Smooth attractors for the quintic wave equations with fractional damping
aa r X i v : . [ m a t h . A P ] J un SMOOTH ATTRACTORS FOR THE QUINTIC WAVEEQUATIONS WITH FRACTIONAL DAMPING
ANTON SAVOSTIANOV AND SERGEY ZELIK
Abstract.
Dissipative wave equations with critical quintic nonlinearity anddamping term involving the fractional Laplacian are considered. The addi-tional regularity of energy solutions is established by constructing the newLyapunov-type functional and based on this, the global well-posedness anddissipativity of the energy solutions as well as the existence of a smooth globaland exponential attractors of finite Hausdorff and fractal dimension is verified.
Contents
1. Introduction 12. Preliminaries 53. Energy solutions 94. Uniqueness and smoothing property 135. The attractors 166. The case of Dirichlet boundary conditions 217. Appendix. Properties of the extension operator 25References 271.
Introduction
Wave equations with the damping term involving the fractional Laplacian(1.1) ∂ t u + γ ( − ∆ x ) θ ∂ t u − ∆ x u + f ( u ) = 0 , x ∈ Ω ⊂ R n , where γ > θ ∈ [0 ,
2] and f ( u ) is a nonlinear interaction function, are of bigpermanent interest. These equations model various oscillatory processes in a lossymedia including the nonlinear elasticity, electrodynamics, quantum mechanics, etc.We mention here also the relatively recent applications to the acoustic waves prop-agation in viscous/viscoelastic media. Then, the coefficient θ in the non-local termof the equation is related with the power law for the dependence of the acousticattenuation a ( ω ) on the angular frequency ω , namely,(1.2) a ( ω ) = a | ω | θ , see [30, 7] and references therein. The classical choices of the exponent θ are θ = 0 and θ = 1 which correspond to the usual damped wave equations and Mathematics Subject Classification.
Key words and phrases. damped wave equation, fractional damping, critical nonlinearity,global attractor, smoothness.This work is partially supported by the Russian Ministry of Education and Science (contractno. 8502). the so-called strongly damped wave equations respectively although an increasingattention is attracted for the general case θ ∈ (0 ,
2) as well, see for instance [26]and the literature cited therein.Mathematical properties of equation (1.1) and the related equations, includingthe well-posedness, regularity and the asymptotic behavior of solutions have beenstudied in many papers, see [1, 2, 4, 5, 6, 8, 9, 11, 12, 16, 18, 19, 22, 23, 24, 25, 29, 33]and references therein. These properties depend strongly on the value of θ and onthe growth rate of the nonlinearity f . For instance, equation (1.1) is hyperbolic when θ = 0 and (under natural assumptions on f , e.g., for f = 0) and is solvableforward and backward in time, so the solution operators S ( t ) generate a C -groupin the proper phase space and do not possess any smoothing property on finitetime, only the asymptotic smoothing as t → ∞ holds, see e.g., [29].In the case, 0 < θ < , equation (1.1) generates the so-called C ∞ -semigroup, soit is already solvable only forward in time and possesses the smoothing property onfinite time (even starting from the non-smooth initial data, the solution becomessmooth for t > analytic , see [8]. In contrastto that, the solution semigroup is analytic for ≤ θ < parabolic , see [9]. For instance, in the borderline case θ = 1 / γ = 2the change of variable v = ∂ t u +( − ∆ x ) / u transforms (1.1) to the following system:(1.3) ( ∂ t u + ( − ∆ x ) / u = v,∂ t v + ( − ∆ x ) / v = f ( u )and the parabolicity is obvious. For general γ = 2, the change of variable is v = ∂ t u + β ( − ∆ x ) / u where β − γβ + 1 = 0, so β will be complex if γ < γ < θ = 1, the solution semigroup remains analytic, but neverthelessequation (1.1) is not parabolic and possesses only partial smoothing on a finite time,see [19, 25] for more details.Let us now discuss the dependence of (1.1) on the growth rate of the nonlinearity f . For simplicity, we restrict ourselves to consider only the case where Ω is abounded domain in R with proper boundary conditions and the nonlinearity f satisfies the following natural assumption:(1.4) − C + κ | u | q ≤ f ′ ( u ) ≤ C (1 + | u | q ) , q > , where κ, C > f ( u ) when u → ∞ is determined by the growth exponent q . Note that (1.1)possesses the following energy equality at least on a formal level:(1.5) ddt (cid:18) k ∂ t u k L (Ω) + 12 k∇ x u k L (Ω) + ( F ( u ) , (cid:19) = − γ k ( − ∆ x ) θ/ ∂ t u k L (Ω) , where F ( s ) = R s f ( v ) dv is the potential of the non-linearity f and ( u, v ) standsfor the scalar product of functions u and v in H := L (Ω). By this reason, the energy solutions are defined in such way that all terms in that inequality havesense (at least after integration in time), namely, for the case of Dirichlet boundary UINTIC WAVE EQUATION 3 conditions, the following regularity of energy solutions is assumed:(1.6) ( u ( t ) , ∂ t u ( t )) ∈ E := [ H (Ω) ∩ L q +2 (Ω)] × L (Ω) , ( − ∆ x ) θ/ ∂ t u ∈ L ([0 , T ] , L (Ω)) . Here and below, H s (Ω) denotes the Sobolev space of distributions whose derivativesup to order s belong to L (Ω) and H s is the closure of C ∞ (Ω) in the metric of H s (Ω).As usual, for sufficiently small q which is less than the critical exponent q ∗ = q ∗ ( θ )(the so-called sub-critical case), the non-linearity is subordinated to the linear partof the equation, so the analytic properties of it remains similar to the linear case f = 0. In particular, we have the existence, uniqueness and dissipativity of the en-ergy solutions in an almost straightforward way. However, nothing similar works inthe super-critical case q > q ∗ ( θ ) where usually only the existence of a weak solutionwithout the uniqueness and further regularity is known. As expected, equationswith super-critical non-linearities may generate singularities in finite time even star-ing with smooth initial data (similarly, e.g., the so-called self-focusing phenomenonin the non-linear Schr¨odingier equation or the gravitational collapse in general rel-ativity) and this, in particular, may cause also the non-uniqueness. Unfortunately,to the best of our knowledge, no such examples are known for equations (1.1) withdissipative non-linearities f satisfying (1.4), so being pedantic, the precise value ofthe critical exponent q ∗ ( θ ) is not known. Moreover, it is a priori not excluded thatsmooth solutions of the concrete equation (1.1) do not blow up in finite time, forinstance, due to the presence of some non-trivial extra Lyapunov type functionalscontrolling the higher norms of the solution, then q ∗ ( θ ) = ∞ and energy solutionsof (1.1) are well-posed and dissipative no matter how fast is the growth of thenon-linearity f .As shown in [19], see also [25] that is indeed the case when θ = 1, so the blowup of smooth solutions is impossible and the uniqueness theorem holds for theenergy solutions which are asymptotically smooth as t → ∞ no matter how fastthe non-linearity f is growing. It is also indicated there that(1.7) q ∗ ( θ ) = ∞ , θ ∈ [ 34 , , so the critical growth exponent does not exist for θ ≥ as well. In the case θ ∈ ( , ), the result of [19] gives the estimate(1.8) q ∗ ( θ ) ≥ θ − θ , θ ∈ ( 12 ,
34 ) . The non-parabolic case θ ∈ [0 , ) is more delicate. In the purely hyperbolic case θ = 0 the assumption q ≤ q ∗ (0) = 4) is expected hereas critical. This conjecture is partially verified in [3] (based on the recent progressin Strihartz-type estimates for bounded domains), where the global well-posednessof weak solutions has been proved for the case of quintic hyperbolic equation. Tothe best of our knowledge, the corresponding attractor theory for that equation inbounded domains is not developed yet. QUINTIC WAVE EQUATION
In the other borderline case θ = , it is also expected that the critical exponent q ∗ (1 /
2) = 4, so the natural conjecture here is q ∗ ( θ ) = 4 , θ ∈ [0 ,
12 ] . The sub-critical case q < θ = 1 / q = 4 are also obtained there based on the so-called mildsolutions. However, to the best of our knowledge, the analog of the so-called non-concentration effect which is typical for the quintic wave equations (with θ = 0), see[3, 28], is not known in the case when θ = , so these results are not very helpful forproving the absence of blow up of smooth solutions and/or building up a reasonableattractor theory. Thus, clarifying the situation with the quintic nonlinearity and θ = was a long-standing open problem.The aim of this paper is to give a solution of this problem. Namely, we willshow that the energy solution of (1.1) in 3D bounded domain with θ = andquintic non-linearity is unique and is smooth (say, u ( t ) ∈ H (Ω)) for all t > E = H (Ω) × L (Ω)is dissipative and possesses a smooth global attractor of finite Hausdorff and fractaldimension. To prove this result, we show that any energy solution possesses thefollowing extra regularity:(1.9) u ∈ L ([0 , T ] , H / (Ω))which is enough to verify the above stated properties in a more or less standardway.Note that the regularity (1.9) does not follow from the energy equality (1.5)and, analogously to [19], some extra Lyapunov-type functionals are necessary forobtaining it. Namely, as not difficult to see that the desired estimated will beobtained if we would succeed to multiply equation (1.1) on ( − ∆ x ) / u (or whichis the same, to multiply the second equation of (1.3) by v ) and estimate the non-linear term with fractional Laplacian: ( f ( u ) , ( − ∆ x ) / u ). To estimate this term,we utilize the well-known formula k u k H s (Ω) ∼ k u k L (Ω) + Z x ∈ Ω Z y ∈ Ω | u ( x ) − u ( y ) | | x − y | s dx dy, s ∈ (0 , , see, e.g., [31]. In particular, for more simple case of periodic boundary conditionsΩ = T := [0 , π ] , we establish the following representation:(1.10) ( f ( u ) , ( − ∆ x ) s u ) == c s Z h ∈ R Z x ∈ T ( f ( u ( x + h )) − f ( u ( x )) , u ( x + h ) − u ( x )) | h | s dx dh, where c s > f and u . Thisestimate together with the left inequality of (1.4) imply that( f ( u ) , ( − ∆ x ) / u ) ≥ − C k u k H / (Ω) and this is enough to verify (1.9), see Sections 2 for the details.The case of a general bounded domain Ω ⊂ R endowed by the Dirichlet bound-ary conditions is a bit more difficult since we do not know the analogue of (1.10)for that case and have to proceed in a different way using the odd extension of UINTIC WAVE EQUATION 5 the solution u through the boundary. Then, the already obtained energy estimateoccurs to be sufficient for estimating the extra terms arising under the extension,so with the help of this trick, we actually reduce the general case to the case ofperiodic boundary conditions considered before, see Section 6.The paper is organized as follows.The definitions of functional spaces as well as assumptions on the non-linearityand external forces used throughout of the paper are given in Section 2. Moreover,the key technical tool (1.10) is verified here for the case of periodic boundaryconditions.The weak energy solutions are introduced and studied in Section 3. In particular,extra regularity (1.9) is verified here in the case of periodic boundary conditions.Moreover, as shown there, that extra regularity is enough to verify that any energysolution satisfies the energy equality (1.5).The well-posedness and smoothing property for the energy solutions are veri-fied in Section 4 under the assumption (1.9) of extra regularity. The existence ofglobal and exponential attractors for the associated solution semigroup are provedin Section 5.Finally, the case of Dirichlet boundary conditions is considered in Section 6. Inthat section, we verify the extra regularity (1.9) for the Dirichlet case using the oddextension of a solution through boundary. Important properties of such extensionoperator are collected in Appendix 7.To conclude, we note that the methods developed in the paper work not onlyfor the borderline case θ = and not only for quintic nonlinearities. We give thefurther applications in the forthcoming paper.2. Preliminaries
In that section, we briefly remind the properties of fractional Sobolev spacesrelated with our problem and verify the key formula (1.10). In order to be able toconsider the cases of Dirichlet and periodic boundary conditions from the unifiedpoint of view, we add the extra damping term α∂ t u to equation (1.1) and willconsider the following problem:(2.1) ( ∂ t u + γ ( − ∆ x ) ∂ t u + α∂ t u − ∆ x u + f ( u ) = g,u (cid:12)(cid:12) t =0 = u , ∂ t u (cid:12)(cid:12) t =0 = u , either in a bounded smooth domain Ω ⊂ R with Dirichlet boundary conditionsor on a torus Ω = T := [0 , π ] with periodic boundary conditions. Here ∆ x isa Laplacian with respect to the variable x , u = u ( t, x ) is an unknown function, γ and α are fixed strictly positive numbers, g ∈ L (Ω) and f ∈ C ( R ) satisfies thefollowing growth and dissipativity assumptions:(2.2) . f ( u ) u ≥ − C + κ | u | , . f ′ ( u ) ≥ − K, . | f ′ ( u ) | ≤ C (1 + | u | ) , where C , κ , and K are given positive constants. Finally, the initial data ξ u (0) :=( u , u ) is assumed belonging to the energy space E := H (Ω) × L (Ω) . QUINTIC WAVE EQUATION
Here and below H s (Ω), s ∈ R , stands for the classical Sobolev spaces of distributionswhose derivatives up to order s belong to L (Ω) and H s (Ω) means the closure of C ∞ (Ω) in the metric of H s (Ω). Recall that, for the non-integer positive values of s the norm in the space H s (Ω) is defined via the interpolation:(2.3) k u k H s := k u k H [ s ] + Z x,y ∈ Ω | D [ s ] u ( x ) − D [ s ] u ( y ) | | x − y | s − [ s ]) dx dy, where [ s ] is the integer part of s and D [ s ] stands for the collection of all partialderivatives of order [ s ]. For negative s , the space H s (Ω) is defined by the duality: H s (Ω) := [ H − s (Ω)] ∗ , see e.g., [31] for the details. The standard inner product in L (Ω) will be denoted by ( u, v ).We now remind the fractional powers of the Laplacian. To this end, let 0 ≤ λ ≤ λ ≤ · · · be the eigenvalues of the Laplacian − ∆ x and { e i } ∞ i =1 ⊂ C ∞ (Ω) be theassociated eigenvectors. Then, due to the Parseval equality, k u k L (Ω) = ∞ X i =1 u i , u i := ( u, e i ) , u = ∞ X i =1 u i e i . The scale of Hilbert spaces H s ∆ , s ∈ R , associated with the Laplacian is defined asa completion of L (Ω) with respect to the following norm:(2.4) k u k H s ∆ := ∞ X i =1 (1 + λ i ) s u i , u := ∞ X i =1 u i e i and the fractional Laplacian ( − ∆ x ) θ , θ ≥
0, acts from H s ∆ to H s − θ ∆ via the follow-ing expression:(2.5) ( − ∆ x ) θ u := ∞ X i =1 λ θi u i e i , u = ∞ X i =1 u i e i . Recall also that, in the case of Dirichlet boundary conditions, λ > − ∆ x ) θ is an isomorphism between H s ∆ and H s − θ ∆ for all θ, s ∈ R . In the case ofperiodic boundary conditions λ = 0, so the operator ( − ∆ x ) θ is not invertible andone should replace it by ( − ∆ x + 1) θ in order to restore the isomorphism.The relations between the Sobolev spaces H s (Ω) and the spaces H s ∆ in the caseof Dirichlet boundary conditions are well-known at least in the case of smoothdomains. Namely, for s > s = 1 / , / , / , · · · one has(2.6) H s ∆ = H s (Ω) ∩ { u (cid:12)(cid:12) ∂ Ω = − ∆ x u (cid:12)(cid:12) ∂ Ω = · · · = ( − ∆ x ) [( s − / u (cid:12)(cid:12) ∂ Ω = 0 } and, in particular, by duality, H s ∆ = H s (Ω) for − / ≤ s ≤ s = − /
2. Incontrast to that, in the case of, say, s = 1 /
2, the space H / is a proper (dense)subset of H / (Ω) = H / (Ω) determined by the following norm:(2.7) k u k H / ∼ k u k H / (Ω) + Z Ω d ( x ) − | u ( x ) | dx < ∞ , where d ( x ) is a distance from x to the boundary ∂ Ω and, by duality, H − / (Ω) ⊂ H − / ⊂ D ′ (Ω) , see [21] or [31] for more details. UINTIC WAVE EQUATION 7
We now consider the special case Ω = T with periodic boundary conditions. Inthat case, since there is actually no boundary, we have the equality H s (Ω) = H s ∆ for all s ∈ R . Moreover, the eigenvectors of the Laplacian ( − ∆ x ) are now thecomplex exponents e ik.x , k ∈ Z (here and below a.b stands for the usual dotproduct in R ), with the corresponding eigenvalues | k | , so the Parseval equalityreads(2.8) k u k L (Ω) = (2 π ) X k ∈ Z | u k | , u ( x ) = X k ∈ Z u k e ik.x , u k := 1(2 π ) ( u, e ik.x )and also, for s ≥ d ( k u k L + k ( − ∆) s u k L ) k u k H s d ( k u k L + k ( − ∆) s u k L ) , where d and d are some positive constants depending on s .The next representation of the norm k ( − ∆ x ) s/ u k L is crucial for what follows. Lemma 2.1.
Let s ∈ (0 , and u ∈ H s ( T ) , then the following identity holds: (2.10) k ( − ∆ x ) s u k L = c Z R Z T | u ( x + h ) − u ( x ) | | h | s dxdh, for some strictly positive c = c s which depends only on s .Proof. Indeed, let u ∈ H s ( T ), where s ∈ (0 ,
1) and(2.11) u ( x ) = X k ∈ Z u k e ik.x . Then by Parseval equality, Z T ( u ( x + h ) − u ( x )) dx = (2 π ) X k ∈ Z | u k | | e ik.h − | == (2 π ) X k ∈ Z | c k | sin ( k.h/ . Consequently,(2.12) Z R Z T ( u ( x + h ) − u ( x )) | h | s dxdh = 32 π X k ∈ Z | u k | Z R sin ( k.h/ | h | s dh = (cid:12)(cid:12)(cid:12)(cid:12) h := z | k | (cid:12)(cid:12)(cid:12)(cid:12) = 32 π X k ∈ Z | u k | | k | s Z R sin (cid:16) k | k | .z (cid:17) | z | s dz. Due to the rotation invariance of | z | , we have(2.13) 4 Z R sin (cid:16) k | k | .z (cid:17) | z | s dz = 4 Z R sin ( y / | y | s dy := c − . Since s ∈ (0 ,
1) the last integral is finite and, consequently, Z R Z T ( u ( x + h ) − u ( x )) | h | s dxdh = c − (2 π ) X k ∈ Z | u k | | k | s = c − k ( − ∆ x ) s/ u k L . Thus, Lemma 2.1 is proved. (cid:3)
QUINTIC WAVE EQUATION
Using (2.10) together with the obvious identity a ( u, v ) = 14 ( a ( u + v, u + v ) − a ( u − v, u − v ))which holds for any bilinear form a ( u, v ), we conclude that, for any u, v ∈ H s ( T ),(2.14) ( v, ( − ∆ x ) s v ) = (( − ∆ x ) s/ u, ( − ∆ x ) s/ v ) == c Z R Z T ( u ( x + h ) − u ( x ))( v ( x + h ) − v ( x )) | h | s dxdh. In particular, taking v = f ( u ) in the last formula and using that, due to the secondassumption of (2.2) and the integral mean value theorem,( f ( u ( x + h )) − f ( u ( x )))( u ( x + h ) − u ( x )) == Z f ′ ( κu ( x + h ) + (1 − κ ) u ( x )) dκ | u ( x + h ) − u ( x ) | ≥ − K | u ( x + h ) − u ( x ) | , we see that(2.15) ( f ( u ) , ( − ∆ x ) s u ) ≥ − K k ( − ∆ x ) s/ u k L ( T ) ≥ − C k u k H s ( T ) hold at least for sufficiently smooth functions u for which the integrals in the leftand right-hand sides of (2.15) have sense and, therefore, the key estimate (1.10) isverified. To be able to apply this estimate to less regular functions u for which theexistence of that integrals is not known a priori, we introduce, for every ε >
0, thecut-off kernels(2.16) θ ε ( z ) := ( | z | , | z | ≥ ε,ε, | z | ≤ ε and the associated bilinear forms(2.17) [ u, v ] s,ε := Z R Z T ( u ( x + h ) − u ( x ))( v ( x + h ) − v ( x )) θ ε ( h ) s dx dh. Then, on the one hand, obviously,(2.18) [ u, v ] s,ε ≤ [ u, u ] s,ε [ v, v ] s,ε , [ u, u ] s,ε ≤ c − k ( − ∆ x ) s/ u k L ( T ) ≤ C k u k H s ( T ) , where C is independent of ε >
0. Moreover, analogously to (2.15), we have(2.19) [ f ( u ) , u ] s,ε ≥ − K [ u, u ] s,ε . On the other hand, since the kernel θ ε ( h ) is no more singular, the integrals in (2.19)have sense, for instance if u ∈ L p ( T ) is such that f ( u ) ∈ L q ( T ) with p + q = 1.In the next section, we will use this estimate in the situation when p = 6.We conclude the section by one more obvious lemma which allows us to estimatethe H s -norms using the smoothed norms (2.17). Lemma 2.2.
