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Kirchhoff equations with strong damping

We consider Kirchhoff equations with strong damping, namely with a friction term which depends on a power of the"elastic"operator. We address local and global existence of solutions in two different regimes depending on the exponent in the friction term. When the exponent is greater than 1/2, the dissipation prevails, and we obtain global existence in the energy space assuming only degenerate hyperbolicity and continuity of the nonlinear term. When the exponent is less than 1/2, we assume strict hyperbolicity and we consider a phase space depending on the continuity modulus of the nonlinear term and on the exponent in the damping. In this phase space we prove local existence, and global existence if initial data are small enough. The regularity we assume both on initial data and on the nonlinear term is weaker than in the classical results for Kirchhoff equations with standard damping. Proofs exploit some recent sharp results for the linearized equation and suitably defined interpolation spaces.


Introduction
Let H be a separable real Hilbert space. For every x and y in H, let |x| denote the norm of x, and let x, y denote the scalar product of x and y. Let A be a self-adjoint linear operator on H with dense domain D(A). We assume that A is nonnegative, namely Ax, x ≥ 0 for every x ∈ D(A), so that for every α ≥ 0 the power A α x is defined provided that x lies in a suitable domain D(A α ).
We consider the second order evolution equation where δ > 0 and σ ≥ 0 are real numbers, and m : [0, +∞) → [0, +∞) is a given function. Equations of this type have long been studied in the literature after Kirchhoff's monograph [26]. Let us give a brief outline of the previous results which are closely related to ours.
Non-dissipative Kirchhoff equations When δ = 0, equation (1.1) reduces to This is the abstract setting of the hyperbolic partial differential equation  [26] after some mathematical simplifications to the full system of (local) equations of nonlinear elasticity. From the mathematical point of view, local existence for (1.3)-(1.2) is known provided that the nonlinearity m(x) is Lipschitz continuous and satisfies the strict hyperbolicity condition m(x) ≥ µ 1 > 0 ∀x ≥ 0, (1.4) and initial data (u 0 , u 1 ) are in the space D(A 3/4 )×D(A 1/4 ). This result was substantially established by Bernstein in the pioneering paper [3], and then refined by many authors (see [1] for a modern version). Local existence results extend easily to the so-called mildly degenerate case, namely when the nonlinear term satisfies just the degenerate hyperbolicity condition m(x) ≥ 0 ∀x ≥ 0, (1.5) but m(|A 1/2 u 0 | 2 ) > 0.
Global existence in Sobolev spaces is no doubt the main open problem in the theory of Kirchhoff equations. Local existence in Sobolev spaces is an open problem as well in at least three situations: when the nonlinear term is not Lipschitz continuous, when m(|A 1/2 u 0 | 2 ) = 0 (the so-called really degenerate case, see [39]), and when initial data are just in the energy space D(A 1/2 )×H or in any space weaker that D(A 3/4 )×D(A 1/4 ). The interested reader is referred to the survey [17] for more details.
From the technical point of view, a key role is played by the linearization of (1.3), namely equation u ′′ (t) + c(t)Au(t) = 0, (1.6) where c(t) is a time-dependent coefficient. It is well-known that (1.6) admits global solutions for all initial data in the energy space D(A 1/2 ) × H, or more generally in D(A α+1/2 ) × D(A α ), provided that c(t) is Lipschitz continuous and satisfies the strict hyperbolicity condition 0 < µ 1 ≤ c(t) ≤ µ 2 . (1.7) In the case of Kirchhoff equations the coefficient c(t) is m(|A 1/2 u(t)| 2 ), hence its Lipschitz continuity is related to the boundedness of This is the point where both the Lipschitz continuity of m(x) and the choice of the phase space D(A 3/4 ) × D(A 1/4 ) come into play.
When the coefficient c(t) is not Lipschitz continuous, solutions to (1.6) can exhibit a severe derivative loss for all positive times, as shown in the seminal paper [8]. In the sequel we call (DGCS)-phenomenon this instantaneous loss of regularity. As a consequence, when c(t) is not Lipschitz continuous the initial value problem for (1.6) is well-posed only for very regular initial data, such as Gevrey or analytic functions. We refer to [8] for the linear theory, and to [14] for its application to Kirchhoff equations.
The existence results for linear equations and the (DGCS)-phenomenon motivate the three main assumptions in the theory of Kirchhoff equations, namely Lipschitz continuity of the nonlinearity m(x), strict hyperbolicity, and initial data at least in D(A 3/4 )×D(A 1/4 ). These three main assumptions become three insurmountable barriers when trying to extend the theory.
Kirchhoff equations with standard damping Equation (1.1) with δ > 0 and σ = 0 has long been studied starting from the 80s, both in the strictly hyperbolic case (see [12,38]), and in the mildly degenerate case (see [33] for nonlinear terms of the form m(x) = x α and [13] for general Lipschitz continuous nonlinear terms).
As for local solutions, the state of the art is exactly the same as in the non-dissipative case. As for global solutions, it was possible to prove that they exist in Sobolev spaces (such as D(A) × D(A 1/2 ) and sometimes D(A 3/4 ) × D(A 1/4 )) provided that initial data are small enough in the same space and m(x) is Lipschitz continuous. The interested reader is referred to the survey [16] for further details.
In other words, a standard damping seems to be impotent against the classical three barriers. As a further evidence, it was recently shown in [19] that the linearized equation with standard dissipation u ′′ (t) + 2δu ′ (t) + c(t)Au(t) = 0 can exhibit the (DGCS)-phenomenon if the coefficient c(t) is not Lipschitz continuous.
