Semilinear damped wave equation in locally uniform spaces

We study a damped wave equation with a nonlinear damping in the locally uniform spaces and prove well-posedness and existence of a locally compact attractor. An upper bound on the Kolmogorov's \begin{document}$\varepsilon$\end{document} -entropy is also established using the method of trajectories.

1. Introduction. We study the semilinear damped wave equation where f and g are nonlinear continuous functions described in more detail in Section 3, with initial conditions We focus on proving the well-posedness of the problem in the context of locally uniform spaces, the existence of a locally compact attractor and mainly on establishing an upper bound on the Kolmogorov's ε-entropy. We use the method of trajectories introduced in [16], which has been previously used in a similar context for showing the finite dimensionality of the global attractor of (1) in bounded domains in [17]. However, the approach applied to the bounded domain problem cannot be used directly due to a different nature of embeddings in weighted spaces and requires a slightly different technique. To this end, we introduce a variant of locally uniform spaces which seems suitable for equations with a finite speed of propagation. Also as usual in locally uniform spaces, the problem has an inherent non-compactness and non-separability. In order to obtain the dissipation of energy, we formulate additional assumptions that allow for the nonlinearities in the equation to be superlinear. One could expect that a suitable control of dispersion could yield dissipative estimates under still weaker growth restrictions on the nonlinearities. This equation has been intensely studied in the setting of bounded domains. The existence of a global attractor with supercritical nonlinearities has been shown in [8] and for critical nonlinearities in [19] with less restrictive conditions on the damping.
The finite dimensionality has been discussed in [17] and has been achieved even for critical nonlinearities in [4] and [12]. In the case of linear damping, the results include the existence of a global attractor on manifolds [11], asymptotic regularity of solutions of perturbed damped wave equation with nonlinearity of arbitrary growth [21] and recently in [10] the existence and smoothness of a global attractor for the equation with critical nonlinearity in d = 3.
In the context of locally uniform spaces, a linearly damped wave equation has been studied in [7] and [22]. In [22] the author has also established an upper bound on the Kolmogorov's ε-entropy of the locally compact attractor. Some results, including well-posedness and the existence of a locally compact attractor, have also been shown for a strongly damped wave equation in [20] and recently for a wave equation with fractional damping in [18]. To the best of our knowledge, nonlinear damping in this setting has not yet been studied.
The paper is organized as follows: In Section 2 we review the locally uniform spaces. In Section 3, the well-posedness of the equation (1) is established. In Section 4, we discuss additional assumptions that lead to dissipative estimates. In Section 5, we introduce the trajectory setting and prove a local variant of squeezing property, which is used in Section 6 to establish the existence of the locally compact attractor and an upper bound on its Kolmogorov's ε-entropy.
2. Function spaces. In this section we review the basic facts about weighted Sobolev spaces and locally uniform spaces. These spaces and their relations have been studied in [23] and [2].
By an admissible weight function of growth rate ν ≤ 0 we understand φ : R d → (0, +∞) measurable and bounded satisfying for some C φ ≥ 1 and every x, y ∈ R d . Forx ∈ R d and ε > 0 we define the weight function φx ,ε with center inx and decay rate ε by φx ,ε (x) = e −ε|x−x| .
(3) Then clearly for every multiindex α there exists C α > 0 such that We emphasize that, thanks to [2, Proposition 4.1], the particular choice of the weight function (3) does not play any role in the definition of the locally uniform spaces below as long as certain decay properties are met. Also note that by the above definition, φx ,ε is an admissible weight function with growth ε.
In the special case p = 2 we use the notation We denote the scalar product on L 2 x,ε (R d ) by (·, ·)x ,ε . The scalar product on L 2 (R d ) will be denoted by (·, ·). We will sometimes omit the space domain R d in the notation of function spaces.