Let the function u ∈ L ( T ) be such that [ u, u ] s,ε ≤ C for all ε > , where C is independent of ε . Then, u ∈ H s ( T ) and (2.20) k ( − ∆ x ) s/ u k L ≤ c lim inf ε → [ u, u ] s,ε . Indeed, the assertion of the lemma is an immediate corollary of the fact that ε → φ ε ( h ) is monotone increasing and the Fatou lemma. UINTIC WAVE EQUATION 9 Energy solutions
The aim of this section is to introduce and study the weak energy solutions forproblem (2.1). In particular, as will be shown below, any such solution possessesan extra space-time regularity at least in the periodic case (the case of Dirichletboundary conditions will be considered later in Section 6). This extra regularity willbe essentially used later in order to verify existence, uniqueness and dissipativityof energy solutions.We start with recalling the basic energy equality for problem (2.1). Indeed,multiplying formally (2.1) by ∂ t u and integrating in x ∈ Ω, after integration byparts (using the Dirichlet or periodic boundary conditions), we get(3.1) ddt E ( u ( t ) , ∂ t u ( t )) + α k ∂ t u ( t ) k L (Ω) + γ k ( − ∆ x ) / ∂ t u ( t ) k L (Ω) = 0 , where(3.2) E ( u, v ) = 12 k v k L (Ω) + + 12 k∇ u k L (Ω) + ( F ( u ) , − ( g, u )and F ( u ) := R u f ( s ) ds . Integrating (3.1) in time, we formally have(3.3) E ( u ( t ) , ∂ t u ( t ))++ Z t α k ∂ t u ( s ) k L (Ω) + γ k ( − ∆ x ) / ∂ t u ( s ) k L (Ω) ds = E ( u (0) , ∂ t u (0)) . As usual, weak energy solutions are expected to have minimal regularity whichguarantees that all terms in (3.3) are well defined. Note that, due to the growthrestriction of (2.2), | F ( u ) | ≤ C (1 + | u | )which together with the embedding H ⊂ L show that the energy functional(3.2) is naturally well-defined and continuous on the energy space ( u, v ) ∈ E . Thisjustifies the following definition. Definition 3.1.
A function ξ u := ( u, ∂ t u ) is a weak energy solution of problem(2.1) if(3.4) ξ u ∈ L ∞ (0 , T ; E ) , ∂ t u ∈ L (0 , T ; H / ) ,ξ u (0) = ( u , u ) and equation (2.1) is satisfied in the sense of distributions. Thelatter means that, for any T > φ ∈ C ∞ ((0 , T ) × Ω),(3.5) − Z T ( ∂ t u, ∂ t φ ) dt + Z T ( ∇ u, ∇ φ ) dt + α Z T ( u t , φ ) dtγ Z T ( ∂ t u, ( − ∆ x ) / φ ) dt + Z T ( f ( u ) , φ ) dt = Z T ( g, φ ) dt. Since ξ u ( t ) is a priori only in L ∞ (0 , T ; E ), the equality ξ u (0) = ( u , u ) requiressome explanations. First, since u, u t ∈ L ∞ (0 , T ; L ), we conclude that u ∈ C ([0 , T ] , L (Ω))and the initial value for u ( t ) has a sense. To verify the continuity in time of ∂ t u ( t ),we need the information on ∂ t u ( t ). To this end, we note that, using the growthrestriction (2.2) on f and Sobolev embedding H ⊂ L one can easily verify that f ( u ) ∈ L ∞ (0 , T ; H − (Ω)) and then from (3.5), we conclude that(3.6) ∂ t u ∈ L ∞ (0 , T ; H − (Ω)) . Indeed, using definition of distributional derivative together with (3.5), we have(3.7) < ∂ t u, φ > = − Z T ( ∇ u, ∇ φ ) dt − Z T ( ∂ t u, αφ + γ ( − ∆ x ) / φ ) dt − Z T ( f ( u ) , φ ) dt + Z T ( g, φ ) dt for any φ ∈ C ∞ ((0 , T ) × Ω). Thus, we see that the distribution ∂ t u can be extendedby continuity to the linear functional on L (0 , T ; H (Ω)) and (3.6) now follows fromthe evident fact that [ L (0 , T ; H (Ω))] ∗ = L ∞ (0 , T ; H − (Ω)). Therefore, ∂ t u ∈ C ([0 , T ] , H − (Ω))and the initial data for ∂ t u is also well-defined. Remark 3.2.
Since ξ u ∈ L ∞ (0 , T ; E ) ∩ C ([0 , T ] , E − ) , where E − := L (Ω) × H − (Ω), we conclude that the value ξ u ( t ) is well defined forany t ∈ [0 , T ] and the function t → ξ u ( t ) is weakly continuous as a function withvalues in E :(3.8) ξ u ∈ C w ([0 , T ] , E ) , see e.g., [20]. Therefore, all terms in the energy equality (3.3) indeed have sensefor any t ≥ u . Nevertheless, the regularity (3.4)is a priori not enough for establishing the validity of the energy equality since wedo not have enough regularity to take φ = ∂ t u in (3.5) (the most difficult is theterm ( f ( u ) , ∂ t u ), see below), so to the best of our knowledge neither the validity ofenergy equality nor the strong continuity of the function t → ξ u ( t ) in E was knownbefore for the energy solutions of (2.1) with quintic nonlinearity. Below we willestablish these facts based on (2.15) and the extra energy type functional.The next theorem gives the existence of energy solutions and their dissipativity. Theorem 3.3.
Let the nonlinearity f satisfies (2.2) , α, γ > , g ∈ L (Ω) and ξ u (0) = ( u , u ) ∈ E . Then problem (2.1) (endowed by Dirichlet or periodic bound-ary conditions) possesses at least one weak energy solution ξ u ( t ) in the sense ofDefinition 3.1 which satisfies the following estimate: (3.9) k ξ u ( t ) k E + Z t +1 t k ∂ t u ( s ) k H / ds ≤ Q ( k ξ u (0) k ) e − βt + Q ( k g k L (Ω) ) , where β > and Q is a monotone increasing function which is independent of t and u . The proof of this theorem is completely standard for the theory of damped waveequations. Indeed, to derive (3.9) formally it is sufficient to multiply equation (2.1)by ∂ t u + εu for some positive ε , integrate by parts and apply the Gronwall inequality.The existence of a solution as well as the justification of estimate (3.9) can be doneusing, say, the Galerkin approximations, see [2, 20] for the details. UINTIC WAVE EQUATION 11
We are now ready to state the main result of this section on the extra regularityof energy solutions. For simplicity, we restrict ourselves to consider first only thecase of periodic boundary conditions (the analogous result for Dirichlet boundaryconditions will be obtained in Section 6).
Theorem 3.4.
Let
Ω := T (with periodic boundary conditions). Then, anyweak energy solution u of (2.1) in the sense of Definition 3.1 belongs to the space L (0 , T ; H / (Ω)) and the following estimate holds: (3.10) k u k L ([ τ,τ +1] ,H / (Ω)) ≤≤ Q ( k ξ u k L ∞ ([ τ,τ +1] , E ) + k ∂ t u k L ([ τ,τ +1] ,H / ) + k g k L (Ω) ) , where the monotone function Q is independent of τ ∈ R + and u .Proof. We give first the formal derivation of estimate (3.10). To this end, wemultiply equation (2.1) by ( − ∆ x ) / u and integrate over x ∈ Ω. Then, integratingby parts and using (2.15), we get(3.11) ddt (cid:16) ( ∂ t u, ( − ∆ x ) / u ) + γ k ( − ∆ x ) / u k L (Ω) + α k ( − ∆ x ) / u k L (Ω) (cid:17) ++ k ( − ∆ x ) / u k L (Ω) ≤ C ( k g k L (Ω) + k u k H (Ω) ) + k ( − ∆ x ) / ∂ t u k L (Ω) ) . Integrating this inequality in time t ∈ [ τ, τ + 1] and using the Cauchy-Schwartzinequality again, we end up with(3.12) Z τ +1 τ k ( − ∆ x ) / u ( s ) k L (Ω) ds ≤ Z τ +1 τ k ( − ∆ x ) / ∂ t u ( s ) k L (Ω) ds ++ C (cid:16) k g k L (Ω) + k u ( τ ) k H (Ω) + k u ( τ + 1) k H (Ω) + k ∂ t u ( τ ) k L (Ω) ++ k ∂ t u ( τ + 1) k L (Ω) + Z τ +1 τ k u ( s ) k H (Ω) ds (cid:19) which gives the desired estimate (3.10). Thus, it only remains to justify the aboveestimates. To this end, we will take the inner product (2.17) with s = 1 / ε > u . This, is equivalent to taking the test function φ h,ε ( x ) := − u ( x + h ) − u ( x ) + u ( x − h ) θ ε ( h ) in (3.7) with further integration with respect to h ∈ R . Since the function φ h,ε ∈ L (0 , T ; H (Ω)), then this integration is justified. Using now (2.18) and (2.19)together with the standard formulas[∆ x u, u ] / ,ε = − [ ∇ x u, ∇ x u ] / ,ε , [ ∂ t u, u ] / ,ε = ∂ t [ ∂ t u, u ] / ,ε − [ ∂ t u, ∂ t u ] / ,ε , we end up with the following analogue of (3.12):(3.13) Z τ +1 τ [ ∇ x u ( s ) , ∇ x u ( s )] / ,ε ds ≤≤ [ ∂ t u ( s ) , u ( s )] / ,ε (cid:12)(cid:12) s = τ +1 s = τ ++ C ( k g k L + k u ( τ ) k H (Ω) ) + C Z τ +1 τ k ∂ t u ( s ) k H / (Ω) + k u ( s ) k H (Ω) ds. We estimate the middle term in (3.13) as follows:(3.14) [ ∂ t u, u ] / ,ε = [( − ∆ x + 1) − / ∂ t u, ( − ∆ x + 1) / u ] / ,ε ≤≤ [( − ∆ x +1) − / ∂ t u, ( − ∆ x +1) − / ∂ t u ] / ,ε +[( − ∆ x +1) / u, ( − ∆ x +1) / u ] / ,ε ≤≤ C ( k ( − ∆ x ) / ( − ∆ x + 1) − / ∂ t u k L + k ( − ∆ x ) / ( − ∆ x + 1) / u k L ) ≤ C k ξ u k E . Inserting this estimate in (3.13), passing to the limit ε → (cid:3) Remark 3.5.