Equations with strong damping Mathematical models with strong damping were proposed in [4], leading to the linear equation u ′′ (t) + 2δA σ u ′ (t) + Au(t) = 0, (1.8) which afterwards was rigorously analyzed by many authors from different points of view, and with different choices of σ. Here we just quote the first papers [5,6,7] (see also the more recent [22]) devoted to analyticity properties of the semigroup associated to (1.8), and the papers [24,25,37] by the dispersive school where (1.8) is considered in the concrete case where Au = −∆u in R n or special classes of unbounded domains. Finally, equation (1.8) was considered in [20] in full generality, namely for every σ ≥ 0 and every nonnegative self-adjoint operator A. Concerning Kirchhoff equations with strong dissipation, the first result we are aware of was obtained by Nishihara in [31]. He considered the case σ = 1, and proved global existence for initial data in D(A) × D(A 1/2 ) provided that the nonlinearity is Lipschitz continuous and satisfies the degenerate hyperbolicity condition (1.5). Some years later, Matos and Pereira [28] improved this result by showing global existence in the energy space D(A 1/2 ) × H with the same assumptions on σ and m(x). In other words, in the case σ = 1 Nishihara overcame one of the three classical barriers (strict hyperbolicity), and then Matos and Pereira overcame one more barrier (regularity of initial data). Both results do not extend to σ < 1, and seem to exploit the Lipschitz continuity of m(x) in an essential way. We discuss this technical point at the beginning of section 6.
As far as we know, references [31] and [28] represented the state of the art up to now, and all subsequent papers on the subject were devoted just to qualitative properties of these solutions, such as decay estimates (see for example [32,34]).
In this paper we consider equation (1.1) in the full range σ > 0. Two different regimes appear.
• When σ > 1/2 (supercritical regime) the dissipation prevails in an overwhelming way. Indeed, in Theorem 2.1 we prove global existence in the energy space assuming only continuity and degenerate hyperbolicity of the nonlinear term.
• When 0 < σ ≤ 1/2 (subcritical regime) the dissipation competes with the continuity modulus of the nonlinear term. In Theorem 2.2 we assume strict hyperbolicity, and we prove -local existence for all initial data in a phase space depending on σ and on the continuity modulus of m.
-global existence provided that initial data are small in the same space.
Just to give an example, if m(x) is β-Hölder continuous for some β ∈ (1 − 2σ, 1), then we obtain local/global existence in phase spaces of the form Note that this condition allows nonlinear terms which are not Lipschitz continuous and initial data with α < 1/4.
Overview of the technique Theorem 2.1 and Theorem 2.2 can be proved in many different ways (here we use a fixed point argument on the coefficient c(t), but also fixed point arguments on u(t) or Galerkin approximations could work) after that three main tools have been developed. The first tool is linearization. The results for the non-dissipative Kirchhoff equation (1.3) are based on the theory of the linearized equation (1.6). In the same way, our results for (1.1) rely on the theory of the linear equation where c(t) is a time-dependent coefficient. This theory was developed in the recent paper [19]. The following two regimes appeared.
Let us consider for example the subcritical regime. In the non-dissipative case, or in the case with standard dissipation, the key point was controlling the Lipschitz constant of c(t). Now the linear theory tells us that it is enough to control the γ-Hölder constant of c(t) for some γ > 1 − 2σ. In turn, this constant depends on the continuity moduli of m(x) and |A 1/2 u(t)| 2 .
In order to control the latter, we develop our second tool. We prove that, when (1.10) generates a continuous semigroup, the continuity modulus of |A 1/2 u(t)| 2 depends on initial data only (and not on c(t)), according to the following scheme: Thus, if the nonlinear term is β-Hölder continuous for some β ∈ (0, 1), and initial data are in D(A α+1/2 ) × D(A α ) for some α ∈ (0, 1/4), then c(t) is Hölder continuous with exponent 4αβ, and when 4αβ > 1 − 2σ the linear theory is well-posed. This is where condition (1.9) appears.
The last tool we develop allows to deal with the equality case 4αβ = 1−2σ. Roughly speaking, the reason is that an initial condition which is "purely" in D(A α+1/2 ) × D(A α ) does not exist, in the same way as a vector which is just in H, and nothing more, does not exist. Indeed, any vector in H lies actually in some better space, and this better space guarantees a better continuity modulus for |A 1/2 u(t)| 2 . We state this point formally in Proposition 5.6.
Conclusions Our results are the best possible ones in the supercritical regime, where all three barriers have been overcome to the maximum extent. In the subcritical regime we extended the theory beyond two of the three barriers, namely to less regular nonlinearities and less regular initial data. The width of this extension depends on σ and it is the best possible one according to the linear theory, because the linearized equation does admit the (DGCS)-phenomenon beyond the spaces we considered in this paper.
Our existence results confirm the general paradigm according to which Kirchhoff equations are well-posed, at least locally in time, whenever the (DGCS)-phenomenon is excluded. What happens beyond remains an open problem, and deep new ideas are likely to be needed (see [14] for a partial result).
We also hope that these results could give an indication about the regularizing effects one can expect when adding a strong dissipation to quasilinear hyperbolic equations.
Structure of the paper This paper is organized as follows. In section 2 we introduce the functional setting and we state our main results concerning local and global solvability for problem (1.1)-(1.2). In section 3 we recall the results we need from the linear theory developed in [19]. In section 4 we study how solutions to the linear problem (1.10) depend continuously on the coefficient c(t). In section 5 we introduce our interpolation spaces. In section 6 we prove a priori estimates for Kirchhoff equations after discussing what goes wrong in the classical ones. Finally, in section 7 we prove our main results, actually in a stronger form based on our interpolation spaces.

