REGULAR SOLUTIONS 1 REGULAR SOLUTIONS FOR WAVE EQUATIONS WITH SUPER-CRITICAL SOURCES AND EXPONENTIAL-TO-LOGARITHMIC DAMPING Lorena Bociu Department of Mathematics, NC State University Raleigh, NC 27695, USA Petronela Radu Department of Mathematics, University of Nebraska-Lincoln Lincoln, NE 68588, USA Daniel Toundykov Department of Mathematics, University of Nebraska-Lincoln Lincoln, NE 68588, USA Communicated by the associate editor name) Abstract. We study regular solutions to wave equations with super-critical source terms, e.g., of exponent p > 5 in 3D. Such high-order sources have been a major challenge in the investigation of finite-energy H 1 L 2 ) solutions to wave PDEs for many years. The well-posedness question has been answered in part, but even the local existence, for instance, in 3 dimensions requires the relation p 6m/m + 1) between the exponents p of the source and m of the viscous damping. We prove that smoother initial data H 2 H 1 ) yields regular solutions that do not necessitate a correlation of the source and the damping. Local existence of such solutions is shown for any source exponent p 1 and any monotone damping including: exponential, logarithmic, or none at all in dimensions 3 and 4 and with some restrictions on p in dimensions n 5). This result extends the known theory which in the context of supercritical sources predominantly focuses on damping of polynomial growth and guarantees local smooth solutions without correlating the damping and the source only if p < 5 in 3D or p < n+2 if n = 3, 4). n 2 Furthermore, if we assert the classical condition that the damping grows at least as fast the source, then these regular solutions are global. 1. Introduction. 2 Mathematics Subject Classification. Primary: 35L5; Secondary: 35L2. Key words and phrases. wave equation, regular solutions, critical exponent, super-critical, nonlinear damping. The second author is supported by NSF grant DMS-98435. The third author is supported by NSF grant DMS-1211232.
2 LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV 1.1. Motivation. The primary goal is to expand results on the long-standing wellposedness problem for a damped wave equation perturbed by a super-critical source term, e.g., of order strictly above 5 in 3 dimensions. To date this question has generated a vast library of research work that is reviewed in more detail below. However, the well-posedness for solutions of finite energy in the presence of high-exponent sources was established relatively recently in the series of papers [4, 7, 6, 8, 23, 26]. For decades the wave equation with nonlinear damping of exponent m ) and source of exponent p ), the first one extending the lifespan of solutions and the other potentially shortening it, has been a benchmark prototype for many 2ndorder hyperbolic problems. It has been long known that the relation p m is necessary for global existence of finite-energy solutions [28, 29]. But already for source exponents above the critical Sobolev embedding level p > 3 in 3 dimensions), even local existence of finite energy solutions has only been verified [26, 23, 4] under the following condition: p 6m m + 1. It is not presently known if this assumption can be removed for weak solutions. However, we prove that the condition is not needed to obtain local existence of solutions with higher regularity. 1.2. Synopsis of new results. In this work we demonstrate local existence of regular, in appropriate sense, solutions for any source exponent p and any damping in dimensions 3 and 4, thus extending the existing research library which provides analogous results for smooth solutions only if p < 5. Moreover, the new theorem also holds for damping of non-polynomial growth e.g. logarithmic or exponential, or none at all for that matter if we are concerned with local existence only. Such general feedbacks naturally arise in sub-gradient interpretation of the damping, e.g. see [2, Section 4.3], but this framework is not directly applicable to supercritical sources since they don t correspond to even locally Lipschitz operators on the state space an extended version of this method appeared in [4], but still for polynomial-like damping only). To our knowledge this is the first paper to deal with non-polynomial damping in context of supercritical sources. And if the damping dominates the source, e.g. grows faster than an m-degree polynomial for some m p, then these regular solutions are global. 1.3. Model. Let Ω R n be an open bounded domain of class C 2 with boundary Γ. The following wave equation with interior interaction of a nonlinear source and monotone viscous damping is the object of our discussion: u tt + gu t ) = u + fu) in Ω [, ) u = in Γ [, ) 1) u) = u and u t ) = u 1 The feedback map s gs) is monotone increasing and represents the effect of viscous dissipation energy loss due to friction). For global well-posedness we will eventually require a quantifiable lower bound on the growth of g e.g.: c s m gs). The map s fs) models nonlinear amplitude-modulated forces that could either have a dissipative effect e.g. nonlinear Hooke s law) or a destabilizing one. The latter is the most interesting scenario, namely when the term fu) is energybuilding and thus counteracts the effect of the damping gu t ). The dynamics of
REGULAR SOLUTIONS 3 the model is fundamentally affected by the behavior of the nonlinear term fu); henceforth, its upper bound will be of polynomial order p, via the estimate fs) C s p. 1.4. Significance of the source exponent p. The well-posedness analysis of 1) may become extremely challenging due to the presence of f which in our scenario is neither monotone, nor dissipative, nor locally Lipschitz on the natural energy space, which for this equation means u, u t ) H 1 Ω) L 2 Ω). In general, such a source can be classified into one of three categories exemplified here for a 3-dimensional scenario: Sub-critical & critical: This range corresponds to p 3, with equality being the critical case. The criticality comes from the 3D) Sobolev embeddings H 1 Ω) L 2 =6 Ω). When f is sub-critical, the map z fz) may be regarded under some differentiability conditions on f) as a locally Lipschitz compact operator H 1 Ω) L 2 Ω). Consequently, if we recast 1) as a first order evolution problem y = Ay on the space H 1 Ω) L 2 Ω), then the nonlinear operator induced by f would be locally Lipschitz on this space. At the critical level p = 3, the locally Lipschitz property persists, but the compactness is lost. Super-critical: In the range 3 < p 5, the Nemytski operator associated to f is no longer locally Lipschitz on the finite energy space, however the potential energy associated to the source, given by ˆfu) dω, with ˆf denoting an Ω anti-derivative of f, is still well-defined for solutions u, u t ) H 1 Ω) L 2 Ω). Upper bound 5 also marks the threshold up to which one may apply the potential well theory e.g. see [22, 9]) to this problem in 3 space dimensions. Super-super-critical: For 5 < p < 6 the source is no longer tractable in the framework of the potential well theory; however, the nonlinear function fu) is still in L 1 Ω) for finite energy solutions. In the presence of sources that are super-critical and above, the role of feedback gu t ), modeling frictional dissipation, is two-fold: it is intended not only to stabilize as in the classical control theory of dissipative systems), but also to ascertain existence, uniqueness, and the life-span of solutions by preventing finite-time blowup. The idea of using damping to positively influence the well-posedness of solutions was introduced over 2 years ago in the context of control theory for dealing with boundary or point controls, which by themselves without any damping do not lead to well-posed dynamics [18]. General energy-building sources, even if locally Lipschitz and conducive to local existence) may cause blow-up of solutions in finite time unless counteracted by sufficiently strong nonlinear damping [4, 5, 13, 22, 3]. 1.5. Previous results on well-posedness. Full analysis of Hadamard wellposedness for weak finite energy H 1 Ω) L 2 Ω)) solutions for 1) with u H 1 Ω), and u 1 L 2 Ω)) was provided recently in [4, 6, 7]. In fact, the authors consider a more general case of super-critical sources that are present simultaneously in the interior and on the boundary of the domain. It was shown that solutions exist locally, are unique, and depend continuously on initial data in the finite-energy topology. These papers introduced new techniques that rely on monotonicity methods combined with suitable truncations-approximations of nonlinear terms, rather than on compactness arguments which had limited the results in the previous attempts [3]. This strategy made it possible to extend the range of Sobolev exponents for which