We emphasize that the above proof works only in the case of periodicboundary conditions although, as we will see later, estimate (3.10) remains true alsofor the case of Dirichlet boundary conditions. On the other hand, the above givenproof essentially uses only the fact that f ′ ( u ) ≥ − K and the growth restrictions on f is nowhere essentially used, so the above result remains true in the supercriticalcase of faster than quintic growth rate as well. Corollary 3.6.
Let the energy solution of problem (2.1) satisfy estimate (3.10) .Then, for every s ∈ [0 , , u ∈ L /s (0 , T ; L / (1 − s ) (Ω)) and the following estimateholds: (3.15) k u k L /s ([ t,t +1] ,L / (1 − s ) (Ω)) ≤ Q s ( k ξ u (0) k E ) e − βt + Q ( k g k L (Ω) ) , where the monotone function Q s depends on s , but is independent of t and u . Indeed, (3.15) follows from (3.10), (3.9) and the interpolation inequality k u k L /s ([ t,t +1] ,L / (1 − s ) (Ω)) ≤ C s k u k − sL ∞ ([ t,t +1] ,H (Ω)) k u k sL ([ t,t +1] ,H / (Ω)) . The next result shows that the extra regularity (3.10) is enough to verify the energyequality.
Corollary 3.7.
Let the energy solution u ( t ) of problem (2.1) (with periodic orDirichlet boundary conditions) satisfy estimate (3.10) . Then the function t → E ( u ( t ) , ∂ t u ( t )) is absolutely continuous and the energy identity (3.1) holds for al-most all t . In particular, ξ u ∈ C ([0 , T ] , E ) .Proof. Indeed, using the (3.15) with s = , the embedding H / ⊂ L (Ω) andgrowth restriction (2.2) on the nonlinearity f , we see that k f ( u ) k L ( t,t +1; H − / ) ≤ C k f ( u ) k L ( t,t +1; L / (Ω)) ≤ C (1 + k u k L ( t,t +1; L / (Ω)) ) ≤ C u and, therefore, f ( u ) ∈ L (0 , T ; H − / ). Moreover, due to estimate (3.10), ∆ x u ∈ L (0 , T ; H − / ) and, due to Definition 3.1, ( − ∆ x ) / ∂ t u ∈ L (0 , T ; H − / ). Thus,from equation (2.1), we derive that ∂ t u ∈ L (0 , T ; H − / ) as well.Since all terms of equation (2.1) belong to the space L (0 , T ; H − / ) and ∂ t u ∈ L (0 , T ; H / ) = [ L (0 , T ; H − / )] ∗ , the multiplication of the equation on ∂ t u (= taking the test function φ = ∂ t u in (3.5)) is allowed. In addition, this regularity implies that the functions t →k ∂ t u ( t ) k L and t → k∇ x u ( t ) k L are absolutely continuous and( ∂ t u ( t ) , ∂ t u ( t )) = 12 ddt k ∂ t u ( t ) k L , ( ∇ x u ( t ) , ∇ x ∂ t u ( t )) = 12 ddt k∇ x u ( t ) k L , UINTIC WAVE EQUATION 13 for almost all t , see e.g., [29]. Finally, approximating the function u by smoothfunctions and passing to the limit, one verifies that the function t → ( F ( u ( t )) ,
1) isabsolutely continuous and ddt ( F ( u ( t )) ,
1) = ( f ( u ) , ∂ t u )for almost all t . Thus, the energy equality is proved. The continuity of ξ u ( t ) asa E -valued function follows in a standard way from the energy equality and thecorollary is proved. (cid:3) Uniqueness and smoothing property
In that section we show that the extra regularity (3.10) is sufficient to verify thewell-posedness and smoothness of energy solutions. Everywhere in that and nextsections we assume that any energy solution satisfies (3.10). As we have alreadyseen that is the case when the periodic boundary conditions are posed (as will beshown below, that is also true in the case of Dirichlet boundary conditions undersome additional technical assumptions). We start with the uniqueness result.
Theorem 4.1.
Let the assumptions of Theorem 3.3 hold and let, in addition, allenergy solutions satisfy (3.10) . Then the energy solution of problem (2.1) is uniqueand for the difference v ( t ) of two energy solutions u ( t ) and u ( t ) the followingestimate is valid: (4.1) k v ( t ) k H (Ω) + k ∂ t v ( t ) k L (Ω) e ˆ Kt ( k v (0) k H (Ω) + k ∂ t v (0) k L (Ω) ) , ∀ t ∈ [0 , T ] , where ˆ K is a positive constant depending on k ξ u (0) k E , k ξ u (0) k E and k g k L (Ω) .Proof. The function v ( t ) as difference of two solutions u ( t ) and u ( t ) solves(4.2) ( ∂ t v + α∂ t v + γ ( − ∆) / ∂ t v − ∆ x v + f ( u ) − f ( u ) = 0 ,ξ v (0) = ξ u (0) − ξ u (0) . Since both u and u belong to L (0 , T ; H / (Ω)), arguing as in Corollary 3.7, wecan justify the multiplication of (4.2) by ∂ t v and obtain the following identity(4.3) 12 ddt k ξ v ( t ) k E + γ k ( − ∆ x ) v t k L + α k ∂ t v ( t ) k L = − ( f ( u ) − f ( u ) , ∂ t v ) . We estimate the right-hand side of (4.3) using the integral mean value theorem,growth restriction (2.2) on f ′ , and the H¨older inequality with exponents 3, 6 and 2:(4.4) | ( f ( u ) − f ( u ) , v t ) | ≤ (cid:18)Z | f ′ ( λu + (1 − λ ) u ) | dλ, | v || v t | (cid:19) C (cid:0) (1 + | u | + | u | ) , | v || v t | (cid:1) ≤ C (1 + k u k L (Ω) + k u k L (Ω) ) k v k H k ∂ t v k L ≤≤ C (1 + k u k L (Ω) + k u k L (Ω) ) k ξ v k E . Combining (4.3) and (4.4), we have(4.5) ddt k ξ v ( t ) k E + 2 γ k ( − ∆ x ) / ∂ t v ( t ) k L (Ω) + 2 α k ∂ t v ( t ) k L (Ω) C (1 + k u ( t ) k L (Ω) + k u ( t ) k L (Ω) ) k ξ v ( t ) k E . Using now (3.15) with s = 1 /
2, we get(4.6) k u i k L (0 ,T ; L (Ω)) ≤ ( Q ( k ξ u i (0) k E ) + Q ( k g k L (Ω) ))( T + 1) , i = 1 , , and the Gronwall inequality applied to (4.5) gives the desired estimate (4.1) andfinishes the proof of the theorem. (cid:3) The next proposition gives us additional smoothness of solutions assuming thatthe initial data is more regular. We start with the estimate which is divergent astime tends to infinity, the analogous dissipative estimate will be obtained later.
Proposition 4.2.
Let assumptions of Theorem 4.1 hold and the initial data (4.7) ξ u (0) ∈ E := [ H (Ω) ∩ H (Ω)] × H (Ω) . Then ξ u ( t ) belongs to E for any positive t and the following estimate holds: (4.8) k ξ ∂ t u ( t ) k E + k ξ u ( t ) k E ≤ Q ( k ξ u (0) k E + k g k L (Ω) ) e Q ( k ξ u (0) k E + k g k L )( t +1) , for some monotone increasing function Q which is independent of t ≥ and u .Proof. We give below only the formal derivation of estimate (4.8) which can bejustified in a standard way using e.g., the Galerkin approximations. To this end,we differentiate (2.1) in time and denote v ( t ) := ∂ t u ( t ). Then this function solves(4.9) ∂ t v + α∂ t v + γ ( − ∆ x ) / ∂ t v − ∆ x v + f ′ ( u ) v = 0 , ξ v (0) := ( ∂ t u (0) , ∂ t u (0)) , where ∂ t u (0) is defined via(4.10) ∂ t u (0) := ∆ x u (0) − α∂ t u (0) − γ ( − ∆ x ) / ∂ t u (0) − f ( u (0)) + g. Using the embedding H ⊂ C , it is not difficult to see that k ∂ t u (0) k L (Ω) ≤ Q ( k ξ u (0) k E + k g k L (Ω) )and, therefore, k ξ v (0) k E ≤ Q ( k ξ u (0) k E + k g k L (Ω) )for some monotone increasing function Q . Analogously, for every t ≥
0, we have(4.11) k ξ v ( t ) k E ≤ Q ( k ξ u ( t ) k E + k g k L (Ω) ) , where Q is independent of t . Vice versa, multiplying equation (2.1) by − ∆ x u ( t )(for every fixed t ), integrating by parts and using that f ′ ≥ − K , we end up with k ∆ x u ( t ) k L (Ω) ≤ C (cid:16) k ξ v ( t ) k E + k u ( t ) k H (Ω) + k g k L (Ω) (cid:17) and, therefore,(4.12) k ξ u ( t ) k E ≤ C (cid:16) k ξ v ( t ) k E + k ξ u ( t ) k E + k g k L (Ω) (cid:17) . Thus, to verify (4.8) it is sufficient to estimate the quantity k ξ v ( t ) k E only. To thisend, we multiply equation (4.9) by ∂ t v and integrate over Ω. Then, arguing as inthe derivation of (4.5), we end up with(4.13) ddt k ξ v ( t ) k E + δ k ∂ t v ( t ) k H / ≤ C (1 + k u ( t ) k L (Ω) ) k ξ v ( t ) k E , for some δ >
0. Applying the Gronwall inequality to (4.13) and using the estimate(4.6) for the L ( L )-norm of u together with (4.12) and (4.11), we end up withthe desired estimate (4.8) and finish the proof of the proposition. (cid:3) UINTIC WAVE EQUATION 15
The next proposition gives the parabolic smoothing property for the energysolutions.
Proposition 4.3.