Notation and statement of main results
Functional spaces Let H be a separable Hilbert space. Let us assume that H admits a countable complete orthonormal system {e k } k∈N made by eigenvectors of A. We denote the corresponding eigenvalues by λ 2 k (with the agreement that λ k ≥ 0), so that Ae k = λ 2 k e k for every k ∈ N. We do not assume the monotonicity of the sequence {λ k }, even if this assumption would be reasonable in many applications. Every u ∈ H can be written in a unique way in the form u = ∞ k=0 u k e k , where u k = u, e k are the Fourier components of u. In other words, the Hilbert space H can be identified with the set of sequences {u k } of real numbers such that ∞ k=0 u 2 k < +∞. We stress that this is just a simplifying assumption, with substantially no loss of generality. Indeed, according to the spectral theorem in its general form (see for example Theorem VIII.4 in [36]), one can always identify H with L 2 (M, µ) for a suitable measure space (M, µ), in such a way that under this identification the operator A acts as a multiplication operator by some measurable function λ 2 (ξ). All definitions and statements in the sequel can be easily extended to the general setting just by replacing the sequence {λ k } with the function λ(ξ), and the sequence {u k } of Fourier components of u with the element u(ξ) of L 2 (M, µ) corresponding to u under the identification of H with L 2 (M, µ).
Powers of the operator A are just defined as It is well-known that D(A α ) is itself a Hilbert space, with norm defined by In concrete applications, H is the space L 2 (Ω) for some open set Ω ⊆ R n , and Au = −∆u, with suitable boundary conditions. If the boundary of Ω is regular enough, D(A α ) is the set of functions in the Sobolev space H 2α (Ω) attaining the boundary conditions in a suitable sense.
(2.2) From (2.1) and (2.2) one can easily deduce that every continuity modulus satisfies and that the composition of two continuity moduli is again a continuity modulus. We exploit these two properties several times in the sequel. A function c : [0, +∞) → R is said to be ω-continuous if More generally, a function c : X → R (with X ⊆ R) is said to be ω-continuous if it satisfies the same inequality for every a and b in X.