4 LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV the analysis is applicable 5 p < 6 for the exponent of the interior source and 3 k < 4 for the boundary in 3 dimensions), and extend previously available results on this problem for the case of interior super-critical sources and boundary sources [13, 12, 3, 26, 23, 17, 3, 1]. Moreover, the results in [4, 5] provide sharp range of parameters for the damping and sources that characterize the exact borderline between well-posedness and blow-up for finite energy solutions, thereby completing the picture for Hadamard well-posedness of weak solutions. This theory has been complemented by [9] which demonstrates i) global existence of potential well solutions without exploiting interior and boundary overdamping i.e. without requiring m p and q k); ii) blow-up in finite time for potential well solutions for initial data of non-negative energy that complements the blow up result in [5] for initial data of negative energy. For an extension of the Hadamard well-posedness of weak solutions to the Cauchy problem on R, see [8]. In this context we will focus on well-posedness of regular solutions: u, u t ) H 2 Ω) H 1 Ω). Such a setting was considered before, e.g. [11, Thm. 2.1] p 3) and [3, Thm. 1.1] with p < 5). More recently, in [24] the author proved local existence of solutions for the entire range 1 < p < n+2 n 2, m in n = 3, 4. The long-standing restriction that governed all previous existence results for weak solutions p + p m < 2n n for n 2 n 2 < p < n + 2 n 2 was eliminated via a new methodology that takes into account higher order estimates, by using first- and second-order potential well arguments. The techniques of [24] also apply to source terms of order p 5 with defocusing good signs that contribute to a decrease of the energy. Thus, the author proved that the presence of damping terms is not essential to handling source terms even if they are energyaccretive) locally in time: the results apply to systems with no damping g ) as well as just, with linear damping m = 1). Another feature of this approach is that it works on unbounded domains well, as it is based on the patching argument previously developed for wave equations in [23, 24]. 1.6. Goals of this paper. Our focus is on the well-posedness of the system 1) with smoother initial data u H 2 Ω) H 1 Ω), and u 1 H 1 Ω). The goal is to show that the correlation between damping and the source can be relaxed, especially in dimensions above 2. This result was demonstrated in [24] provided p < n+2 n 2 n = 3, 4). The argument presented here eliminates this restriction and accommodates any p 1, except in high-dimensional cases n 5 where p has to be restricted. A summary of previous work in 3 space dimensions is illustrated in Figure 1A): existence of local weak solutions was proven for the range of exponents to the left of the dashed curve in [26, 23, 24, 4, 6, 7, 8], existence of global weak solutions was proven for the region above the line m = p, for p < 5 in [26, 4], existence of local strong solutions was established for the entire box 1 < p < 5, m in [25]. In comparison, the new results of this paper demonstrate: existence of local regular solutions for all interior sources with exponent p 1 without correlation with the damping. For example, if gs) s m then Figure
REGULAR SOLUTIONS 5 A) B) Figure 1. Assuming dimension n = 3 and that the damping has a polynomial like growth gs) s m, these graphs depict various ranges of the source and damping exponents p and m) as described in Section 1.6. The sloped line corresponds to p = m and the dashed curve in A) to p = 6m m+1 for p 3. 1B) shows that we obtain existence of local regular solutions for all values of m >. Moreover, the conclusion holds for damping of exponential or logarithmic growth, or even without any damping at all g ). existence of global regular solutions whenever the damping grows at least as fast as the source, for example, when m p if g has a lower polynomial bound of order m. 1.7. Notation. The form, ) Ω will indicate the L 2 Ω) inner product with the corresponding norm denoted. Inner products and norms on other spaces will be indicated by suggestive subscripts. The space C w [, T ], Y ) will denote weakly continuous functions with values in a Banach space Y, namely such functions v : [, T ] Y that the map t vt), y is continuous for every y Y. For two functions the relation as) bs) indicates that as) Cbs) for a constant C > independent of s. Whenever invoked it will also be assumed that C is independent of the solution trajectories in question. If as) bs) and bs) as) we may write as) bs). The calculations will often take place on a space-time cylinder, T ) Ω where T > will be understood from the context. For a shorthand we will denote :=, T ) Ω. 2. Preliminaries on Orlicz spaces. To concisely formulate the analysis involving the monotone feedback map g we, inspired by the approach in [26], will work in Orlicz spaces which extend reflexive Lebesgue space. The benefit of this framework
6 LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV is that it does not necessitate to impose a polynomial growth condition on the function g. Here we summarize some of the basic facts concerning these spaces. For a more extensive reference on the background theory see [15] or [1, Ch. 8]. Let g be a continuous strictly monotone increasing function vanishing at and having lim s gs) =. Then we can define the following N -functions: Φt) = t gs)ds and Ψt) = t g 1 s)ds for t. 2) For T > and a function vx, t) on introduce the nonlinear functionals ρu; Φ) := Φ u )dq and ρu; Ψ) := Ψ u )dq. 3) It can be shown that Φ and Ψ are complementary N -functions, in particular they are convex conjugates of each other. The functions for which the quantity ρ is finite form the Orlicz classes K Φ ) and K Ψ ) respectively. The linear hulls under pointwise addition and scalar multiplications) of these classes are the Orlicz spaces L Φ ) and L Φ ), which are Banach spaces of measurable functions under with the norms u Φ = sup uv and u Ψ = sup uv. ρv;ψ) 1 ρv;φ) 1 The Orlicz class K Φ is always contained in the Orlicz space L Φ [15, 9.12), p. 23]) as readily follows from u Φ ρu, Φ) + 1. 4) The closure of bounded functions in L Φ ) yields a subspace E Φ ). In general we have the strict inclusions E Φ ) K Φ ) L Φ ), but the equivalence holds provided the following condition is satisfied: Definition 1 2 -condition [15, p. 23]). A function F : R + R + is said to satisfy the 2 -condition if there exists C > such that for all large s > we have F 2s) CF s). Under the 2 -condition the Orlicz classes and Orlicz spaces coincide. Moreover, if both Φ and Ψ have the 2 property then the spaces L Φ ), L Ψ ) are reflexive Banach spaces dual to each other. However, if g is exponential or logarithmic, then respectively either g or g 1 violates the 2 -condition. Henceforth we will not a priori assume the 2 condition, neither on g nor on g 1 as neither is necessary for local existence. In the result concerning global existence, the asserted sufficiently rapid growth of g at infinity will, however, imply that g 1 should satisfy the 2 requirement. The main properties of Orlicz spaces are summarized in the next proposition. Proposition 1 On Orlicz spaces). Let Φ and Ψ be complementary N -functions. Then a) A set U is bounded in L Φ ) if and only if it is bounded in the class K Φ ), i.e. { Φ u ) dq : u U} is bounded.