Let assumptions of Theorem 4.1 hold. Then any energy solution u ( t ) of problem (2.1) possesses the following smoothing property for t ∈ (0 , : (4.14) t (cid:0) k ξ ∂ t u ( t ) k E + k ξ u ( t ) k E (cid:1) ≤ Q ( k ξ u (0) k E ) + Q ( k g k L (Ω) ) , for some monotone increasing function Q which is independent of t and u .Proof. Again, we give only the formal derivation of estimate (4.14) which can bejustified, say, by the Galerkin approximations. Multiplying inequality (4.13) for v := ∂ t u by t and using that2 t k∇ x v ( t ) k L (Ω) = 2 ddt ( t ( ∇ x u ( t ) , ∇ x v ( t ))) − ∇ x u ( t ) , ∇ x v ( t )) −− t (( − ∆ x ) / u ( t ) , ( − ∆ x ) / ∂ t v ( t ))and that2 t k ∂ t v ( t ) k L (Ω) ≤ Ct k ∂ t v ( t ) k H − / k ∂ t v ( t ) k H / ≤≤ δt k ∂ t v ( t ) k H / + C k ∂ t u ( t ) k H − / , after the elementary estimates, we get(4.15) ddt (cid:0) t k ξ v ( t ) k E − t ( ∇ x u ( t ) , ∇ x v ( t )) (cid:1) −− C (1 + k u ( t ) k L (Ω) )( t k ξ v ( t ) k E ) ≤ C (cid:16) k ∂ t u ( t ) k H − / + k u ( t ) k H / (Ω) (cid:17) . Integrating this inequality in time and using that the norms k ∂ t u k L (0 , H / ) and k ∂ t u k L (0 , H − / ) are under the control (see the proof of Corollary 3.7 concerningthe second norm) as well as the inequality | t ( ∇ x u ( t ) , ∇ x v ( t )) | ≤ t k ξ v ( t ) k E + 2 k ξ u ( t ) k E , we end up with t k ξ v ( t ) k E ≤ C Z t (1 + k u ( s ) k L (Ω) ) s k ξ v ( s ) k E ds + Q ( k ξ u (0) k E + k g k L (Ω) ) . Using finally that the L (0 , L (Ω))-norm of u is also under the control, see (4.6),and applying the Gronwall inequality to the last estimate, we end up with thedesired estimate (4.14) for the E -norm of ξ v ( t ). The analogous estimate for the E -norm of ξ u ( t ) is now an immediate corollary of (4.12) and the proposition isproved. (cid:3) We are now ready to establish the dissipativity of (2.1) in E . Corollary 4.4.
Let the assumptions of Theorem 4.1 hold and let, in addition, ξ u (0) ∈ E . Then, the following estimate holds: (4.16) k ξ u ( t ) k E + k ξ ∂ t u ( t ) k E ≤ Q ( k ξ u (0) k E ) e − βt + Q ( k g k L (Ω) ) , where β > and the monotone function Q are independent of t and ξ u (0) . Proof.
Indeed, according to Proposition 4.3,(4.17) k ξ u ( t + 1) k E ≤ Q ( k ξ u ( t ) k E ) + Q ( k g k L (Ω) )for some monotone function Q independent of t and u . Combining this estimatewith the dissipative estimate (3.9), we get the desired estimate (4.16) for t ≥
1. Toobtain estimate (4.16) for t ≤
1, it is sufficient to use Proposition 4.2. Thus, thecorollary is proved. (cid:3) The attractors
In that section, we prove the existence of global and exponential attractors forproblem (2.1) under the additional assumption that any energy solution possessesthe additional regularity (3.10). We start with summarizing the important prop-erties of the energy solutions obtained above. First, according to Theorem 4.1, inthat case problem (2.1) generates a semigroup S ( t ) in the energy phase space E via(5.1) S ( t ) : E → E , S ( t ) ξ := ξ u ( t ) , where ξ u ( t ) is a unique energy solution of (2.1) such that ξ u (0) = ξ ∈ E . Second,this semigroup is dissipative due to estimate (3.9) and is locally Lipschitz continuousdue to estimate (4.1). Third, due to estimates (4.16) and (4.17), the ball B R of asufficiently large radius R of E will be a compact in E (since E is reflexive and B R is convex, B R is closed in E as well) absorbing set for this semigroup, i.e., for everybounded set B in E there is time T = T ( B ) such that S ( t ) B ⊂ B R , ∀ t ≥ T. As usual, based on B R one can construct the semi-invariant compact absorbing setfor S ( t ) via(5.2) B := ∪ t ≥ S ( t ) B R , S ( t ) B ⊂ B . Indeed, since B R ⊂ B , the set B is also an absorbing set for S ( t ). This set isbounded in E due to estimate (4.16) and its closedness in E (and, therefore, in E as well) is evident. Thus, B is a compact set in E and is a semi-invariant absorbingset for the semigroup S ( t ).For the convenience of the reader, we now recall the definition of a global attrac-tor, see [2, 23, 29] for more details. Definition 5.1.
A set
A ⊂ E is a global attractor for the semigroup S ( t ) in E if:1) The set A is compact in E .2) The set A is strictly invariant: S ( t ) A = A , t ≥ A is an attracting set for the semigroup S ( t ) in E , i.e., for anybounded set B in E and any neighborhood O ( A ) of the attractor A in E there istime T = T ( B, E ) such that S ( t ) B ⊂ O ( A ) , t ≥ T. Remind that the third property in Definition 5.1 can be reformulated in theequivalent way using the so-called non-symmetric Hausdorff distance between sets,namely,(5.3) lim t →∞ dist E ( S ( t ) B, A ) = 0 , UINTIC WAVE EQUATION 17 for any bounded set B ⊂ E . Here and below the Hausdorff distance between sets X and Y in the space E is defined as follows:dist E ( X, Y ) := sup x ∈ X inf y ∈ Y k x − y k E . Proposition 5.2.
Let the assumptions of Theorem 4.1 hold. Then the solutionsemigroup S ( t ) in E of problem (2.1) possesses the global attractor A which isbounded in E and is generated by all trajectories of S ( t ) which are defined for all t ∈ R and bounded in E : (5.4) A = K (cid:12)(cid:12) t =0 , where K ∈ C b ( R , E ) is the set all bounded solutions of (2.1) defined for all t ∈ R .Proof. According to the abstract attractor’s existence theorem, see e.g., [2], weneed to check that a) The solution semigroup S ( t ) possesses a compact absorbingset; b) the operators S ( t ) are continuous in E for every fixed t . Since both of theseassertions are already verified above, the existence of the global attractor is alsoverified. Since the constructed absorbing set B is bounded in E and the attractoris a subset of B , it is also bounded in E . Finally, the representation (5.4) is alsoa standard corollary of the attractor’s existence theorem. Thus, the proposition isproved. (cid:3) As the next step, we intend to verify the finite-dimensionality of the globalattractor A . To this end, we recall the definition of the fractal (box-counting)dimension. Definition 5.3.
Let K be a compact set in a metric space E . By Hausdorffcriterium, for every ε > K can be covered by finitely-many balls of radius ε in E . Let N ε ( K, E ) be the minimal number of such balls which is enough to cover E .Then, the fractal dimension of K is defined as follows:(5.5) dim f ( K, E ) := lim sup ε → log N ε ( K, E )log ε . Of course, dim f ( K, E ) a priori can be infinite if E is infinite-dimensional. Wealso remind that, in the case when K is a finite-dimensional Lipschitz manifold in E , the fractal dimension coincides with the usual dimension, but for irregular sets(which is often the case where K = A is a global attractor), the fractal dimensionmay be not integer, see e.g., [27] for more details.We check the finite-dimensionality of the global attractor A for problem (2.1) byconstructing one more important object - the so-called exponential attractor whichhas been introduced in [13] in order to overcome the main drawback of a globalattractor, namely, the absence of control for the rate of convergence in (5.3). Definition 5.4.
A set M is an exponential attractor for the semigroup S ( t ) in E if the following conditions are satisfied:1) The set M is compact in E .2) The set is M is semi-invariant: S ( t ) M ⊂ M .3) The set M has finite fractal dimension in E .4) The set M attracts exponentially the images of bounded sets, i.e., for everybounded set B in E ,(5.6) dist E ( S ( t ) B, M ) ≤ Q ( k B k E ) e − βt , t ≥ , for some positive β and monotone function Q which are independent of t .Roughly speaking, the exponential rate of attraction (5.6) is achieved by addingto the global attractor a number of ”metastable” trajectories (which approach ittoo slowly) and the non-trivial result of the exponential attractors theory is thepossibility to do that without destroying the finite-dimensionality (in almost allcases where the finite-dimensionality of the global attractor is established, see [23]for more details). The control of the rate of convergence (5.6), in particular, makesan exponential attractor much more robust with respect to perturbations. For in-stance, in contrast to a global one, an exponential attractor is as a rule upper andlower semincntinuous (and even H¨older continuous) with respect to the perturba-tion parameter, see [23]. As the price to pay, an exponential attractor is not unique (similar to center/inertial manifolds), although this drawback can be partially over-came using the proper selection of one-valued branches of exponential attractors independence of the perturbation parameters, see [15, 23] and the references thereinfor more details.The next theorem, which establishes the existence of an exponential attractorfor problem (2.1), can be considered as the main result of the section. Theorem 5.5.
Let assumptions of Theorem 4.1 hold. Then the solution semigroupassociated with problem (2.1) possesses an exponential attractor M which is boundedin the space E .Proof. As usual, it is enough to construct an exponential attractor for the semigroup S ( t ) restricted to the semi-invariant absorbing set B defined by (5.2) only. Alsoas usual, we start with constructing the exponential attractor M d for the map S = S (1) : B → B which will be upgraded after that to the desired exponentialattractor for the case of continuous time. To this end, we need the following lemma.
Lemma 5.6.