Main results
The main results of this paper concern local and global existence for problem (1.1)-(1.2). The first one gives a full answer in the case σ > 1/2, which we call supercritical dissipation. • σ > 1/2 and δ > 0 are two real numbers, Then the problem admits at least one global solution Our second result addresses the case σ ≤ 1/2, which we call subcritical dissipation. • σ ∈ (0, 1/2] and δ > 0 are two real numbers, for the same α ∈ [0, 1/4) which appears in (2.5).
Then the following conclusions hold true.
(1) (Local existence) There exists T > 0 such that problem (1.1)-(1.2) admits at least one local solution (2) (Global existence) There exists ε 0 > 0 such that, if initial conditions satisfy then there exists at least one global solution Remark 2.3. In Theorem 2.2, the ω-continuity assumption on m(x) can be easily weakened to local ω-continuity, meaning that we can limit ourselves to assume that for every R > 0 the function m(x) has continuity modulus ω R (x) in [0, R], and all these continuity moduli satisfy (2.5) with the same exponent α (of course the limsup may depend on R in an arbitrary way). We spare the reader from this standard generalization.
If β ∈ (0, 1 − 2σ), then there is no α ≤ 1/4 for which (2.5) is satisfied, so that our theory does not apply to this case.
In the limit case β = 1 − 2σ, one should take α = 1/4 in order to fulfil (2.5). This case in not included in Theorem 2.2 above. Nevertheless, a careful inspection of the proof reveals that in this limit case one can obtain both local and global solvability in D(A 3/4 ) × D(A 1/4 ) provided that initial data are small enough (so that the smallness is required also for local existence). We spare the reader from this subtlety.
Remark 2.5. The following pictures provide a rough description of the state of the art. In the horizontal axis we represent the regularity of m(x). With some abuse of notation, values β ∈ (0, 1) mean that m(x) is β-Hölder continuous, β = 1 means that it is Lipschitz continuous, β > 1 means even more regular. The value α in the vertical axis represents the phase space In the case σ = 0 (not addressed in this paper), both local existence and global existence for small data are known in the region where α ≥ 1/4 and β ≥ 1.
In the case 0 < σ < 1/2, Theorem 2.2 gives the same conclusions in a larger region, which is bounded by the line α = (1 − 2σ)/4, the line β = 1 − 2σ, and the curve α = (1 − 2σ)/(4β). The vertical part of the boundary is not included, or at least it is just partially included as explained at the end of Remark 2.4.
In the case σ > 1/2, Theorem 2.1 guarantees global existence in the phase space, independently of the size of initial data, even in the degenerate hyperbolic case.
Remark 2.6. Let us discuss in detail the borderline case σ = 1/2. First of all, it fits in the framework of Theorem 2.2, where for σ = 1/2 assumption (2.5) holds true for free, even if α = 0. Thus we obtain both local existence and global existence for small data in the energy space D(A 1/2 ) × H, provided that m(x) is continuous and satisfies the strict hyperbolicity condition (1.4).
Concerning the degenerate hyperbolic case, the limit exponent σ = 1/2 is not included in Theorem 2.1 as stated above. Nevertheless, a careful inspection of the proof reveals that some conclusions hold true also when σ = 1/2. Indeed, we can prove that a local solution exists for all initial data (u 0 , u 1 ) in the energy space such that 4δ 2 > m(|A 1/2 u 0 | 2 ), and this solution survives as long as 4δ 2 > m(|A 1/2 u(t)| 2 ). This could be always the case for suitable choices of m(x) and (u 0 , u 1 ), for example in the trivial case where m(x) < 4δ 2 for every x ≥ 0. We spare the reader from this further subtlety.
This discussion shows that in the critical case σ = 1/2 we have coexistence of two statements, one dealing with strictly hyperbolic nonlinearities and one dealing with degenerate hyperbolic nonlinearities. This coexistence reflects the analogous coexistence in the linear case (see [19,Remark 3.7]) and suggests the existence of a more general unifying statement, which could probably deserve further investigation.
Remark 2.7. Solutions to (1.1) are also solutions to (1.10), hence they inherit all properties of solutions to linear equations. In particular, multiple choices of the phase space are possible when σ > 1/2, as observed in [20,19]. Therefore, under the assumptions of Theorem 2.1, the problem is well-posed not only in the energy space or more generally

The linearized equation -Previous results
In this section we collect all the results we need from the linear theory developed in [19]. To this end, we introduce some further notation. Given any ν ≥ 0, we write H as an orthogonal direct sum where H ν,− is the closure of the subspace generated by all eigenvectors of A relative to eigenvalues λ k < ν, and H ν,+ is the closure of the subspace generated by all eigenvectors of A relative to eigenvalues λ k ≥ ν. For every vector u ∈ H, we write u ν,− and u ν,+ to denote its components with respect to the decomposition (3.1). We point out that H ν,− and H ν,+ are A-invariant subspaces of H, and that A is a bounded operator when restricted to H ν,− , and a coercive operator when restricted to H ν,+ if ν > 0.
Here we state the results in local form, namely with coefficients c(t) defined in some time-interval [0, T ]. They follow immediately from the corresponding results proved in [19] for coefficients defined in [0, +∞) (it is enough to extend the coefficient by setting c(t) = c(T ) for every t ≥ T ).

