end{array}$. (35) implies that (20) is satisfied as stated in this theorem just by renaming r to be R and η to be ((1 + 1/C3)(2 + M)η)1/2 + ((1 + 1/C3)(2 + M)η)1/p. The proof is completed.
Asymptotic Compactness in H1(ℝn) × L2(ℝn)
In this section, we shall prove the asymptotic compactness in the space H1(ℝn) × L2(ℝn) of the solution semiflow Φ associated with the nonlinear damped wave equation (4).
Lemma 5
The following statements hold for Lp(ℝn).
1) For 1 ≤ p < ∞, let {ψm} be a sequence and ψ be a function in Lp(ℝn) such that ||ψm − ψ||Lp → 0 as m → ∞. Then there exists a subsequence {ψmk} such that
Proof. Since ℝn with the Lebesgue measure is a σ-finite measure space, the first item is a standard result in Real and Functional Analysis.
For the second item, since Lp(ℝn) is a reflexive Banach space for 1 < p < ∞, the boundedness of {ψm} in Lp(ℝn) implies that there is φ ∈ Lp(ℝn) such that ψm → φ weakly as m → ∞. By Mazur’s lemma, this weak convergence implies there exists a sequence {ζm} ⊂ Lp(ℝn) such that
From the condition ψm → ψ a.e. and ${\zeta _m} \in {\rm{\;conv }}(\cup_{i = m}^\infty {\psi _i})$, it follows that
$$\begin{array}{}
\displaystyle
{\zeta _m} \to \psi \,\,{\text{a}}{\rm{.e}}{\rm{. in }}{\mathbb{R}^n}.
\end{array}$$
On the other hand, by the first statement in this lemma, the strong convergence in (38) implies that there exists a subsequence {ζmk} such that ζmk → σ a.e. as k → ∞. Therefore, (39) leads to ψ = φ a.e. on ℝn so that ψm → ψ weakly as m → ∞. The third item is a known result in Functional Analysis, cf. [5, Chapter 4]. Thus the proof is completed.
Let us define the following energy functional on E: for (u, v) ∈ E,
For every bounded set B ⊂ E and any integer k > 0, there exists a constant M1 = M1(B, k) > 0 such that for all m > M1 one has tm > k with the property that
for all t ∈ [0, k] and (u0, m, v0,m) ∈ B, where the constant R is the same as in (19).
Proof. Integrate the inequality (14) over the time interval [0, t] ⊂ [0, k], where δ ≥ σ by (13). Similar to (18), there exists M1 = M1(B, k) > 0 such that for all m > M1 one has tm > k and
For every bounded set B and for any sequences tm → ∞ and g0,m = (u0,m, v0,m) ∈ B, the sequence$\begin{array}{}
\displaystyle
\{\Phi(t_m, g_{0,m})\}^\infty_{m=1}
\end{array}$has a strongly convergent subsequence in H1(ℝn) × L2(ℝn), where Φ is the solution semiflow generated by the nonlinear damped wave equation (4).
Proof. The proof goes through the following steps.
Step 1
By Lemma 3, there is a constant M2 = M2(B) > 0 such that for all m ≥ M2 and g0,m ∈ B, we have
where R > 0 is given by (19). Then there is $\begin{array}{}
\displaystyle
(\tilde{u}, \tilde{v}) \in E
\end{array}$ such that, up to a subsequence and relabeled as the same,
If so, then (46) and (48) will lead to $\lim\nolimits_{m\rightarrow \infty}\|\Phi(t_m, g_{0,m})\|_{H^1 \times L^2} = \|(\tilde{u}, \tilde{v})\|_{H^1 \times L^2}$. By the item 3 of Lemma 5, we shall obtain (47).
Step 2
By Lemma 6 and (7), there exists a constant C > 0 such that, for any given integer k > 0 and all m ≥ M1(B, k), one has tm > k and
for any (u0,m, v0,m) ∈ B. In particular, (49) is satisfied for t = k.