Clearly the embedding L p The weighted Sobolev spaces are defined in an obvious manner and allow the continuous embedding The weighted spaces also allow certain compact embeddings. More precisely, let k ≥ l and q ≥ p be such that W k,p (B(0, 1)) → → W l,q (B(0, 1)). Then we have the compact embedding x,ε (R d ) withε > εp/q, which gives that for example the embedding where 1 < m < ∞ and 1 ≤ s < 2d/(d − 2), is compact and continuous for s = 2d/(d − 2). Let C k denote a closed unit cube in R d centered at The weighted locally uniform Lebesgue space L p b,φ for p ∈ [1, ∞) and an admissible weight function φ is defined by If φ ≡ 1, we omit the subscript and write for example L 2 b instead of L 2 b,1 . For p = 2 we use a simplified notation are neither separable nor reflexive. Locally uniform Sobolev spaces are again constructed in a straightforward manner. The standard embeddings holding on bounded domains also hold for locally uniform spaces, namely However, none of these embeddings are compact.
The weighted Lebesgue spaces and the locally uniform spaces are connected through the equivalence of the locally uniform norm. The following lemmata are standard and their proofs can be found e.g. in [2,9] and [23]. Lemma 2.1. Let ε > 0, 1 ≤ p < ∞, k ∈ N 0 and let φ be an admissible weight function with growth ν < ε.
Moreover, the norm is equivalent to the original W k,p b,φ (R d )norm.
Then there exist constants C 1 and C 2 such that Finally we define so-called parabolic locally uniform space These spaces and their weighted variants have been studied in [9].
3. Well-posedness for locally uniform data. In this section we prove the existence and uniqueness of weak solutions of (1) for infinite energy data. We will make use of the following energy spaces which arise in the case of (1) in unbounded domains We consider Φ b as the phase space for the asymptotic analysis. However, it is well known that the locally uniform spaces are not separable, hence there are problems with attaining the initial conditions and approximating less regular data. There are at least two ways how to overcome this inconvenience. The first one is to consider Sobolev spaces with the weight functions like φx ,ε with better properties. The second way is to use a phase space which is defined as closure of smooth functions in · Φ b (such approach was considered e.g. in [15]). Both settings combined with the finite speed of propagation property of wave equations lead to the uniqueness and existence result. We have chosen the second approach.
Let us denote L p loc (I; W k,p loc (Ω)) the set of measurable functions u on I × Ω such that for any compact J ⊆ I and K ⊆ Ω we have u ∈ L p (J; W k,p (K)). Particularly, u ∈ L ∞ loc (0, ∞; X) means that u is strongly (Bochner) measurable on (0, ∞) and u ∈ L ∞ loc (0, T ; X) for any T > 0. We impose the following requirements on the nonlinearities of studied equation: for every r ∈ R : where γ j , β > 0. In what follows, we consider the following set of parameters: where 2 * = 2d/(d − 2). Also from (G1) we observe holds for every u, v ∈ R.
The identity (11) has an equivalent version closely connected to the energy space Φ ε,x . By using ψφ ε,x with ψ ∈ D((0, ∞) × R d ) as a test function in (11), we obtain By a standard density argument, the identity (12) holds also for any function ψθ where θ = θ(t) is a smooth function compactly supported in (0, ∞) and is satisfied for every T ∈ (0, ∞). To this end, we test (12) by ψθ n with , θ n (0) = 0 where η is the standard non-negative mollifier compactly supported in (−1, 1) and (14) by letting n → ∞ and using the continuity of ψ with respect to time. The weak formulation is therefore equivalent to (14) with test functions (13).
The lack of regularity of u t with respect to the space variables prevents us from using it as a test function in (14). On the other hand, one can test the weak formulation (14) by the time difference x,ε ) we have for a fixed t ∈ (0, ∞) In the rest of the paper, with an obvious abuse of terminology, we will use the phrase "testing by u t " instead of taking the time differences as test functions and sending τ → 0 + . For more details see e.g. [13].