Existence and estimates in the energy space
As always, we distinguish the supercritical and the subcritical case. In both cases, we limit ourselves to initial data in the energy space.
Theorem A (Supercritical dissipation, see [19,Theorem 3.1]). Let T > 0, and let us consider problem (1.10)-(1.2) under the following assumptions: • A is a self-adjoint nonnegative operator on a separable Hilbert space H, • the coefficient c : [0, T ] → R is measurable and satisfies the degenerate hyperbolicity assumption (1.11), • σ and δ are two positive real numbers such that either σ > 1/2, or σ = 1/2 and 4δ 2 ≥ µ 2 , Then the problem has a unique solution Moreover, there exists a constant K 1 (δ, µ 2 ), depending only on δ and µ 2 (and in particular independent of T ), with the following property. For every ν ≥ 1 such that it turns out that

4)
and more precisely for every t ∈ [0, T ] it turns out that Theorem B (Subcritical dissipation, see [19,Theorem 3.2]). Let T > 0, and let us consider problem (1.10)-(1.2) under the following assumptions: • A is a self-adjoint nonnegative operator on a separable Hilbert space H, • the coefficient c : [0, T ] → R satisfies the strict hyperbolicity assumption (1.7) and the ω-continuity assumption (2.4) for some continuity modulus ω(x), • σ ∈ [0, 1/2] and δ > 0 are two real numbers such that where we set Then the problem has a unique solution with the regularity stated in (3.2). Moreover, there exists a constant K 2 (δ, µ 1 , µ 2 ), depending only on δ, µ 1 and µ 2 (and in particular independent of T ), with the following property. For every ν ≥ 1 such that it turns out that and more precisely for every t ∈ [0, T ] it turns out that The key tool in the proof of Theorem A and Theorem B are some estimates on the Fourier components of the solution. We recall these estimates because we need them in the sequel. Let {u k (t)} denote the components of u(t) with respect to the usual orthonormal system {e k }, and let {u 0k } and {u 1k } denote the corresponding components of initial conditions. It turns out that u k (t) is a solution to Estimates on low-frequency components are quite general (see [19,Remark 3.3]). Let us assume only that c(t) is a measurable function satisfying the degenerate hyperbolicity assumption (1.11) in [0, T ], and that δ ≥ 0, σ ≥ 0, ν ≥ 0 and λ k ≤ ν. Then it turns out that Summing over all indices k with λ k < ν we obtain both (3.5) and (3.10).
Estimates on high-frequency components are more delicate, and different in the two cases. In the supercritical case, let us assume that c(t), δ and σ are as in Theorem A, that ν ≥ 1 satisfies (3.3), and that λ k ≥ ν. Then it turns out that where the constant K 1 (δ, µ 2 ) depends only on δ and µ 2 . This was established in [19,Lemma 5.1]. Since the constant is independent of k, summing over all indices k with λ k ≥ ν we obtain (3.6). Finally, (3.4) follows from (3.5) and (3.6) (in this point we need that K 1 (δ, µ 2 ) ≥ 1, which in turn is a consequence of (3.6) with t = 0). In the subcritical case, let us assume that c(t), δ and σ are as in Theorem B, that ν ≥ 1 satisfies (3.9), and that λ k ≥ ν. Then it turns out that where the constant K 2 (δ, µ 1 , µ 2 ) depends only on δ, µ 1 and µ 2 . This was established in [19,Lemma 5.2]. Since the constant is independent of k, summing over all indices k with λ k ≥ ν we obtain (3.11).

The regularizing effect
The strong dissipation has a regularizing effect in the range σ ∈ (0, 1). This effect was studied in [19] in terms of Gevrey spaces. Here we just state what we need in the sequel, without relying on the theory of abstract Gevrey spaces. Let us start with the supercritical case.
Theorem C (Supercritical dissipation -Regularizing effect). Let u(t) be a solution to problem (1.10)-(1.2) under the same assumptions of Theorem A. Let us assume in addition that either σ ∈ (1/2, 1), or σ = 1/2 and 4δ 2 > µ 2 . Let us consider the function and let us distinguish three cases.
In the second case it turns out that also and therefore the regularity (3.16) follows from the regularizing effect of parabolic problems.
In the third case the regularity in (S 0 , T ] follows from [19,Theorem 3.8] as in the first case, while the regularity of u ′ (t) in (0, S 0 ) follows from the parabolic problem as in the second case (indeed c(t) = 0 for every t ∈ [0, S 0 ]).
The last result concerns the regularizing effect in the subcritical case, and it is an immediate consequence of [19,Theorem 3.9].