REGULAR SOLUTIONS 7 b) The Orlicz norm on L Φ ) is equivalent to the Luxemburg norm [15, p. 8] { u Φ inf k > : Φ k 1 u ) } 1. 5) c) The inequality uv u Φ v Ψ 6) holds for any pair of functions u L Φ ) and v L Ψ ). [15, Thm. 9.3, p. 74]. From here follows an embedding L Ψ ) [L Φ )]. But integration against an L Ψ function is not the general form of a bounded linear functional on L Φ unless both the N -functions satisfy the 2 condition. However, it is the general form of such a functional if restricted to the subspace E Φ ) as the next property states. d) The space E Φ ) is separable and its dual is isomorphic and homeomorphic to L Ψ ) [1, Thm. 8.19, p. 273]. For a functional l [E Φ )] the norm therefore is l [EΦ ] = sup lu). u Φ 1, u E Φ e) Orlicz space L Ψ ) is E Φ -weakly complete and E Φ -weakly compact. In particular, a sequence {ψ n } that is bounded in L Ψ ) has a E Φ )-weakly convergent subsequence {ψ nk } in the sense that for a unique χ L Ψ ) ψ nk v χv as k for every v E Φ ) [15, Thm. 14.4, p. 131]. And analogously L Φ ) is E Ψ -weakly complete and E Ψ -weakly compact). f) If Ψ and Φ both satisfy the 2 condition, then the corresponding Orlicz classes and Orlicz spaces coincide with E Ψ and E Φ respectively. Remark 1 Lebesgue spaces). If gs) = s m 1 s, m > 1, then the Orlicz spaces L Ψ ) and L Φ ) are topologically isomorphic to the Lebesgue spaces L m+1 m ) and L m ) respectively. 3. Assumptions and definitions. Here we present and explain the assumptions on the maps f, g and specify a variational interpretation of solutions. Assumption 1. With reference to system 1) suppose that g-1) g is a continuous odd strictly monotone increasing function with g) = and lim s gs) =. g-2) Let the N -functions Φ and Ψ be defined using g as in 2). Assume that for any set V of measurable functions on the boundedness of scalars { } gv)v : v V implies that the family {gv) : v V} is bounded in L Ψ ) see Section 3.1 for examples). f-1) f C 1 R), f) = the latter is inessential and merely for convenience). f-2) f satisfies the following growth condition: f s) s p 1, where p 1 for dimω) = n 4, and 1 p < n 2 n 4 for dimω) = n 5.
8 LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV Remark 2. The condition on p < n 2 n 4 in dimensions n 5 can, in fact, be relaxed as it will be demonstrated in Theorem 4.3. 3.1. Explaining the conditions g-1), g-2) on damping. The benefit of the N -function characterization of gs) is that it naturally accommodates bounds arising from the monotonicity of this map without the need to relate its growth to Lebesgue spaces analogous approach, but for Φ, Ψ both satisfying the 2 -condition, was employed in [26, Sec. 3]). The odd property of g mentioned in g-1) can be relaxed to gs) g s). Regarding the property g-2) we give several examples. 3.1.1. Polynomially bounded dmaping. The assumption g-2) is satisfied in the canonical case gs) = sgns) s m for m >. Then g v ) Ψ gv) ) = τ 1/m dτ = m v m + 1 τ m m+1 m = m m + 1 v m+1 =m v τ m dτ = mφ v ) = m mg v ) v = mgv)v. v gτ)dτ Integrating over shows that gv)v being bounded also yields an estimate on gv) in the class K Ψ ), which in turn gives a bound on the norm in L Ψ ). This scenario illustrates the classical estimate for polynomially growing damping of degree m: gv)v C = v L m+1 ) and gv) L m+1 m QT ). 3.1.2. Exponential damping. Let gs) = sgns)e s 1) and g 1 s) = sgns) ln s + 1). Given that gv)v = v e v v ) 7) is bounded we would like to deduce a bound on gv) in L Ψ. Let s directly estimate gv) in the Orlicz class K Ψ : ρgv), Ψ) = gv) e v 1 g 1 s)dsdq = lns + 1)dsdQ = v e v e v + 1)dQ Because s e s s ) s e s e s + 1) for s 1, then since is a bounded domain) the boundedness of 7) gives a bound on ρgv), Ψ), which controls via 4) the norm gv) Ψ. 3.1.3. Logarithmic damping. Let gs) = sgns) ln s +1) whence g 1 s) = sgns)e s 1) for s. Starting with a bound on gv)v = ln v + 1) v 8)
REGULAR SOLUTIONS 9 compute ρgv), Ψ) = gv) g 1 s)dsdq = = ln v +1) e s 1)dsdQ ) v + 1 ln v + 1) dq. Since ln s + 1) s dominates s + 1 ln s + 1) as s, then from a bound on 8) we infer a bound on gv) in the class K Ψ. 3.2. Weak and regular solutions. Consider the Laplace operator on functions with vanishing traces: Au = u with DA) := { u H 2 Ω) u = on Γ } = H 2 Ω) H 1 Ω). 9) This operator is positive self-adjoint on L 2 Ω) and maximal accretive L 2 Ω) L 2 Ω). So we can define fractional powers of A and identify via the topological isomorphism DA 1/2 ) H 1 Ω). Definition 2 Weak solution). Weak solutions will be considered when the exponent p of the source satisfies 1 p < if n 2, or 1 p 2 if n 3 where 2 = 2n n 2 is the critical Sobolev exponent for the embedding H1 Ω) L 2 Ω). By a weak solution of 1) defined on some interval [, T ], we mean functions u C w [, T ]; H 1 Ω) ) L 2, T ; H 1 Ω) ), u t C w [, T ]; L 2 Ω) ) L 2 ) with the following properties: i) Let r be sufficiently large to ensure the embedding DA r ) L Ω). For any test-function φ H 1, T ; DA r )) the following identity must hold T u t φ t + u φ) dq + gu t )φ dq = u t, φ) Ω + fu)φ dq 1) ii) In addition, for ψ H 1 Ω), ψ 1 L 2 Ω) lim ut) u, ψ = and limu t t) u 1, ψ 1 ) Ω =. t t For weak solutions we introduce the finite energy functional on H 1 Ω) L 2 Ω) Eu, v) := 1 2 u 2 + 1 2 v 2, with Et) := Eut), u t t)). 11) Definition 3 Regular Solution). Regular solutions can be defined for a larger range of source exponents: 1 p < if n 4, or 1 p 2 if n > 4, where 2 = 2n n 4 is the critical Sobolev exponent for the embedding H2 Ω) L 2 Ω). A regular solution of 1) is a weak solution in the sense of Definition 2 with the additional regularity u, u t ) C w [, T ]; H 2 Ω) H 1 Ω) ) L, T ; H 2 Ω) H 1 Ω) ). Remark 3 Test functions). If gu t ) L Ψ ), then from the variational identity 1) it follows that the space of test functions can be closed in the graph topology of H 1, T ; L 2 Ω) ) L 2, T ; H 1 Ω) ) L 2, T ; L q Ω)) L Φ ) where q is the conjugate exponent of either 2 /p for weak solutions) or 2 /p for regular solutions). In lower dimensions the membership in L 2, T ; L q Ω)) is automatic as follows from Sobolev embeddings. The choice of a sufficiently large parameter r in Definition 2 ensures that the original space H 1, T ; DA r )) is contained in