Theorem 3.2. Let µ and p be as in (6) there exists a unique weak solution of (1) which satisfies the energy equality We remark that both existence and uniqueness of solutions can be shown even in the so-called super-critical case, particularly when The existence part remains as is below. Uniqueness follows by combination of the approach presented in [3] for bounded domains with the localisation technique developed in [1, Section 7]. As usual in the context of locally uniform spaces, one cannot expect the strong time continuity of solutions in the phase space Φ b . Taking d = 1, one can check that is a weak solution of However, is not continuous as the range of u is not separable. Moreover, the function u is not strongly (Bochner) measurable.
Proof of Theorem 3.2. Assume that ε > 0 andx are given. It is sufficient to show existence of solutions on (0, T ) for fixed T ∈ (0, ∞) independent on the initial data together with time continuity, particularly (u, u t ) ∈ C([0, T ]; Φx ,ε ). The existence of global solutions then follows from a continuation argument.
Step 1 -approximations and solutions on bounded domains. We approximate the nonlinear term f by Lipschitz functions and the initial data by compactly supported data. Let {f k } k be a sequence of functions such that for every k ∈ N the For k ∈ N, we define function as a direct consequence of approximation by mollifiers and decay of φx ,ε . Existence and uniqueness of strong solutions on bounded domains is a well known result (see e.g. [14]). The finite speed of propagation holds as the source term f k is Lipschitz. Hence, for every k ∈ N we can construct which is a global strong solution (the equation (18) is satisfied almost everywhere satisfying the initial conditions Moreover, u k (t, ·) is compactly supported for any t ∈ [0, ∞) and Step 2 -uniform estimates in weighted Lebesgue spaces. Let us multiply both sides of (18) by u k t φx ,ε and integrate the resulting equality w.
Step 3 -stability in C([0, T ]; Φx ,ε ) and existence. Let us subtract the equation for u l from the equation for u k , multiply the difference by (u k t − u l t )φx ,ε and integrate over R d with respect to x. Using the monotonicity of g and standard estimates, we obtain for every τ ∈ (0, T ). From Gronwall's lemma, we infer that Observe that (25) together with (16), (17), (21) and (24) gives Finally, we conclude that u k t → u t almost everywhere in (0, T ) × R d which in combination with (21) implies thatḡ = g(u t ). Summing up the results on convergence given above and noting that u k satisfies (11), we can pass to the limit in (14).
The energy equality (15) holds for u k and using the convergence results above, in particular (26), it follows that it holds also for u. The relation (10) follows from Gronwall's lemma and taking supremum overx ∈ R d (see also estimates leading to (23)).
Step 4 -uniqueness of weak solutions. Let us test the weak formulation for u 1 and u 2 by (u 1 − u 2 ) t keeping in mind that actually we test by D τ [u 1 − u 2 ] and send τ → 0. Subtracting both equalities. we obtain the energy equality in the following form: for any τ ∈ (0, T ); hence, we have according to the monotonicity of g. Using assumptions (F2) and (10), we get Indeed, the inner integral containing f can be estimated using Lemma 2.2 as follows: ) . Hence, u 1 (t) = u 2 (t) ∈ Φx ,ε almost everywhere in [0, T ] as a consequence of Gronwall's lemma.
Proof. Assume that (u 0 , u 1 ), (v 0 , v 1 ) ∈ B. Following the same line as in the proof of uniqueness, we obtain (27). Standard application of Gronwall's lemma gives Finally, applying supremum overx ∈ R d on both sides of (28), from Lemma 2.1 we infer

Dissipation of energy.