Let the above assumptions hold. Then, for any ξ , ξ ∈ B , the fol-lowing is true: (5.7) k S (1) ξ − S (1) ξ k E / ≤ L k ξ − ξ k E , where E / := [ H / (Ω) ∩ H (Ω)] × H / and the constant L is independent of ξ , ξ ∈ B .Proof of the lemma. Indeed, let ξ u i ( t ) := S ( t ) ξ i , i = 1 ,
2, be two trajectories start-ing from ξ , ξ ∈ B and let v ( t ) = u ( t ) − u ( t ). Then, this function satisfies equa-tion (4.3). Moreover, using that B is bounded in E and the embedding H ⊂ C ,analogously to (4.4), we derive that(5.8) k f ( u ) − f ( u ) k L (Ω) ≤ C k u − u k L (Ω) , where C is independent of ξ i ∈ B . Applying Cauchy-Schwartz inequality to theterm on the right of (4.3), using(5.8), we have ddt k ξ v ( t ) k E + k ∂ t v ( t ) k H / ≤ C k ξ v ( t ) k E and, due to the Gronwall inequality, we end up with(5.9) k ξ v ( t ) k E + Z t k ∂ t v ( t ) k H / dt ≤ C k ξ v (0) k E e Kt UINTIC WAVE EQUATION 19 for some positive C and K which are independent of ξ i ∈ B . In addition, multi-plying equation (4.2) by ( − ∆) v and using (5.8) and (5.9), analogously to (3.12),we get Z t k v ( s ) k H / (Ω) ds ≤ C k ξ v (0) k E e Kt and, therefore,(5.10) Z t k ξ v ( s ) k E / ds ≤ C k ξ v (0) k E e Kt . Finally, multiplying equation (4.2) by t ( − ∆ x ) / ∂ t v and using again (5.8), we derive ddt ( t k ξ v ( t ) k E / ) ≤ C k ξ v ( t ) k E / . Integrating this inequality in time and using (5.10), we have t k ξ v ( t ) k E / ≤ C k ξ v (0) k E e Kt and the lemma is proved. (cid:3) We are now ready to prove the theorem. Indeed, since the embedding E / ⊂ E is compact, Lemma 5.6 gives the existence of an exponential attractor M d for thediscrete semigroup generated by the map S = S (1) on B , see [14]. The exponentialattractor M for the continuous semigroup can be then obtained via the standardformula(5.11) M := ∪ t ∈ [0 , S ( t ) M d . To guarantee that this set has finite fractal dimension (the other properties of theexponential attractor follow immediately from the fact that M d is an exponentialattractor for discrete semigroup), it remains to check that the map ( t, ξ ) → S ( t ) ξ is uniformly Lipschitz continuous on [0 , × B . The uniform Lipschitz continuitywith respect to ξ is guaranteed by (5.9) and the Lipschitz continuity in time followsfrom the fact that ξ ∂ t u ( t ) is uniformly bounded in E for any trajectory ξ u ( t ) startingfrom ξ u (0) ∈ B , see estimate (4.16). Thus, (5.11) is indeed the desired exponentialattractor and the theorem is proved. (cid:3) To conclude this section, we consider the case when α = 0 in (2.1). In the caseof Dirichlet boundary conditions, it does not change anything since, due to thePoincare inequality, k ∂ t u k H / ≤ C k ( − ∆ x ) / ∂ t u k L , and the term γ ( − ∆ x ) / ∂ t u is enough for energy dissipation. Thus, in that case,all results obtained above remain true for α = 0 as well.The case of periodic boundary conditions is more delicate. Indeed, in that casethe dissipation vanishes at the spatially homogeneous mode and the dissipativeestimate (3.9) can be a priori lost. Moreover, it is indeed lost in two elementarycases. First is the case where the nonlinearity g is spatially homogeneous: g ≡ const . Then equation (2.1) possesses spatially homogeneous solutions ¯ u ( t, x ) := ¯ u ( t )and they solve the ODE(5.12) d dt ¯ u ( t ) + f (¯ u ( t )) = g which is clearly not dissipative and does not possess a global attractor. The second one is the case when f ( u ) := Lu , L >
0, is linear . In that case, thespatial average ¯ u ( t ) := R Ω u ( t, x ) dx of the solution (2.1) satisfies equation (5.12)with the spatial average ¯ g of the external force g in the right-hand side (instead of g ), thus the dissipation is again lost. As the next proposition shows, (2.1) will benevertheless dissipative in other cases. Proposition 5.7.
Let α = 0 , Ω = T and the rest of conditions of Theorem 3.3 besatisfied. Let, in addition, the right-hand side g be not a constant identically andthe graph of nonlinearity f do not contain flat segments, i.e., for any a ∈ R , theset ( f ′ ) − ( a ) be nowhere dense in R . Then, the energy functional (3.2) is a globalLyapunov function for the solution semigroup S ( t ) generated by (2.1) in the energyspace E .Proof. Indeed, due to (3.3), the energy functional is non-increasing along the tra-jectories of (2.1). Thus, we only need to check that the equality(5.13) E ( u ( T ) , ∂ t u ( T )) = E ( u (0) , ∂ t u (0)) , for some solution u ( t ) and some T >
0, implies that u ( t ) ≡ u is an equilibrium.To prove this fact, we note that, due to (3.3), Z T k ( − ∆ x ) / ∂ t u ( t ) k L (Ω) dt = 0 . Thus, ∂ t u ( t ) is spatially homogeneous, that is ∂ t u ( t ) does not depend on x andtherefore, the solution u ( t, x ) has the form u ( t, x ) = h u ( t ) i − h u i + u ( x ) , where h u ( t ) i = | Ω | ( u ( t ) ,
1) and u is our initial data. Differentiating (2.1) in time,we see that(5.14) f ′ ( u ( t )) ddt h u ( t ) i = − d dt h u ( t ) i . Assume now that h u ( t ) i is not a constant. Then, there exists time t ∈ (0 , T ) suchthat ddt h u ( t ) i 6 = 0. From (5.14), we conclude that f ′ ( h u ( t ) i − h u i + u ( x )) = a := − d dt h u ( t ) i / ddt h u ( t ) i for all x ∈ T . Moreover, due to the smoothing property, u ( x ) ∈ H ( T ) ⊂ C ( T ).Since ( f ′ ) − ( a ) is nowhere dense, we conclude that u ( x ) ≡ const . But then g ( x )also must be a constant which contradicts the assumptions of the proposition. Thiscontradiction proves that ∂ t u ( t ) ≡
0, so the energy functional is a global Lyapunovfunction of (2.1). Proposition 5.7 is proved. (cid:3)
Remark 5.8.
The existence global Lyapunov function together with the evidentfact that the set of equilibria is bounded in E and with the asymptotic compactness(which is also immediate in our case due to the smoothing property) implies thedissipativity and the existence of a global attractor, see e.g., [17] for the details.Thus, under the assumptions of Proposition 5.7, we a posteriori have the dissipativeestimate (3.9) as well as the global and exponential attractors existence. However,in contrast to the case of α >
0, we do not know how to obtain (3.9) directly fromthe energy-type estimates.
UINTIC WAVE EQUATION 21 The case of Dirichlet boundary conditions
In that section, we verify that the extra regularity (3.10) is available for energysolutions in the case of Dirichlet boundary conditions as well. In order to avoid thetechnicalities, we pose slightly stronger than (2.2) conditions on the nonlinearity f ,namely, we assume that f satisfies the following conditions:(6.1) . f ∈ C ( R , R ) , . f ′ ( u ) ≥ − C + κ | u | , | f ′ ( u ) | ≤ C (1 + | u | ) , . f ( − u ) = f ( u ) , for some positive constants C and κ . Note that, in contrast to (2.2), assumptions(6.1) exclude the non-linearities f with subcritical (less than quintic) growth rate.However, it does not look as a big restriction since the subcritical case is mucheasier and the desired extra regularity of energy solutions is straightforward there.The following analogue of Theorem 3.4 is the main result of the section. Theorem 6.1.