1 LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV L ). Consequently, the closure in the above topologies L Φ ) in particular) would ensure that the resulting test elements belong to the subspace E Φ ) whose dual is isomorphic to L Ψ ). 4. Main Results. Our main theorem demonstrates propagation of the initial regularity which allows for local regular solutions. The solutions are also global under the additional assumption that the damping dominates the source at infinity. Theorem 4.1 Local existence of unique regular solutions). Consider equation 1) under the Assumption 1. Suppose u H 2 Ω) H 1 Ω), u 1 H 1 Ω). Then there is T M > such that 1) has a unique regular solution on [, T ] for any T < T M. The result holds even without damping g ). Theorem 4.2 Global existence of regular solutions). With Assumption 1 suppose further s m gs) for all s 1 for some m p. Then the regular solution obtained in Theorem 4.1 is global. Recall that in dimensions n 5 the Assumption 1 asks for p < n 2 n 4. In the context of local well-posedness this result can be further extended: Theorem 4.3 Relaxing p < n 2 n 4 in n 5 for local wellposedness). Consider equation 1) under Assumption 1, but replace part f-2) with: i) f s) s p 2 for all s R. ii) f) = necessary whereas in Assumption 1 it was just for convenience). Then in dimensions n 5 for any p < 2n 4 n 4 = 1 + n n 4 there exists a sufficiently large m > such that if s m gs) for all s 1, then the local existence result of Theorem 4.1 holds. More precisely it requires: m > If in addition g satisfies then this solution is unique. pn 4) + 4 2n 4) pn 4) or p < 2mn 2) 4 n 4)m + 1). s r m+1 gs) gr))s r) 12) 5. Proofs. The above theorems are verified in a series of steps: I. The Galerkin approximations used to construct local regular solutions are presented in Section 5.1. The investigation of the limit of these approximations is given in Sections 5.2 and 5.3 with 5.3.1 handling the nonlinear damping term). II. The uniqueness of regular solutions is addressed in Section 5.5. III. The alternate version of local existence and uniqueness Theorem 4.3) is outlined in Section 5.6. IV. Local regular solutions are extended globally in Section 5.7. V. Finally, Section 5.8 mentions an alternative approach to local existence of regular solutions using semigroup theory in the simpler scenario when the function g is linear.
REGULAR SOLUTIONS 11 5.1. Galerkin approximations and a priori bounds. We consider the standard Galerkin scheme that constructs solutions of 1) via a limit of finite-dimensional approximations based on the eigenfunctions of the Laplacian. The set {e k } k N of eigenfunctions of the Dirichlet Laplacian A = forms an orthonormal basis for L 2 Ω), and orthogonal basis for every DA r ) H 2r Ω), r >. We let u n t) := n k=1 un,k t)e k, where u n t) satisfies the following system of ordinary differential equations: u n tt, v) Ω + u n, v) Ω + gu n t ), v) Ω = fu n ), v) Ω 13) u n ), v) Ω = u, v) Ω, u n t ), v) Ω = u 1, v) Ω 14) for all v span{e k } n k=1. By our choice of e k it follows that u n ) u strongly in H 1 Ω) u n t ) u 1 strongly in L 2 Ω). Moreover, since u n ) and u n t ) are projections of smooth data u and u 1, then the former are bounded in H 2 and H 1 respectively. Note that 13) 14) is an initial value problem for a system of n second-order ordinary differential equations with continuous nonlinearities in the n unknown functions u n,k and their derivatives. From the Cauchy-Peano theorem it follows that for every n 1, 13) 14) has a solution u n,k C 2 [, T n ], for some T n >. Proposition 2 A priori bounds on approximate solutions). There exists T M > such that for any T < T M the functions {u n, u n t )} are defined for t [, T ] and bounded in the space L, T ; DA) H 1 Ω) ). In addition, {u n t } are bounded in L Φ ) and {u n tt} are bounded in Orlicz-Bochner space L Ψ, T ; [DA r )] ) for r such that DA r ) L Ω). Proof. Step 1: u n and u n t. Pick n > and let [, T n ) be the maximal right-interval of existence for the solution to 13). Let s temporarily fix some T, T n ). Define the higher-order energy functional of n-th approximation: E 1,n t) := 1 2 un t) 2 H 2 Ω) + 1 2 un t t) 2 H 1 Ω). Use the test function v = u n t in 13) and integrate over t [, T ]. If g u n t ) exists and is integrable then Hence the multiplier u n t E 1,n T ) + T gu n t ), u n t ) Ω = g u n t ), u n t ) Ω. ultimately gives the identity g u n t ), u n t 2 ) Ω = E 1,n ) + T fu n ), u n t ) Ω, where the second term on the left is non-negative and we get 17) listed below. If g does not exist a.e. or has no suitable regularity, then we can approximate it by smooth functions taking advantage of the fact that g corresponds to a maximal monotone graph in R R. Define the operator B : DB) L 2 Ω) L 2 Ω) by 15) Bux) = gux)) a.e. x Ω 16) with DB) = {u L 2 Ω) gu) L 2 Ω)}. Since g is continuous monotone increasing with g) = then g I +g) 1 s) s. Consequently, the operator I +B) 1 maps L 2 Ω) into DB), and thus B is a maximal monotone operator e.g. see [2,