Passing to the limit in linear problems
In this section we consider a sequence of linear problems We assume that the sequence of coefficients c n : [0, T ] → [0, +∞) pointwise converges to a limit coefficient We investigate the convergence of solutions to (4.1)-(4.2) to the solution of the limit problem u ′′ Our goal is proving uniform convergence of solutions in the energy space, namely We obtain the following two results, which are probably interesting in themselves. Let us assume that • A is a self-adjoint nonnegative operator on a separable Hilbert space H, • the coefficients c n : [0, T ] → R are measurable and satisfy the degenerate hyperbolicity assumption (1.11) with the same µ 2 , • c n (t) pointwise converges to c ∞ (t), • σ and δ are two positive real numbers such that either σ > 1/2, or σ = 1/2 and 4δ 2 ≥ µ 2 , Then u n → u ∞ in the sense of (4.6).
Theorem 4.2 (Convergence in the subcritical case). Let u n (t) be the sequence of solutions to the approximating problems (4.1)-(4.2), and let u ∞ (t) be the solution to the limit problem (4.4)-(4.5). Let us assume that • A is a self-adjoint nonnegative operator on a separable Hilbert space H, • the coefficients c n : [0, T ] → R satisfy the strict hyperbolicity assumption (1.7) with the same constants µ 1 and µ 2 , and the ω-continuity assumption (2.4) with the same continuity modulus ω(x), • c n (t) pointwise converges to c ∞ (t), • σ ∈ [0, 1/2] and δ > 0 are two real numbers satisfying (3.7), Then u n → u ∞ in the sense of (4.6).
We observe that in both cases the limit coefficient c ∞ (t) satisfies the same assumptions of the approximating coefficients c n (t).
The rest of this section is devoted to the proof of these results. A careful inspection of the argument reveals that in both cases the pointwise convergence assumption (4.3) can be weakened to convergence in L 2 ((0, T )).

Estimates on components
In this section we provide estimates for solutions to the family of non-homogeneous linear ordinary differential equations with null initial data Our interest is motivated by the fact that Fourier components of the difference u n (t) − u ∞ (t) are solutions to problems of this type. Then there exist two constants Γ 1 and Γ 2 such that The constants Γ 1 and Γ 2 depend only on δ, σ, and µ 2 (in particular they are independent of λ, f , and T ).
The left-hand side of (4.15) is a quadratic form in w ′ (t) and w(t). The coefficients A and B are positive. Therefore, this quadratic form is nonnegative for all values of w ′ (t) and w(t) if and only if AB ≥ C 2 . With some algebra, this condition turns out to be equivalent to If σ = 1/2 and 4δ 2 ≥ µ 2 , then and hence even the first term in the left-hand side of (4.16) is greater than or equal to the right-hand side, independently of k 3 . The same is true if σ > 1/2 and 4δ 2 λ 4σ−2 ≥ µ 2 . If σ > 1/2 and 4δ 2 λ 4σ−2 ≤ µ 2 , then λ is bounded in terms of δ and µ 2 , and hence the term k 2 3 alone in the left-hand side of (4.16) is greater than or equal to the right-hand side, provided that k 3 is suitably chosen. This completes the proof of (4.14).
Then there exist two constants Γ 3 and Γ 4 such that The constants Γ 3 and Γ 4 depend only on δ, σ, µ 1 , and on the continuity modulus ω(x) (in particular they are independent of λ, f , and T ).
Proof Let us extend c(t) beyond T by setting c(t) := c(T ) for t ≥ T . For every ε > 0, let us introduce the regularized coefficient It is easy to see that c ε ∈ C 1 ([0, T ], R) and satisfies the following estimates: The constants ν, and k 1 , . . . , k 4 we introduce in the sequel are positive numbers depending only on δ, σ, µ 1 , and on the continuity modulus ω(x).
To begin with, we fix ν ≥ 1 such that (3.9) holds true. Such a value exists due to assumption (3.7). Then we set and we consider the energy The last term can be estimated as in (4.10). Therefore, keeping (4.17) into account, it follows that (4.21) Let L 1 , L 2 , L 3 denote the three lines in the expression of E ′ λ (t). To begin with, we observe that Since λ 4σ ≤ 1 + λ 2 (because σ ≤ 1/2), from (4.20) we deduce that Now we claim that Indeed, this inequality is equivalent to where for the sake of shortness we set The left-hand side of (4.24) is a quadratic form in w ′ (t) and w(t). The coefficient A is positive. Therefore, this quadratic form is nonnegative for all values of w ′ (t) and w(t) if and only if AB ≥ C 2 . With some algebra, this condition turns out to be equivalent to (4.25) When λ ≥ ν we chose ε(λ) = 1/λ, so that (4.18) and (4.19) read as Therefore, since (3.9) holds true for λ ≥ ν, it follows that As a consequence, when λ ≥ ν the sum of the first two terms in the left-hand side of (4.25) is greater than or equal to the whole right-hand side, independently of k 3 .
As a consequence, the third term in the left-hand side of (4.25), namely k 2 3 , is greater than or equal to the whole right-hand side, provided that k 3 is large enough. This completes the proof of (4.23).
Plugging (4.22) and (4.23) into (4.21), we deduce that The conclusion follows exactly as in the proof of Lemma 4.3