In contrast to the bounded domain case, the energy of the solutions does not necessarily decrease over time. This may be attributed to the last element in (12) and the absence of the embeddings between the weighted spaces of the same weight. Thus, it seems that an additional assumption has to be made in order to show any dissipation of energy. As we will see below, we can either have linearly bounded g and possibly superlinear function f , or we can ensure the dissipation by connecting the growths of the functions f and g. By a dissipation assumption, we understand one of the following: ) and there exists κ ∈ (0, 1) and C > 0 such that −g(r)s ≤ κf (s)s + C (g(r)r + 1) ∀r, s ∈ R.
The assumption (D1), i.e. linearly bounded damping, is well studied in the case of the bounded domain. The assumption (D2) is a variant of an assumption from [5] and allows for example the use of the functions where µ ∈ [1, 3) and p ∈ [µ, 3) (29) with d = 3 and 0 < a < α. However, we remark that the assumption (D2) implicitly gives restriction on the admissible growth exponent µ, for instance in the example (29) we observe that µ ≤ 3, though formally we allow µ < 5. The assumption (D2) seems to be unnatural since it prohibits the following simple choice of nonlinearities f (s) = 0, g(r) = r|r|. In the future, we hope that a less restrictive condition than (D2) will be obtained. We emphasize that the upper entropy bound established the last section does not depend on the particular choice of the dissipation condition.

5.
Locally uniform squeezing property. In this section we introduce the trajectory setting and prove that the solution operator in the space of trajectories satisfies a local variant of the so-called squeezing property (cf. [6]), which will in turn lead to the asymptotic compactness and an upper bound on Kolmogorov's ε-entropy. To achieve this, we require additional assumptions on µ and the damping nonlinearity g. We note that one can obtain the asymptotic compactness required for the existence of a locally compact attractor also without these additional assumptions by means of a standard decomposition argument. From now on, let h ≡ 0 and for simplicity we assume d = 3. In addition, we require µ ∈ [1, 7/3), p ∈ [0, 3), These assumptions and the properties of f lead to the estimates Let > 1 and v > 1 be fixed and let φ be an admissible weight function. We define the space of trajectories by where we denote Note that the half-cone {(t, x) ∈ K(0); 0 < t < } can be covered by a finite number of cones K(x j ), j ∈ N , x j ∈ B(0, 3v ). We emphasize that the size of N is independent of .
We define the operators e : B → Φ b and L(t) : B → B by e((χ, χ t )) = (χ( ), χ t ( )), Let O ⊆ R 3 and let φ be an admissible weight function. We define Assume for u is a sufficiently smooth solution of (1). Then using a standard result on differentiation, integration by parts and the equation (1) where n denotes the outward normal to Z k (t) in the space domain R 3 . Let χ 1 , χ 2 ∈ B and let u 1 , u 2 be the respective weak solutions. Set w = u 1 − u 2 and let 0 < t 1 < t 2 < 2 . Integrating the identity (38) over t ∈ (t 1 , t 2 ), approximating by more regular data and by mollification, we get to the equation where n is denotes the outward normal to ∂ Z k (t). Since v > 1, the boundary integral is non-positive and using (G1) and a similar estimate on the first element on the right-hand side of the previous equation as in the proof of uniqueness, we arrive to Invoking Gronwall's lemma we get for 0 < s < t < 2 . Integrating over s ∈ (0, ) and t ∈ ( , 2 ) leads to We multiply the equation by φ(x k ) and use the property (2) to get The Lipschitz continuity of L in E ,v b,φ follows by taking supremum over k ∈ N and estimating the maximum on the right-hand side by the supremum over j ∈ N. The Lipschitz continuity of e can be obtained in a similar manner.
The positive invariance of B follows immediately from the definitions.