12 LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV Thm. 1.2 on p. 43, or p. 82]). In addition, the domain of B contains L Ω). The Yosida approximations of B are defined by B λ = λ 1 I I + λb) 1) = BI + λb) 1 for λ >, and B λ u)x) coincides a.e. x Ω with g λ ux)) where g λ are regularizations of g: g λ s) = λ 1 s I + λg) 1 s) ). Each g λ is a monotone increasing globally Lipschitz function. For v DB) we have B λ v) Bv) in L 2 Ω) as λ [2, Prop. 1.3, p. 49]. Consequently, since u n t is smooth in particular in L Ω) DB)) then for any fixed n and a.e. t [, T ] gu n t ), u n t ) Ω =Bu n t ), u n t ) Ω = lim λ B λ u n t ), u n t ) Ω = lim λ g λ u n t ), u n t ) Ω = lim λ g λu n t ), u n t 2 ) Ω, which likewise leads to the inequality E 1,n T ) E 1,n ) fu n ) u n t 17) bypassing the direct integration by parts of the product gu n t ), u n t ) Ω. Next, from 17) we proceed to estimate the product involving the source term. We demonstrate it in the case of n 5 lower dimensions don t place any restrictions on p). Let q = 2n n 2 be the critical Sobolev exponent for the embedding H1 Ω) L q Ω). A Hölder estimate with three conjugate exponents 2, q and 2q/q 2) gives fu n ), u n t ) = Ω f u n ) u n u n t u n p 1 L 2qp 1)/q 2) u n L q u n t. 18) By the choice of q we have u n L q Ω) u n H 2 Ω). Note also that in dimensions below 5 the L 2qp 1) q 2 Ω) norm can be estimated via the H 2 Ω) norm. Whereas if n 5 this estimate holds provided or equivalently if p n 2 n 4 and 2qp 1) q 2 2n n 4, as a fortiori guaranteed by f-2). Thus fu n ), u n ) [E 1,n ] p+1)/2 E 1,n T ) E 1,n ) + C T [E 1,n t)] p+1 2 dt. 19) The resulting optimal bound on E 1,n t) may blow up in finite time since 1 2 p+1) > 1; however, a comparison theorem see for instance [19, Thm. 1.4.1, p. 15]) shows that E 1,n t) can blow up no faster than the solution to z = cz p+1)/2 where c and the initial condition z) depend only on C, p and an upper bound on E 1,n ). Thus, there exists T M > such that if T < T M then E 1,n t) must remain a priori bounded on [, T ]. In particular, the time-dependent coefficients of u n,k are a priori bounded on [, T ], and therefore u n t, x) = n j=k un,k t)e k x) is bounded on spacetime domain. Consequently, the maximal right-time of existence T n must be at least T M and the energy E 1,n is uniformly bonded on [, T ] for any T < T M independently of n.
REGULAR SOLUTIONS 13 Step 2: u n t in Orlicz space. To prove that u n t is bounded in L Φ ) uniformly in n it suffices to prove that for Φ u n t ). Recall the quadratic energy 11) and let E n t) := Eu n t), u n t t)). Use u n t as the test function in 13) to obtain E n T ) + gu n t )u n t dq = E n ) + fu n )u n t dq. 2) According to the uniform bounds on on u n and u n t it follows that the product gu n t )u n t is bounded uniformly in n. On the other hand, since g is odd and monotone increasing then Φ v ) = v gτ)dτ g v ) v = gv)v, whence Φ u n t ) gu n t )u n t Cu, u 1 ), from which the desired conclusion follows. Step 3: bound on u n tt. We proceed by duality. Let r be large enough to guarantee the continuous embedding DA r ) L Ω). 21) Consider a test φ function in a DA r )-valued Orlicz-Bochner space φ X := L Φ, T ; DA r )), consisting of Bochner-measurable functions such that T Φk f DA r ))dt < for some k >. Then it forms a Banach space under the Luxemburg norm { T φ X = inf k > : Φ } k 1 ) φ DA r ) dt 1. Due to 21) we also have the inclusion X L 1, T ; DA r )) L Φ ). 22) Next, since {e k } form an orthogonal basis for DA r ) we can define orthogonal projections: P n : DA r ) span{e k : k = 1, 2,..., n} DA r ). In particular, {P n } are bounded on DA r ) uniformly in n and hence bounded uniformly on L Φ, T ; DA r )). Then from the Galerkin variational identity 13), by virtue of 22) we have u n ttφ = u n φ gu n t )P n φ) + fu n )P n φ) u n L,T ;H 1 Ω)) φ L1,T ;H 1 Ω)) + gu n t ) LΨ ) P n φ LΦ ) + fu n ) L,T ;L Ω)) P 2 /p n φ L1,T ;L {2 /p} Ω)) u n L,T ;H 1 Ω)) + gu n t ) Ψ + fu n ) ) L,T ;L 2 p Ω) ) φ X. The restriction on p in f-2) a fortiori implies that fu n ) is bounded in L 2 /p Ω) provided u n is bounded in H 2 Ω). Then the previously verified bounds on u n and u n t imply that integration against u n tt defines a bounded family of functionals in
14 LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV X. In particular, these functionals are bounded when restricted to the subspace E Φ, T ; DA r )). Since DA r ) is reflexive, then it has the Radon-Nikodym property e.g. [2, Cor. 3.6.12, p. 23]), and so [E Φ, T ; DA r ))] is topologically isomorphic to L Ψ, T ; [DA r )] ) via [14, Thm. 2] which yields the desired conclusion. 5.2. Limits of the approximations. The a priori bounds shown in the previous section lead via compactness to the following convergence results: Proposition 3. Let T M be as in Proposition 2. For any T < T M there exists a function u L, T ; H 2 Ω) H 1 Ω)) with time derivative u t L, T ; H 1 Ω)) such that on a subsequence re-indexed again by n) the approximate solutions defined by 13), 14) satisfy: u n t u t u n u strongly in L, T ; H 2 ε Ω) H 1 Ω)) 23) strongly in L q, T ; H 1 ε Ω)) any ε > and 1 q <. 