Proof of Theorem 4.1
The result is established in two steps. In the first one, we prove the conclusion under the more restrictive assumption that (u 0 , u 1 ) ∈ D(A) × D(A 1/2 ). In this case a stronger result holds true, in the sense that the norm of u n − u ∞ in the energy space can be estimated in terms of the norm of c n − c ∞ in L 2 ((0, T ), R). In the second step we apply an approximation procedure in order to obtain the conclusion for all initial data, abandoning the possibility to estimate the convergence rate.
Convergence for more regular data Let us assume that (u 0 , u 1 ) ∈ D(A) × D(A 1/2 ). We prove that there exists a constant Γ, depending only on δ, σ, µ 2 and T , such that for every t ∈ [0, T ].
To this end, we introduce the difference w n (t) := u ∞ (t) − u n (t), which turns out to be a solution to with null initial data w n (0) = w ′ n (0) = 0. We also consider the components w n,k (t) of w n (t) with respect to the orthonormal system {e k }. If u n,k (t) are the corresponding components of u n (t), it turns out that w n,k (t) is the solution to the ordinary differential equation with null initial data w n,k (0) = w ′ n,k (0) = 0. Therefore, we can apply Lemma 4.3 with We obtain that where Γ 1 and Γ 2 do not depend on k. Therefore, summing over all k's we deduce that In order to estimate |Au n (s)|, we choose ν ≥ 1 satisfying (3.3) (we point out that ν depends only on δ, σ and µ 2 ), and we apply Theorem A to the function A 1/2 u n (t), which is again a solution to equation (4.1) thanks to linearity. We obtain that Plugging this estimate into (4.27), we finally deduce (4.26).
Convergence for general data Let us consider now an initial condition (u 0 , u 1 ) ∈ D(A 1/2 ) × H. We show that, for every η > 0, there exists n 0 ∈ N such that To this end, we exploit a classical approximation argument. We choose ν ≥ 1 satisfying (3.3), and then we choose (v 0 , v 1 ) ∈ D(A) × D(A 1/2 ) such that where K 1 (δ, µ 2 ) is again the constant of Theorem A.
Let v n (t) and v ∞ (t) denote the solutions to (4.1) and (4.4), respectively, with initial is again a solution to (4.1), from (3.4) and (4.29) it follows that

Proof of Theorem 4.2
Same proof of Theorem 4.1, the only difference being that the key estimates come from Theorem B and Lemma 4.4, instead of Theorem A and Lemma 4.3, and the constant Γ in (4.26) now depends on δ, σ, µ 1 , µ 2 , T , and on the continuity modulus ω.

Interpolation spaces
In this section we introduce a family of interpolation spaces between the energy space D where K 0 denotes the set of nonnegative integers k such that λ k > 0, and u 0k and u 1k denote the components of u 0 and u 1 with respect to the usual orthonormal system. Then we define the space It is easy to see that Moreover, V ω is actually a vector space, (u 0 , u 1 ) ω is a seminorm, and the full norm induces a Hilbert space structure on V ω . When ω(x) = x 4α for some α ∈ [0, 1/4], the space V ω is just D(A α+1/2 ) × D(A α ).
Since in this paper we are dealing with continuity moduli in several different contexts, from now on we write ω d (x) to denote continuity moduli involved in interpolation spaces (here "d " stands for "data").

Propagation of regularity
When problem (1.10)-(1.2) generates a continuous semigroup in the energy space, then every pair of initial conditions in V ω d gives rise to a solution lying in the same space for all positive times.
Let us start with the case σ ≥ 1/2. Let us assume in addition that (u 0 , u 1 ) ∈ V ω d for some continuity modulus ω d (x).
Proof Let {u k (t)} denote the components of u(t) with respect to the usual orthonormal system {e k }, and let {u 0k } and {u 1k } denote the corresponding components of initial conditions. Since we are in the assumptions of Theorem A, we can estimate these components as in (3.12) and (3.13). Since K 1 (δ, µ 2 ) ≥ 1, we can combine these two estimates and deduce that independently of k. Dividing by ω(1/λ k ), and summing over K 0 (the set of indices k with λ k = 0), we obtain (5.1). The same estimate shows also the uniform convergence of the series which proves the continuity of the pair (u, u ′ ) with values in V ω d .
The result for the subcritical case is analogous. Here we state explicitly a time independent estimate for high-frequency components in the same spirit of (3.11). This estimate is crucial in the proof of the global existence statement of Theorem 2.2. An analogous estimate holds true also in the supercritical case, but we do not need it in the sequel. Let us assume in addition that (u 0 , u 1 ) ∈ V ω d for some continuity modulus ω d (x).
and for every ν ≥ 1 satisfying (3.9) it turns out that where K 2 (δ, µ 1 , µ 2 ) is the constant of Theorem B. Moreover, the high-frequency component of u(t) satisfies Proof Same proof of Proposition 5.2, this time starting from estimates (3.12) and (3.14), which hold true under the assumptions of Theorem B.
Remark 5.4. From (2.1) it follows that ω d (1/λ k ) ≥ ω d (1/ν) whenever λ k ≤ ν. Therefore, both in the supercritical and in the subcritical case, the low-frequency components can be estimated in terms of the usual energy as follows Also this estimate is crucial in the proof of the global existence statement of Theorem 2.2.