Definition 5.2. We say that the mapping L : B → B has a locally uniform squeezing property (LUSP) for an admissible weight function φ if for every θ > 0 there exists > 1, v > 1, κ > 0 and N ⊆ N such that x j ∈ B(0, 3v ) ⊆ R d for every j ∈ N and for every k ∈ N and χ 1 , χ 2 ∈ B and the respective solutions u 1 , u 2 we have where The above definition contains a slight abuse of terminology as one has to first choose θ > 0 and only then find suitable and v to get the squeezing property of L = L( ) : B → B . However, this will not be of any concern later on as θ > 0 will be chosen only once. Let τ ∈ (0, ), χ 1 , χ 2 ∈ B with the respective solutions u 1 , u 2 and denote w = u 1 − u 2 . Similarly as in the proof of Lemma 5.1 we get Using the estimates (37) in (41) we have where We estimate the first element on the right-hand side using the dissipation of energy by where we put r 1 = 6/(p−1) and 1/r 1 +1/r 2 = 1/2, therefore r 2 ∈ (2, 6). Combining (43) and (44) we arrive to Returning to (42), by the estimates (37) we have Similarly as in (44) we estimate the fourth element on the right-hand side of (46) as where we use the dissipation of energy and set s 1 = 2/(µ − 1) and 1/s 1 + 1/s 2 = 1, therefore 2s 2 ∈ (2, 6). Similarly the fifth element (46) is estimated by where we again used the dissipation estimate and set z 1 = 6/(p−1) and 1/z 1 +1/z 2 = 1, therefore 2z 2 ∈ (2, 3). Set s = max(2s 2 , 2z 2 ). Combining the estimates (46-48) we obtain Define r = max(s, r 2 ). Multiply (49) by δ > 0, add it to (45) and choose v ≥ (1 + δ)/(1 − δ) and δ > 0 small enough to get for some ζ > 0 and integrate by τ from 0 to to obtain and divide the equation (50) by ζ . Next we employ Ehrling's lemma, namely for Ω = B(x, R) ⊆ R d with x ∈ R d , R > 0, γ > 0 arbitrary and C = C(γ, R), on the arguments of the split integrals. Indeed, this is possible since the diameters of the domains in question, i.e. Z k (t) for t ∈ (0, 2 ), are bounded. Combining these estimates with (50) we obtain where N (k) ⊆ N is a finite set of size N such that the union of cones K(x j ) over j ∈ N covers the cone K(x k ). Now letθ > 0 be such thatθC φ exp(ν3v ) < θ, where ν > 0 is the growth of the admissible function φ. By choosing sufficiently large and γ sufficiently small we get It remains to insert the weight function with sufficiently small growth which is easily done by multiplying (52) by ψ(x k ), invoking (2) and using the restriction onθ.
The critical case p = 3 contains essential difficulties and would be an interesting problem for the consideration in future. 6. Locally compact attractor and entropy estimate. Let M be a metric space and K ⊆ M be relatively compact. Let N ε (K, M ) denote the smallest number of balls of radii ε that cover K in M . We define the Kolmogorov's ε-entropy by H ε (K, M ) = ln N ε (K, M ).
A number of typical examples of upper and lower bounds on the Kolmogorov's ε-entropy in various situations can be found e.g. in [22].
The following lemma is crucial for the estimate of Kolmogorov's ε-entropy and considerably simplifies the proof of asymptotic compactness. We note that an estimate of this kind may be used to establish an infinite dimensional exponential attractor. We postpone this issue to a subsequent paper together with an abstract criterion and applications to other equations.
Proof. The proof adapts the techniques from [17, Lemma 4.1] and [9, Lemma 2.6], the main difference being working with hyperbolic trajectories space instead of parabolic ones.
Let k ∈ I(O). Define We equip the space X with the norm where y = (y j ; j ∈ N (k)) ∪ (z) ∈ X. Since B (and thus B) is uniformly bounded on every cone K(x i ), i ∈ N, by Aubin-Lions lemma there exists N ∈ N and (χ i , χ i t ) ∈ B, i = 1, . . . , N , such that It is important to note that N is independent of k and ε, which follows from the estimate y 2 X ≤ Cε 2 holding uniformly for ε > 0 and k ∈ I(O) with C depending only on the Lipschitz constant of L.