24) Proof. From Proposition 2 it follows on a subsequence reindexed for convenience again by n we have u n u weakly* in L, T ; H 2 Ω) H 1 Ω)) u n t u t weakly* in L, T ; H 1 Ω) ). The extension due to Simon [27] of Aubin s compactness result now readily implies 23). For the convergence of u t use the fact that u n tt is bounded in L Ψ, T ; [DA r )] ) L 1, T ; [DA r )] ) for some r. Then the same compactness theorem implies that u n t converges to some v L q, T ; H 1 ε Ω)) any finite q 1. In particular, from the identity u n t) = u n ) + t u n t s)ds in L 2 Ω) for a.e. t [, T ] it readily follows that u has an absolutely continuous in time t version u C[, T ]; L 2 Ω)) with the time derivative a.e. equal to v, which confirms 24). 5.3. Limit n in the approximate equation. Let N N and consider φ C 1 [, T ]; span{e k } N ) k=1 25) then for all n N the Galerkin approximation 13) satisfies t=t T T u n, φ) u n t, φ t ) Ω + u n, φ) Ω + gu n t )φ = fu n )φ. 26) t= From the convergence results in Proposition 3 we have as n on a subsequence) T u n t, φ t ) Ω T u t, φ t ) Ω, T u n, φ) Ω T u, φ) Ω 27) Because the exponent p in assumption f-2) is sub-critical in dimensions 4 that is automatic) while u n bounded in H 2 it follows that on a subsequence) fu n ) converges weakly to some η in L, T ; L q Ω)), q > 1. Since is a bounded domain then fu n ) η also weakly in L 1 ). But from the convergence of u n we
REGULAR SOLUTIONS 15 have fu n ) fu) pointwise a.e. x, t), and hence in measure. Consequently fu n ) converges strongly in L 1 ) and therefore the weak limit η agrees with fu): fu n )φ fu)φ. 28) Moreover, from 15) we get u n t ), φ)) Ω u t ), φ)) Ω. 29) Finally, the uniform bound on u n tt L Ψ, T ; [DA r )] ) [E Φ, T ; DA r ))] via [14, Thm. 2] and the reflexive property of [DA r )] ) as verified in Proposition 2 permits to work with a subsequence weakly* convergent to some element ζ L Ψ, T ; [DA r )] ). All functions in L Ψ, T ; [DA r )] ) are weakly* scalarly measurable since [DA r )] is separable. From here we conclude that for a smooth space-dependent test function ψ = ψx) the duality pairing ψ, ζ ) is integrable. So for t [, T ] the identity u n) t t), ψ) Ω = u n t ), ψ) Ω + t ψ, u n) tt s) ds 3) in the limit n implies that the mapping t u t t), ψ) Ω is well-defined and continuous on [, T ]. Choosing ψ = φt ) which is smooth since φ comes from 25)) implies u n) t T ), φt )) Ω u t T ), φt )) Ω as n. 31) 5.3.1. Passing to the limit in the damping term. The limits 27) 31) addressed almost all of the terms in the approximate variational formulation 26). It remains to analyze the products involving the damping g as n. From 2) we see that gu n t )u n t is bounded. Then {gu n t )} is bounded in L Ψ ) according to Assumption g-2). Using E Φ -weak compactness and completeness of L Ψ ) conclude that there is a unique function χ L Ψ ) such that on a subsequence): lim gu n t )φ dq = χφ dq. n for every φ E Φ ) a fortiori satisfied if φ L )). Moreover, because u n t converges to u t pointwise a.e.-t, x) in the bounded domain, then gu n t ) converges in measure to gu t ). Hence [15, Thm. 14.6, p. 132] the E Φ -weak limit χ must coincide with gu t ) as desired. 5.4. Obtaining local regular solutions. Let T M be the maximal time of existence provided by Proposition 3. As it was verified above, the passage to the limit n on a subsequence) in Galerkin approximation gives for any T < T M t=t T T u, φ) u t, φ t ) Ω + u, φ) Ω + gu t )φ = fu)φ. 32) t= This conclusion holds for every test function φ C 1 [, T ]; span{e k } k=1). These functions are dense in H 1, T ; DA r )) for any r which yields the desired variational identity 1). Moreover: u L, T ; DA)) u t L, T ; H 1 Ω)) as well as
16 LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV u C[, T ]; H 1 Ω)) u t L Φ ). From the first two bullets one infers e.g. [21, Lemma 8.1, p. 275]) that u, u t ) C w [, T ]; H 2 Ω) H 1 Ω)). In addition, these solutions have the following property: Proposition 4 Inequality for finite energy). If u, v are two regular solutions, ũ := u v and Eũt) := Eũt), ũ t t)), then EũT ) + gu t ) gv t ))ũ t Eũ) + fu) fv))ũ t. 33) Proof. Use the variational Galerkin identity 26) for the initial data u n, u n 1 ) and v n, v1 n ) and take the difference of the resulting equations: T T ũ n tt, φ) Ω + ũ n, φ) Ω + gu n t ) gvt n ))φ = fu n ) fv n ))φ. Substitute φ = ũ n t and integrate by parts Eu ñ T ) + gu n t ) gvt n ))ũ n t = Eu ñ ) + fu n ) fv n ))ũ n t. 34) Recall from Proposition 3 that ũ n t converges strongly in every L q 1, T ; H 1 ε Ω)), hence strongly in every L ρ ), ρ < 2n n 2. Therefore fu n ) fv n ))ũ n t will converge on a subsequence to some ξu t provided fu n ) and fv n ) are bounded in L ρ ) where ρ 2n > min{1, n+2 }. Specifically, in dimensions n 5 it necessitates p < n+2 n 4 which is implied by f-2). Since fun ) fv n ) converges pointwise a.e., it does so in measure on the bounded domain, hence we can identify ξ = fu) fv). Next take the limit inferior as n in 34), use strong convergence of the energy functionals, and invoke Fatou s lemma for the damping integral the integrand is non-negative for each n), to get the desired result. 5.5. Uniqueness of solutions. In 3 dimensions the uniqueness follows immediately if p 6m m+1 since in this region we have the uniqueness result for weak solutions from [6], and the regular solution described by Theorem 4.1 coincides with the unique weak solution. Now we extend the uniqueness result for regular solution in the complementary region of exponents p. Let u and v be any two regular solutions of 1) defined on [, T M ). Then for t [, T ] [, T M ) find a constant R T such that for all t [, T ] ut) H 2 Ω) + u t t) H 1 Ω) R T. From 33) by the monotonicity of g we have EũT ) Eũ) + fu) fv))ũ t. 35) Since pointwise f u) u p 1 then fu) fv) C u v u p 1 + v p 1).