According to Banach-Alaoglu theorem, there exists a sequence $\begin{array}{}
\displaystyle
\{\tilde{u}_k, \tilde{v}_k\}^\infty_{k=1}
\end{array}$ in the space E and subsequences of $\begin{array}{}
\displaystyle
\{t_m\}_{m=1}^\infty
\end{array}$ and $\begin{array}{}
\displaystyle
\{(u_{0, m},
v_{0, m})\}^\infty_{m=1}
\end{array}$ again relabeled as the same, such that for every integer k ≥ 1,
as m → ∞, which can be extracted through a diagonal selection procedure as in Real Analysis.
By the weakly continuous dependence on the initial data of the weak solutions stated in Lemma 2, here the weak convergence (50) together with the concatenation
2) For the second term on the right-hand side of (57), by (50) and the weakly continuous dependence of solutions on the initial data stated in Lemma 2, we find that for any ξ ∈ [0, k], when m → ∞,
A) For any given η > 0, by the proof of Lemma 4 adapted to the time interval [k, ∞), there exist M3 = M3(B, η) > M2 and K = K(B, η) ≥ 1 such that for ξ ∈ [0, k], whenever r > K and m > M3, one has
by the weakly continuous dependence of solutions on intial data stated in Lemma 2, by the weak lower-semicontinuity of the L2 and LP norms, it yields from (64) that
Since $\begin{array}{}
\displaystyle
H^1(\mathbb{B}_r)
\end{array}$ is compactly embedded in $\begin{array}{}
\displaystyle
L^2(\mathbb{B}_r)
\end{array}$, it follows that for any ξ ∈ [0, k],
Same as the second inequality in (46), from the weak convergence shown by (45), for any sequence $\begin{array}{}
\displaystyle
\{g_{0,m} = (u_{0,m}, v_{0,m})\}_{m=1}^\infty \subset B
\end{array}$, we have
Finally, for the Hilbert space H1(ℝn) × L2(ℝn), the weak convergence (45) and the norm convergence (77) imply the strong convergence. Therefore, up to finite steps of subsequence selections (always relabeled as the same), we reach the conclusion that
In this section we shall first prove an instrumental convergence theorem in the space Lp(X,ℳ, μ) of Vitalitype. It will pave the way to prove asymptotic compactness of the first component of the semiflow Φ in the space Lp(ℝn) for any exponent $\begin{array}{}
\displaystyle
1 \le p \le \frac{{n + 2}}{{n - 2}}
\end{array}$. This is the crucial and final step to accomplish the proof of the existence of a global attractor for this dynamical system F for the nonlinear damped wave equation (1).
Theorem 8
Let (X, ℳ, μ) be a σ-finite measure space and assume that a sequence$\begin{array}{}
\displaystyle
\left\{ {{f_m}} \right\}_{m = 1}^\infty \subset {L^p}(X,{\mathscr M},\mu )
\end{array}$with 1 ≤ p < ∞ satisfies
Statement (b): By the absolutely continuous property of Lebesgue integral on a σ-finite measure space, for any given ε > 0, there exists δ0 = δ0(ε) > 0 such that whenever μ(Y ) < δ0 one has
Next we prove the sufficiency. Suppose the two conditions (a) and (b) are satisfied. First of all, by the condition (a) and Fatou’s Lemma, for an arbitrarily given ε > 0 there exists a set Aε of finite measure with
By Egorov’s theorem on Lebesgue integral over such a set Y of finite measure in the space (X,ℳ, μ), there exists a measurable subset B ⊂ Y with μ(Y\B) < δ1 such that
Therefore, (87) is proved. The proof is completed.
Finally we present and prove the main result of this work on the existence of a global attractor for this semiflow Φ generated by the nonlinear damped wave equation (1) on the product Banach space with critical exponent and arbitrary space dimension.
Theorem 9
Under the Standing Assumption, the semiflow Φ generated by the nonlinear damped wave equation (1) in the converted problem (4) on the space E = (H1(ℝn) ⋂ Lp(ℝn)) × L2(ℝn) has a global attractor A in E.
Proof.Lemma 3 shows that there exists an absorbing set, the K = BE(0, R) in the space E for the semiflow Φ. It suffices to prove that the semiflow F is asymptotically compact in E.