Let the problem (2.1) be equipped by Dirichlet boundary condi-tions, the non-linearity f satisfy assumptions (6.1) , Ω be a smooth bounded do-main, α, γ > , g ∈ L (Ω) and let u ( t ) be a weak solution of problem (2.1) . Then, u ∈ L ([0 , T ] , H / (Ω)) and the following estimate holds: (6.2) k u k L ([ t,t +1] ,H / (Ω)) ≤≤ C (cid:16) k ξ u k L ∞ ([ t,t +1] , E ) + k ∂ t u k L ([0 ,T ] ,H / ) + k g k H − / (cid:17) for some positive constant C which is independent of t and u .Proof. We first rewrite equation (2.1) as follows(6.3) ∂ t u − ∆ x ( u + v ) + f ( u ) = 0 , where v ( t ) := γ ( − ∆ x ) − / ∂ t u + α ( − ∆ x ) − ∂ t u − ( − ∆ x ) − g . Then, due to estimate(3.9)(6.4) k v k L ([ t,t +1] ,H / (Ω)) ≤≤ C (cid:16) k ξ u k L ∞ ([ t,t +1] , E ) + k ∂ t u k L ([0 ,T ] ,H / ) + k g k H − / (cid:17) . Applying the extension operator Ext to both sides of (6.3) and using (7.5) togetherwith the fact that f ( u ) is odd, we have(6.5) ∂ t ˜ u − ∆ x (˜ u + ˜ v ) + f (˜ u ) = ˜ h, where ˜ u := Ext( u ) and ˜ v := Ext( v ) (see Appendix for the definition and propertiesof the operator Ext) and˜ h := X i,j =1 ∂ x i ( a ij ( x ) ∂ x j (˜ u + ˜ v )) + X i =1 b i ( x ) ∂ x i (˜ u + ˜ v ) , where a ij ( x ) and b i ( x ) are the same as in Lemma 7.2, see Appendix.Moreover, due to Lemma 7.1 and Corollary 7.3 together with the growth restric-tions on f , all terms in (6.5) are well defined as elements of L ∞ (0 , T ; H − (Ω δ )).Thus, we have extended equation (2.1) initially defined in Ω to equation (6.5) which is defined in a larger domain Ω δ . As the next step, we extend this equation to thewhole space R by introducing the cut-off function ψ ( x ) = ψ ε ( x ) such that(6.6) ψ ( x ) = 1 , x ∈ Ω ε/ and ψ ( x ) = 0 , x / ∈ Ω ε , where ε ≪ δ is a small parameter which will be fixed below and setting ¯ u = ψ ˜ u ,¯ v = ψ ˜ v . Then, these functions satisfy(6.7) ∂ t ¯ u − ∆ x (¯ u + ¯ v ) + ψf (˜ u ) = ¯ h with ¯ h := X i,j =1 ∂ x i ( a ij ( x ) ∂ x j (¯ u + ¯ v )) + X i =1 ¯ b i ( x ) ∂ x i (˜ u + ˜ v ) + ¯ c ( x )(˜ u + ˜ v )for some ¯ b i , ¯ c ∈ L ∞ ( R n ) with the support in Ω ε . Then, analogously to the space-periodic case, see Lemma 2.1, we introduce the inner product(6.8) [ U, V ] := Z h ∈ R Z x ∈ R ( U ( x + h ) − U ( x ))( V ( x + h ) − V ( x )) | h | dx dh == c ( U, ( − ∆ x ) / R V ) R = c (( − ∆ x ) / R U, ( − ∆ x ) / R V ) R , where ( U, V ) R and ( − ∆ x ) R are the inner product and the Laplacian in the wholespace respectively. Then, obviously,(6.9) . | [ U, V ] | ≤ C k U k L ( R ) k V k H ( R ) , . | [ U, V ] | ≤ C k U k H / ( R ) k V k H / ( R ) , . k U k H / ( R ) ∼ k U k L ( R ) + [ U, U ] . For simplicity, we restrict ourselves to the formal derivation of estimate (6.2) whichcan be justified exactly as in the proof of Theorem 3.4 (important that all terms in(6.7) belong to H − ( R )) and, therefore, the approximated inner product [ · , · ] / ,ε ,see (2.17), of the equation with ¯ u ∈ H ( R ) is well-defined).As in the periodic case, we take inner product (6.8) of equation (6.7) with ¯ u andintegrate over time interval [ t, t + 1]. After integration by parts this gives(6.10) [ ∂ t ¯ u ( t + 1) , ¯ u ( t + 1)] − [ ∂ t ¯ u ( t ) , ¯ u ( t )] − Z t +1 t [ ∂ t ¯ u ( s ) , ∂ t ¯ u ( s )] ds ++ Z t +1 t [ ∇ x ¯ u ( s ) , ∇ x ¯ u ( s )] ds + Z t +1 t [ ∇ x ¯ v ( s ) , ∇ x ¯ u ( s )] ds ++ Z t +1 t [ ψf (˜ u ( s )) , ¯ u ( s )] ds = Z t +1 t [¯ h ( s ) , ¯ u ( s )] ds. Thus, we only need to estimate the terms in (6.10). First, due to Lemma 7.1,(6.11) k ∂ t ¯ u k L ( R ) ≤ C k ∂ t u k L (Ω) , k ¯ u k H ( R ) ≤ C k u k H (Ω) , k ∂ t ¯ u k H / ( R ) ≤ C k ∂ t u k H / and, therefore, due to (6.8) and (3.9), first 3 terms in (6.10) is controlled by theenergy norm of the solution u . Second, according to Lemma 7.1, we also have(6.12) k ¯ v k H / ( R ) ≤ C k v k H / ≤ C ( k ∂ t u k H / + k g k H − / ) UINTIC WAVE EQUATION 23 and, therefore, the 5th term in the left-hand side of (6.10) can be estimated asfollows(6.13) [ ∇ x ¯ v, ∇ x ¯ u ] ≤
12 [ ∇ x ¯ u, ∇ x ¯ u ]++ 12 [ ∇ x ¯ v, ∇ x ¯ v ] ≤
12 [ ∇ x ¯ u, ∇ x ¯ u ] + C k∇ x ¯ v k H / ( R ) and since the H / ( R )-norm of the gradient is controlled by the H / ( R )-norm,the 5th term is also controlled by the 4th one and the energy norm of the solution u ( t ).As the third step, we estimate the right-hand side of (6.10). To this end, wenote that all terms in ¯ h which does not contain second derivatives in space can bestraightforwardly controlled by the energy norm of the solution using the first esti-mate of (6.9) and (6.11). Moreover, the terms which contain the second derivativesof ¯ v can be estimated analogously to the 5th term using the fact that a ij ∈ W , ∞ are the multipliers in H / ( R ), see also the estimate of the next term below. Thus,we only need to estimate the terms[ a ij ∂ x i ¯ u, ∂ x i ¯ u ] = I + I :== Z h ∈ R Z x ∈ R a ij ( x ) ( ∂ x i ¯ u ( x + h ) − ∂ x i ¯ u ( x ))( ∂ x j ¯ u ( x + h ) − ∂ x j ¯ u ( x )) | h | dx dh ++ Z h ∈ R Z x ∈ R ∂ x i ¯ u ( x + h ) ( a ij ( x + h ) − a ij ( x ))( ∂ x j ¯ u ( x + h ) − ∂ x j ¯ u ( x )) | h | dx dh. To estimate I , we remind that a ij ∈ W , ∞ ( R n ) and the integration in h can be donefor | h | ≤ K only ( K > | h | > K is controlledby the L -norm of ∇ x ¯ u ), consequently, by the Cauchy-Schwartz inequality I ≤ Z | h |≤ K Z x ∈ R | ∂ x i ¯ u ( x + h ) | | ∂ x j ¯ u ( x + h ) − ∂ x j ¯ u ( x ) || h | dx dh + C K k ¯ u k H ( R ) ≤≤ [ ∂ x j ¯ u, ∂ x j ¯ u ] / Z | h |≤ K Z x ∈ R | ∂ x i ¯ u ( x + h ) | | h | dx dh ! / + C K k ¯ u k H ( R ) ≤≤ C [ ∂ x j ¯ u, ∂ x j ¯ u ] / k ¯ u k H ( R ) + C K k ¯ u k H ( R ) . Thus, this term is controlled by the 4th term of the left-hand side of (6.10) and theenergy norm. To estimate I , we first note that, obviously, I ≤ Z | h | <ε Z x ∈ R a ij ( x ) ( ∂ x i ¯ u ( x + h ) − ∂ x i ¯ u ( x ))( ∂ x j ¯ u ( x + h ) − ∂ x j ¯ u ( x )) | h | dx dh ++ C ε k ¯ u k H ( R ) . To estimate the first integral, we remind that a ij ( x ) = 0 if x ∈ Ω and both ∇ x ¯ u ( x + h ) and ∇ x ¯ u ( x ) equal to zero if x / ∈ Ω ε (here we used the definition (6.6) of thecut-off function ψ and the restriction | h | < ε ). Thus, the integrand is non-zerowhen x belongs to the 2 ε -layer Ω ε \ Ω only. Since a ij ∈ W , ∞ and a ij (cid:12)(cid:12) ∂ Ω = 0, theyare of order ε in that layer. Thus, I ≤ Cε [ ∇ x ¯ u, ∇ x ¯ u ] + C ε k ¯ u k H ( R ) . Combining the obtained estimates, we see that(6.14) Z t +1 t [¯ h ( s ) , ¯ u ( s )] ds ≤ Cε Z t +1 t [ ∇ x ¯ u ( s ) , ∇ x ¯ u ( s )] ds ++ C ε Z t +1 t ( k ¯ u ( s ) k H + k ¯ v ( s ) k H / ) ds. Thus, all the terms in (6.10) except of the one containing the nonlinearity areestimated. Inserting the obtained estimates to the equality (6.10) and fixing ε tobe small enough, we end up with the estimate(6.15) Z t +1 t [ ∇ x ¯ u ( s ) , ∇ x ¯ u ( s )] ds + Z t +1 t [ ψf (˜ u ( s )) , ψ ˜ u ( s )] ds ≤≤ C (cid:16) k ξ u k L ∞ ([ t,t +1] , E ) + k ∂ t u k L ([ t,t +1] ,H / ) + k g k H − / (cid:17) . Therefore, we only need to estimate the second term in the left hand side of (6.15).Note that, by adding the linear term to f , we may assume without loss of generalitythat f ′ ( u ) ≥ κ (1 + u ) and, consequently,(6.16) ( f ( a ) − f ( b ))( a − b ) ≥ β (1 + | a | + | b | ) ( a − b ) . On the other hand,(6.17) | ( f ( a ) − f ( b ) | ≤ C (1 + | a | + | b | ) | a − b | , | f ( a ) | ≤ C (1 + | a | ) . Using these formulas and the fact that ψ is smooth, we get (cid:18) ψ ( x + h ) f (˜ u ( x + h )) − ψ ( x ) f (˜ u ( x )) (cid:19)(cid:18) ψ ( x + h )˜ u ( x + h ) − ψ ( x )˜ u ( x ) (cid:19) == (cid:18) ψ ( x + h )( f (˜ u ( x + h )) − f (˜ u ( x ))) + f (˜ u ( x ))( ψ ( x + h ) − ψ ( x )) (cid:19) ×× (cid:18) ψ ( x + h )(˜ u ( x + h ) − ˜ u ( x )) + ˜ u ( x )( ψ ( x + h ) − ψ ( x )) (cid:19) ≥≥ βψ ( x + h ) (cid:18) | ˜ u ( x + h ) | + | ˜ u ( x ) | (cid:19) (cid:18) ˜ u ( x + h ) − ˜ u ( x ) (cid:19) −− C | h | ψ ( x + h ) (cid:18) | ˜ u ( x ) || f (˜ u ( x + h )) − f (˜ u ( x )) | + | f (˜ u ( x )) || ˜ u ( x + h ) − ˜ u ( x ) | (cid:19) −− C | h | | f (˜ u ( x ))˜ u ( x ) | ≥≥ βψ ( x + h ) (cid:18) | ˜ u ( x + h ) | + | ˜ u ( x ) | (cid:19) (cid:18) ˜ u ( x + h ) − ˜ u ( x ) (cid:19) −− C | h | ψ ( x + h ) (cid:18) | ˜ u ( x + h ) | + | ˜ u ( x ) | (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ˜ u ( x + h ) − ˜ u ( x ) (cid:12)(cid:12)(cid:12)(cid:12) −− C | h | (cid:18) | ˜ u ( x + h ) | + | ˜ u ( x ) | (cid:19) ≥≥ β ψ ( x + h ) (cid:18) | ˜ u ( x + h ) | + | ˜ u ( x ) | (cid:19) (cid:18) ˜ u ( x + h ) − ˜ u ( x ) (cid:19) −− C | h | (cid:18) | ˜ u ( x + h ) | + | ˜ u ( x ) | (cid:19) . UINTIC WAVE EQUATION 25
That estimates give[ ψf (˜ u ) , ψ ˜ u ] ≥ − C (1 + k ˜ u k L (Ω δ ) ) ≥ − C (1 + k ˜ u k H (Ω δ ) ) ≥ − C (1 + k u k H (Ω) ) . Inserting this estimate to (25), we end up with(6.18) Z t +1 t [ ∇ x ¯ u ( s ) , ∇ x ¯ u ( s )] ds ≤≤ C (cid:16) k ξ u k L ∞ ([ t,t +1] , E ) + k ∂ t u k L ([ t,t +1] ,H / ) + k g k H − / (cid:17) . Estimate (6.2) is an immediate corollary of this estimate and the obvious fact that k u k H / (Ω) ≤ k u k H (Ω) + Z Ω Z Ω |∇ x u ( x ) − ∇ x u ( y ) | | x − y | dx dy ≤ k u k H (Ω) +[ ∇ x ¯ u, ∇ x ¯ u ] . Thus, Theorem 6.1 is proved. (cid:3)
Remark 6.2.
As it has been shown before, the extra regularity of energy solutionsestablished in Theorem 6.1 is enough to verify the well-posedness of energy solutionsfor problem (2.1) as well as their dissipativity, smoothing property and the existenceof finite-dimensional global and exponential attractors. Thus, these results areproved for the case of Dirichlet boundary conditions under the assumptions ofTheorem 6.1.7.