REGULAR SOLUTIONS 17 Therefore, using Hölder estimates with conjugate exponents as in 18) gives for each t [, T ] fu) fv))ũ t dω fu) fv) ũ t dω ũ u p 1 + v p 1) ũ t dω Ω Ω Ω ) u p 1 H 2 Ω) + v p 1 H 2 Ω) Eũt) CR T )Eũt). Substitute this result into 35) and use Gronwall s inequality to conclude that Eũ) = would imply that the solutions u and v coincide. 5.6. Relaxing the condition p < n 2 n 4 in dimensions n 5. Here we outline how the above proof of local existence and uniqueness can be modified to accommodate higher values of p in dimensions n 5 under the extended Assumptions of Theorem 4.3. To verify this result we must essentially refine the proof of a priori bounds claimed in Proposition 2. 5.6.1. Adjusting the proof for the bounds u n H 2 Ω), u n t H 1 Ω). Because the sought result is local in time, then without loss of generality up to a constant dependent on g1), Ω and T ) we can assume that s m gs) holds for all s R as opposed to just s 1. Let E n denote the finite energy of the approximate solution u n, u n t ). Then plugging test-function u n t into 13) gives u n m+1 L m+1 ) En t) + gu n t )u n t = E n ) + fu n )u n t 36) We can estimate E n ) by higher energy E n ) and add this inequality to 17). Because fu n ) Γ, then from the symmetry of the operator A = we get: u n m+1 L m+1 ) + E 1,n T ) E 1,n ) fu n ))u n t + fu n )u n t I II III { }}{ {}}{ {}}{ =E 1,n ) f u n ) u n 2 u n t f u n ) u n u n t + fu n )u n t Recall that we are interested in dimensions n 5. For the integral I use repeated Hölder s inequality with a triple of conjugate exponents α := m + 1)n 2m + 1) n, n n 2, m + 1. Since the assumption on f is f s) s p 2 then for any ε > I T T C ε u u p 2 L αp 2) Ω) u 2 L 2n n 2 Ω) u t L m+1 Ω) m+1 p 2) m L αp 2) Ω) 2m+1) m u dt + ε u L n 2 2n t m+1 L Ω) m+1 ). The term ε u t m+1 L m+1 ) can be absorbed into the left-hand side of 37) for small enough ε >. The rest of the quantities can be expressed in terms of the energy 37)
18 LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV E 1 t) as in 19) using Sobolev embeddings of H 2 Ω) into Lebesgue spaces provided: αp 2) = m + 1)n 2m + 1) n p 2) 2 = 2n n 4. If we are allowed to take the damping exponent m arbitrarily large depending on p), then effectively it suffices to have p 2 < 2n n 4 If we express it more precisely, then we require m > p < 2n 4 n 4. 38) pn 4) + 4 2n 4) pn 4) For the integral II in 37) use the triple of conjugate exponents 2m + 1) m 1, 2, m + 1. Then the same conclusion holds, provided 2m + 1) p 1) < 2n m 1 n 4 which leads to the same condition on m. To estimate the integral III using u n H 2 Ω) and u n t H 1 Ω) one can appeal to the condition p 2 = 2n n 4, which is automatically satisfied by 39). 5.6.2. Proving the bounds on u n t in L Φ ). In order to establish that u n t was bounded in L Φ ) we invoked 2) along with the boundedness of the product fu n )u n t uniformly in n. Observe that by 36) the functions u n t are bounded uniformly in L m+1 ). Consequently, it suffices to have fu n ) bounded in L m+1 m ) independently of n. Since u n us bounded in L, T ; H 2 Ω)), then it remains to guarantee the embedding H 2 Ω) L p m+1 m Ω) 4) which requires p 2n m ) n 4 m + 1 or m The resulting condition is already implied by 39). 39) pn 4) 2n n 4)p. 41) Remark 4. This relation between p and m essentially becomes the H 2 analog of the restriction p 6m m+1 in 3D as in [26, 23, 4]) for finite-energy solutions. 5.6.3. Obtaining a regular solution to the problem. When passing to the limit n the exponent p also plays a role in identifying the limit of the source term as in 28)). In this case it suffices to have p < 2 = 2n/n 4), which follows from 38). The rest of the argument is analogous to the original proof. Note that in this scenario the solution ultimately satisfies u t L Φ ) L m+1 ). 42)
REGULAR SOLUTIONS 19 5.6.4. Adjusting the proof of uniqueness. The uniqueness result relied, first of all, on the inequality for the energy in Proposition 4. That conclusion in turn necessitates the convergence of fu n ) fv n ))u n t vt n ). From 36) we may assume that u n t and vt n converge weakly in L m+1 ). Thus we require strong convergence of fu n ) and fv n ) in the dual space L m+1 m ). Note the estimate fu n ) fu) m+1 m 1 f u n s + 1 s)u)dsu n u) u n p 1 + u p 1) m+1 m u n u m+1 m u n α L αρ ) + u α L αρ ) m+1 m ) u n u m+1 m L ρ m+1 m ) where α = p 1)m+1)/m. As a consequence of their convergence in L, T ; H 2 ε ), the functions u n converge strongly in every L 2 δ ), δ >. Thus we may choose ρ m+1 m arbitrarily close from below to 2. So we should only ensure that the norm of u n in L αρ ) is bounded, meaning αρ 2. The resulting inequality is p 1)m + 1) ρ 2n m n 4 where ρ 2mn can be chosen arbitrarily close from above to 2mn n 4)m+1). This inequality gives a strict version of 41), which, as it has been mentioned, is already satisfied via 39). These steps verify the desired strong convergence of fu n ) in L m+1 m ) and analogously for fv n )). Now with the help of the assumption 12) we get at our disposal the following inequality for the difference Eũ of two solutions u and v: EũT ) + c u t v t m+1 L m+1 ) Eũ) + fu) fv))u t v t ) Eũ) + C ε fu) fv) m+1 m + ε u L m+1 t v t m+1 L m ) m+1 ). The parameter ε is chosen small enough to absorb the velocity term into the lefthand side. The source term is estimated as in 43) with powers above 1 factored into a constant dependent on E) and time T. That step ultimately gives T fu) fv))u t v t ) CR T ) Eũt)dt as in Section 5.5 wherefrom the uniqueness result follows. The above modifications to the original proof of existence and uniqueness imply the result of Theorem 4.3. 5.7. From local regular to global regular solutions. Now we return to the original Assumption 1 and proceed to verify global existence claimed in Theorem 4.2 by building upon the result of Theorem 4.1. At this stage we will demonstrate that both the finite H 1 L 2 ) and higher H 2 H 1 ) energies cannot blow up as t T M provided p m which excludes the scenario with sublinear damping since p 1, unless of course the source is entirely absent). Then for any T M < the 43)