(1) Theorem 7 shows that for any given bounded set B ⊂ E and any sequences tm → ∞ and {g0,m = (u0,m, v0,m)} ⊂ B, the sequence $\begin{array}{}
\displaystyle
\{\Phi (t_m, g_{0,m})\}_{m=1}^\infty
\end{array}$ has a convergent subsequence, which is relabeled by the same, such that
(2) Applying the first item in Lemma 5 to the space L2(ℝn), it follows from (91) that there exists a subsequence $\begin{array}{}
\displaystyle
\{\Phi(t_{m_k}, \, g_{0, m_k})\}_{k=1}^\infty
\end{array}$ of $\begin{array}{}
\displaystyle
\{\Phi(t_m, \, g_{0, m})\}_{m=1}^\infty
\end{array}$ such that
Therefore, the assumption (78) in Theorem 8 is satisfied by the sequence of functions $\begin{array}{}
\displaystyle
\{\mathbb{P}_u \Phi (t_{m_k}, \, g_{0, m_k}) (x)\}_{k=1}^\infty
\end{array}$ in Lp(ℝn).
(3) By Lemma 4, for any ε > 0, there exists an integer k0 = k0(B, ε) > 0 and V = V (ε) ≥ 1 such that for all k > k0 one has
Thus (95) and (96) confirm that with Aε = Bmax{V, V0} the condition (a) in Theorem 8 is satisfied by the sequence of functions $\begin{array}{}
\displaystyle
\{\mathbb{P}_u \Phi (t_{m_k}, g_{0, m_k}) (x)\}_{k=1}^\infty
\end{array}$ in Lp(ℝn).
(4) Finally we prove that the uniform absolutely continuous condition (b) of Theorem 8 is also satisfied by the sequence of functions $\begin{array}{}
\displaystyle
\{\mathbb{P}_u \Phi (t_{m_k}, g_{0, m_k}) (x)\}_{k=1}^\infty
\end{array}$ in Lp(ℝn).
According to the Standing Assumption, for any measurable set Y ⊂ ℝn, we have
Due to the absolute continuity of the respective Lebesgue integrals of the functions ϕ1(x), ϕ2(x), ϕ3(x) and g involved in the above inequality (99), for an arbitrarily given η > 0, there exists μ0 = μ0(η) > 0 such that for any measurable set Y ⊂ ℝn with μ(Y ) < μ0 one has
whenever a measurable set Y ⊂ ℝn satisfies μ(Y) < min{μ0, μ1}. Therefore,
$$\begin{array}{}
\displaystyle
\mathop {\lim }\limits_{\mu (Y) \to 0} {\int _Y}|{\mathbb{P}_u}\Phi ({t_{{m_k}}},{g_{0,m}})(x){|^p}dx = 0\,\,\,{\text{uniformly for all }}\,k \ge 1,
\end{array}$$
so that the condition (b) of Theorem 8 is also satisfied by the sequence of functions $\begin{array}{}
\displaystyle
\{\mathbb{P}_u \Phi (t_{m_k}, \, g_{0,m_k})(x)\}_{k=1}^\infty
\end{array}$ in Lp(ℝn).
As checked by the above steps (2), (3) and (4) in this proof, all the conditions in Theorem 8 are satisfied by the sequence of functions $\begin{array}{}
\displaystyle
\{\mathbb{P}_u \Phi (t_{m_k}, \, g_{0, m_k}) (x)\}_{k=1}^\infty
\end{array}$ in Lp(ℝn). Then we apply Theorem 8 to obtain
$$\begin{array}{}
\displaystyle
\mathop {\lim }\limits_{k \to \infty } {\mathbb{P}_u}\Phi ({t_{{m_k}}},{g_{0,{m_k}}}) = \tilde u \,\,\,{\text{strongly in }}\,{L^p}({\mathbb{R}^n}).
\end{array}$$
Finally, combination of (91) and (105) shows that there exists a convergent subsequence $\begin{array}{}
\displaystyle
\{\Phi(t_{m_k}, g_{0, m_k})\}_{k=1}^\infty
\end{array}$ of the sequence $\begin{array}{}
\displaystyle
\{\Phi (t_m, g_{0,m})\}_{m=1}^\infty
\end{array}$ in the space E = (H1(ℝn) ⋂ Lp(ℝn)) × L2(ℝn). Therefore, the semiflow Φ on the Banach space E is asymptotically compact.
According to Theorem 1, we conclude that there exists a global attractor 𝒜 in E for this semiflow Φ generated by the original nonlinear damped wave equation (1). The proof is completed.