Appendix. Properties of the extension operator
The aim of this appendix is to define and study the odd extension operator forfunctions defined in a smooth bounded domain of R which vanish at the boundary ∂ Ω. This operator is a crucial technical tool for proving the additional regularityof energy solutions for the case of Dirichlet boundary conditions.Namely, since Ω is smooth, any point x in the small δ neighborhood O δ ( ∂ Ω) of ∂ Ω can be presented in a unique way the form(7.1) x = x ′ + s~n, where x ′ ∈ ∂ Ω, ~n is a normal to ∂ Ω at x ′ and s ∈ ( − δ, δ ). Thus, (7.1) realizesa diffeomorphism of O δ ( ∂ω ) and ∂ Ω × ( − δ, δ ) and we will treat the pair ( x ′ , s ) ascoordinates in the neighborhood O δ ( ∂ Ω) of the boundary.In that coordinates the reflection R Ω with respect to the boundary reads(7.2) R Ω : O δ ( ∂ Ω) → O δ ( ∂ Ω) , ( x ′ , s ) → ( x ′ , − s )which corresponds to the C ∞ -map y = R Ω ( x ) in the initial coordinates. Note alsothat R Ω ( x ) maps points which are inside of Ω to the points outside of Ω and R Ω ( R Ω ( x )) ≡ x. Moreover, for any x ∈ ∂ Ω, that the derivative R ′ Ω ( x ) is the usual (linear) reflection with respect to the tangent plane to ∂ Ω at x .We define the desired extension operator Ext as follows:(7.3) Ext( u )( x ) := ( u ( x ) , x ∈ Ω , − u ( R Ω ( x )) , x ∈ O δ (Ω) \ Ω . The next lemma shows that Ext defines indeed a proper extension of functions H s ∆ , − ≤ s ≤ Lemma 7.1.
Let Ω be a smooth domain. Then Ext is a linear continuous operator (7.4) Ext : H s ∆ → H s (Ω δ ) for all − ≤ s ≤ (here and below, Ω δ := { x ∈ R , dist( x, Ω) < δ } ).Proof. Indeed, let u ∈ H (Ω) ∩ H (Ω). Then, since u (cid:12)(cid:12) ∂ Ω = 0, we have u (cid:12)(cid:12) ∂ Ω = Ext( u ) (cid:12)(cid:12) ∂ Ω = 0 . Moreover, since the extension is odd, ∂ ~n u (cid:12)(cid:12) ∂ Ω = ∂ ~n Ext( u ) (cid:12)(cid:12) ∂ Ω . Therefore, there are no jumps of first derivatives on ∂ Ω and, consequently, nosingular parts for the second derivatives as well. Thus, Ext( u ) ∈ H (Ω δ ) and (7.4)is proved for s = 2. For s = 1 and s = 0 it is evident and for the non-integerexponents s it follows then by interpolation. Thus, (7.4) is verified for non-negative s ∈ [0 , s ∈ [ − , u ∈ C ∞ (Ω) and ϕ ∈ C ∞ (Ω δ ) anduse the following identity Z Ω δ Ext( u ) ϕ dx = Z Ω u ( x ) ϕ ( x ) dx − Z Ω δ \ Ω u ( R Ω ( x )) ϕ ( x ) dx = Z Ω u ( x ) ϕ ( x ) dx −− Z Ω u ( y ) ϕ ( R Ω ( y )) | det R ′ Ω ( y ) | dy == Z Ω u ( x )( ϕ ( x ) − | det R ′ Ω ( x ) | ϕ ( R Ω ( x ))) dx := Z Ω u Ext ∗ ( ϕ ) dx, where, by definition, ϕ ( R Ω ( x )) ≡ x / ∈ O δ ( ∂ Ω). This identity shows thatExt ∗ ( ϕ ) ∈ H (Ω) (since | det R ′ Ω ( x ) | = 1 when x ∈ ∂ Ω) and k Ext ∗ ( ϕ ) k H (Ω) ≤ C k ϕ k H (Ω δ ) . Thus, by density arguments, Ext is a linear continuous operator from H − (Ω) to H − (Ω δ ). For non-integer s ∈ ( − , (cid:3) The next lemma which gives the expression for the commutator of Ext and theLaplacian is crucial for what follows.
Lemma 7.2.
Let Ω be a smooth domain and let u ∈ C ∞ (Ω) be a smooth functionsatisfying u (cid:12)(cid:12) ∂ Ω = 0 . Then (7.5) Ext(∆ x u ) − ∆ x (Ext( u )) = X i,j =1 ∂ x i ( a ij ( x ) ∂ x j Ext( u )) + X i =1 b i ( x ) ∂ x i Ext u, where a ij ∈ W , ∞ (Ω δ ) , b i ∈ L ∞ (Ω δ ) and (7.6) a ij (cid:12)(cid:12) Ω ≡ . Proof.
Indeed, since R ′ Ω ( x ) is a reflection and the reflections preserve the Laplacian,we have(7.7) ∆ x ( u ( R Ω ( x ))) = (∆ x u )( R Ω ( x )) + X ij ˜ a ij ( x )( ∂ x i ∂ x j u )( R Ω ( x ))++ X i ˜ b i ( x )( ∂ x i u )( R Ω ( x )) , UINTIC WAVE EQUATION 27 where ˜ a ij (cid:12)(cid:12) ∂ Ω = 0 and the assertion of the lemma is an immediate corollary of (7.7)and the definition of the extension operator Ext. Lemma 7.2 is proved. (cid:3) Corollary 7.3.
Formula (7.5) remains valid if u ∈ H s ∆ , ≤ s ≤ . Indeed, the assertion of the corollary follows by the standard density argumentsfrom the facts that C ∞ (Ω) ∩ H (Ω) is dense in H s ∆ and that both right and lefthand sides have sense for u ∈ H s ∆ for that values of s . References [1] J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equation with criticalexponent, Comm. Partial Differential Equations (1992), 841–866.[2] A. V. Babin, M. I. Vishik, ”Attractors of evolutionary equations”, North Holland, Amster-dam, 1992.[3] N. Burq, G. Lebeau, and F. Planchon, Global Existence for Energy Critical Waves in 3DDomains. J. of AMS, vol
21, no. 3 3, (2008), 831–845.[4] A. Carvalho and J. Cholewa, Local well-posedness for strongly damped wave equations withcritical nonlinearities. Bull. Austral. Math. Soc., (2002), 443–463.[5] Attractors for Strongly Damped Wave Equations with Critical Nonlinearities, Pacific J.Math., vol. , no. 2, (2002), 287–310.[6] A. Carvalho, J. Cholewa and T. Dlotko, Strongly damped wave problems: bootstrappingand regularity of solutions. J. Diff. Eqns., (2008), no. 9, 2310–2333.[7] W. Chen and S. Holm, Fractional Laplacian time-space models for linear and nonlinear lossymedia exhibiting arbitrary frequency power-law dependency. J. Accoust. Soc. of Am., ,no. 4, (2004), 1424–1430.[8] S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentledissipation: the case 0 < α < . Proc. Amer. Math. Soc. (1990), no. 2, 401–415.[9] S. Chen and R. Triggiani,Proof of extensions of two conjectures on structural damping forelastic systems. Pacific J. Math. 136, no. 1, (1989), 15–55.[10] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems. Acta,English transl.: Acta, Kharkov, 2002.[11] I. Chueshov, I. Lasiecka, ”Von Karman Evolution Equations”, Springer, 2010.[12] I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural non-linear damping, Journal of Abstract Differential Equations and Applications, vol. , no. 1(2010), 86–106.[13] A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Exponential attractors for dissipativeevolution equations.
Research in Applied Mathematics, 37. Masson, Paris; John Wiley &Sons, Ltd., Chichester, 1994.[14] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in R , C. R. Math. Acad. Sci. Paris, (2000), 713–718 .[15] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional re-duction for non-autonomous dynamical systems. Proc. Roy. Soc. Edinburgh Sect. A (2005), no. 4, 703–730.[16] J.-M. Ghidaglia and A. Marzocchi, Longtime behaviour of strongly damped wave equations,global attractors and their dimension, SIAM J. Math. Anal., (1991), 879–895.[17] J. K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Mono-graphs no. 25, Amer. Math. Soc., Providence, R. I., 1988.[18] V. K. Kalantarov, Attractors for some nonlinear problems of mathematical physics, Zap.Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), (1986), 50–54.[19] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equation. (2009), 1120–1155.[20] J. L. Lions, ”Quelques methodes de resolution des problemes aus limites non linearies”,Paris: Dunod, 1969.[21] J. L. Lions, E. Magenes, ”Nonhomogeneous Boundary Value Problems and Applications”,Springer-Verlag, New York, 1972.[22] P. Massatt, Limiting behavior for strongly damped nonlinear wave equations, J. DifferentialEquations, 48 (1983), 334–349. [23] A. Miranville, S. Zelik, ”Attractors for dissipative partial differential equations in boundedand unbounded domains”. In: C. M. Dafermos, M. Pokorny, eds., Handbook of DifferentialEquations: Evolutionary Equations, vol. 4, Amsterdam: North-Holland, 2008.[24] V. Pata, S. Zelik, A remark on the weakly damped wave equation, Communications on Pureand Applied Analysis, vol. , no. 3 (2006), 611–616.[25] V. Pata, S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, vol. , no. 7 (2006), 1495–1506.[26] T. Pritz, Frequency power law of material damping. Applied Acoustics, , (2004), 1027–1036.[27] J. Robinson, Dimensions, embeddings, and attractors.
Cambridge Tracts in Mathematics,186. Cambridge University Press, Cambridge, 2011.[28] J. Shatah and M. Struwe. Regularity results for nonlinear wave equations. Ann. of Math.,vol. , no. 3, (1993), 503–518.[29] R. Temam, ”Infinite-dimensional dynamical systems in Mechanics and Physics”, Springer,1988.[30] B. Treebya and B. Cox, Modeling power law absorption and dispersion for acoustic propaga-tion using the fractional Laplacian. J. Accoust. Soc. of Am., , no. 5, (2010), 2741–2748.[31] H. Triebel,
Interpolation theory, function spaces, differential operators.
North-Holland,Amsterdam-New York, 1978.[32] K. Yosida, ”Functional analysis”, 6th ed., Springer, Berlin, 1980.[33] G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear waveequation, Canad. J. Math., (1980), 631–643.[34] S. Zelik, Asymptotic regularity of solutions of nonautonomous damped wave equation witha critical growth exponent, Communications on Pure and Applied Analysis, vol. , no. 4,(2004), 921–934. University of Surrey, Department of Mathematics,Guildford, GU2 7XH, United Kingdom.
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