2 LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV solutions obtained by the Galerkin approximations converge on a neighborhood of t = T M and therefore can be extended past any finite interval of existence. Proposition 5 Non-blowup of finite energy). Suppose u, u t ) is a local regular solution defined on the maximal interval of existence [, T M ). Assume that p m, then the finite energy remains uniformly bounded on [, T M ]. Proof. We need to show that lim sup t T M Et) <. That follows if we prove that Galerkin approximations have energy E n t) that is bounded on [, T M ] independently of n. For brevity we will temporarily suppress the superscripts: u := u n and u t := u n t for some n N. Following the argument in [13] introduce a modified energy functional: V t) := Et) + 1 p + 1 ut) p+1. Use 2) which is justified since we re working with smooth approximations) to conclude t t V t) + gu t )u t = E) + fu)u t + 1 p + 1 ut) p+1. Differentiation in time gives V t) = gu t t))u t t) + Ω Ω Ω Ω fut))u t t) + ut) p 1 ut)u t t). Ω Using the lower bound on the damping and the upper bound on the source we obtain V t) C 1 u t t) m+1 L + C m+1 2 ut) p u t t) C 1 u t t) m+1 L + C m+1 2 ut) p L u p+1 t L p+1 C 1 u t t) m+1 L + C m+1 2,ε ut) p+1 L + ε u p+1 t p+1 L p+1 for any ε >. Since 1 p m, then for ε C 1 we obtain Ω V t) C 2;ε V t). Integrate in time and apply Gronwall s inequality to conclude recalling the temporarily omitted dependence on n): E n t) V n t) V n )e Cεt. The right-hand side is uniformly bounded in n because u n p+1 is uniformly bounded, which in dimensions n 5 is enforced by f-2). Since the finite energy converges to that of the true solution as n we conclude that Et) is bounded on [, T M ]. Proposition 6 Non-blow up of higher energy). Suppose u, u t ) is a regular solution on [, T M ) such that the finite energy is uniformly bounded in time on [, T M ]. Then the higher energy is uniformly bounded on [, T M ]. In particular, for any T 1 T M there exists C T1 > dependent only on the H 1 L 2 norms over [, T 1 ], such that ) ut) 2 H 2 Ω) + u tt) 2 H 1 Ω) u 2 H 2 Ω) + u 1 2 H 1 Ω) + C T 1. 44) sup t [,T 1]
REGULAR SOLUTIONS 21 Proof. What follows is an estimate using a higher-order multiplier u t analogous to the one used to prove the stability of the Galerkin scheme. We will work with Galerkin approximations, but for brevity will temporarily suppress the superscripts: u := u n and u t := u n t for some n N. Fix any T T 1 T M. The test function u t gives, as in 17), u t T ) 2 H 1 + ut ) 2 H 2 u 1 2 H 1 + u 2 H 2 fu) u t. 45) For any t [, T 1 ] estimate the product involving the source term exactly as in 2n 18) where the exponent q stood for n 2 if n 3). In dimensions n 5 we take advantage of the strict inequality p < n 2 n 4 in order to estimate the L2qp 1)/q 2) Ω) norm via an interpolation of H 2 Ω) and H 1 Ω). Specifically, there exists θ, 1) dependent on p such that for all θ, θ ) u L 2qp 1)/q 2) Ω) u H 2 θ Ω) u θ H 2 Ω) u 1 θ H 1 Ω). In lower dimensions this estimate holds a fortiori with the L 2qp 1)/q 2) Ω) norm replaced by the norm in L r Ω) for arbitrarily large r. Thus we infer fu), u t ) = fu) u t ) = f u) u u t ) u θ ) p 1 H 2 u 1 θ H 1 u L q u t u 1+θp 1) H u 1 θ)p 1) 2 H u 1 t H 1. The estimate with p = 1 f const) is much simpler so we focus on p > 1. Let 1 α, β, γ > 1 be a conjugate triple, i.e. α + 1 β + 1 = 1. Then two applications of γ Young s inequality give for any ε > and any t [, T 1 ] fu), u t ) ε u α1+θp 1)) H 2 Ω) + ε u t β H 1 Ω) + C ε u γ1 θ)p 1) H 1 Ω). 1 Choose β = 2, θ < min{θ, p 1 }, α = 2 1+θp 1), and γ 1 = 1 2 α 1. Note that the right-most term depends on the finite energy only hence by hypothesis there exists some constant C ε,t1 such that C ε u γ1 θ)p 1) H 1 Ω) C ε,t1 for all t [, T 1 ]. Then we get from 45) using the fact that T 1 T ) u t T ) 2 H 1 + ut ) 2 H 2 T1 u 1 2 H + u 2 1 H + ε ut) 2 2 H + u tt) 2 2 H 1)dt + T 1C ε,t1 u 1 2 H + u 2 1 H + εt 2 1 sup ut) 2 H 2 + u t t) 2 ) H + 1 T1 C ε,t1. t [,T 1] This estimates holds for all T T 1, so take the supremum on the left over all T [, T 1 ] obtaining 1 εt 1 ) sup t [,T 1] ut) 2 H 2 + u t t) 2 ) H 1 u1 2 H + u 2 1 H + T 1C 2 ε,t1. Choose ε < 1/T 1 to get the desired estimate.
22 LORENA BOCIU, PETRONELA RADU, AND DANIEL TOUNDYKOV Now recall that we have been in fact working with Galerkin approximations. Thus what we have proven reads ) u n t) 2 H 2 Ω) + un t t) 2 H 1 Ω) sup t [,T 1] u n 2 H 2 Ω) + un 1 2 H 1 Ω) + C T 1 { u n H 1 Ω), u n 1 L 2 Ω)} for n N. Since the right-hand side is bounded uniformly in n, then u n, u n t ) are confined to a closed ball in L, T ; H 2 Ω) H 1 Ω)). Hence so is the weak* limit u, u t ). We have demonstrated that the higher energy of solutions does not blow-up near any finite terminal time T M. It means that a priori bounds of Proposition 2 and therefore existence of regular solutions can be established over any finite time interval. This step completes the proof of global existence stated in Theorem 4.2. 5.8. Semigroup approach to local regular solutions for linear damping. We conclude the exposition by mentioning a simpler approach to regular solutions when the damping g is linear and the source is twice-differentiable: Assumption 2. Suppose and for p as in f-2). gs) = γs, γ > f C 2 R) with f s) s p 2 The linearity of g is required for the monotonicity of the evolution generator on the higher-energy space. The stronger assumption on f ensures that its associated Nemytski operator is locally Lipschitz H 2 Ω) H 1 Ω). Let A be the Laplace operator as in 9). Based on it we define the evolution generator A : DA ) DA) DA 1/2 ) DA) DA 1/2 ) with domain DA ) = ) I A = A g ) u u or A = t ut) Au + gu t ) { u, v) DA) DA) Au + gv) DA 1/2 ) Remark 5. This definition of DA ) can be used even if g is nonlinear. In the case when g is linear or linearly bounded at infinity DA ) is a subset of H 3 Ω) H 2 Ω). Then system 1) is equivalent to: ) ) u u t + A u t Here we notice that the operator u B = ut) u t }. ) =. 46) fu) ) fu) is locally Lipschitz continuous on DA) DA 1/2 ). This conclusion follows from the fact that u fu) is locally Lipschitz H 2 Ω) H 1 Ω) via Assumption 2. Our goal is to show that A is maximal monotone on the Hilbert space H = DA) DA 1/2 )