Interpolation spaces vs time regularity
It is well-known that space regularity and time regularity of solutions to hyperbolic problems are strongly related. The following result clarifies the connection between the interpolation space V ω d and the continuity modulus of the function t → |A 1/2 u(t)| 2 . Let us assume that (u(t), u ′ (t)) ∈ V ω d for every t ∈ [0, T ], and there exists a constant L such that Then it turns out that Proof Let u k (t) denote the components of u(t) with respect to the usual orthonormal system. For every ε > 0, let K − ε denote the set of nonnegative integers k such that 0 < λ k ≤ 1/ε, and let K + ε denote the set of nonnegative integers k such that λ k > 1/ε. Let us set for simplicity The time-derivative of g ε (t) exists and is given by For every k ∈ K − ε it turns out that 1/λ k ≥ ε, and hence λ k ω d (1/λ k ) ≤ ω d (ε)/ε because of the monotonicity property (2.2) of the continuity modulus. Therefore, from (5.7) and assumption (5.5) we deduce that for every t ∈ [0, T ]. In particular, from the mean value theorem it follows that for every a and b in [0, T ]. On the contrary, for every k ∈ K + ε it turns out that 1/λ k < ε, and hence ω d (1/λ k ) ≤ ω d (ε) because of the monotonicity property (2.1) of the continuity modulus. It follows that for every t ∈ [0, T ]. From this estimate and (5.8) we deduce that for every a and b in [0, T ], and every ε > 0. Now it is enough to choose ε := |a − b|, and (5.6) is proved.

An intermediate space for each initial condition
In the next result we show that every pair (u 0 , u 1 ) which is in D(A α+1/2 ) × D(A α ) for some α ∈ [0, 1/4) lies actually in a "better" space, namely a space V ω d corresponding to a continuity modulus ω d (x) which tends to 0 faster than x 4α . It is a refinement of the classical calculus result according to which the terms of a converging series can always be multiplied by a diverging sequence obtaining again a converging series. We can also control the (semi)norm in the interpolation space in terms of the (semi)norm in the original space.
Proposition 5.6 (Customized interpolation spaces). For every α ∈ [0, 1/4) and every (u 0 , u 1 ) ∈ D(A α+1/2 ) × D(A α ), there exists a continuity modulus ω d (x) such that (u 0 , u 1 ) ∈ V ω d and ω d (1) = 1, (5.9) Proof Let u 0k and u 1k denote the components of u 0 and u 1 with respect to the usual orthonormal system. Let us set for simplicity For every n ∈ N, let us set there exists an increasing sequence n h of nonnegative integers such that n 0 = 0 and n≥n h a n ≤ E 4 h ∀h ∈ N.
Since B h ⊆ ∪ n≥n h A n , from (5.13) it turns out that Let us consider the sequence ϕ h defined by ϕ 0 = 1, ϕ 1 = 1, and From this definition it follows that for every h ∈ N. Moreover it turns out that lim h→+∞ ϕ h = +∞. (5.16) Indeed, (5.16) is obvious if ϕ h+1 = 2 h for infinitely many indices. If not, it means that ϕ h+1 = ϕ h n h+1 /n h for every h greater than or equal to some h 0 ≥ 1. In this case an easy induction gives that so that (5.16) follows from the fact that n h → +∞. Let us consider now the piecewise affine function ϕ : [0, +∞) → R such that ϕ(n h ) = ϕ h for every h ∈ N, namely the function defined by From the first relation in (5.15) it follows that Finally, from the third relation in (5.15) one can prove that is nondecreasing (5.20) (it is enough to show that its derivative is nonnegative in each interval (n h , n h+1 )). Let us finally set We claim that ω d (x) is a continuity modulus, and that (5.9) through (5.12) hold true. Equality (5.9) follows from the fact that ϕ(1) = 1, while estimate (5.10) follows from (5.18). Since 4α < 1, from (5.19) we deduce that ω d (x) → 0 as x → 0 + , which proves that ω d is continuous also in x = 0, the only point in which continuity was nontrivial. The limit (5.11) follows from (5.19) because 4α < 1. The monotonicity property (2.1) of ω d (x) follows from the fact that ϕ(x) is nondecreasing, while the monotonicity property (2.2) is equivalent to (5.20) after the variable change y := 1/x. It remains to prove (5.12). To this end, we begin by observing that Now for every k ∈ B h it turns out that n h ≤ λ k < n h+1 . From (5.17) and the fact that 4α < 1 it follows that Keeping (5.14) into account, we deduce that which proves (5.12).

A priori estimates for Kirchhoff equations
In this section we derive a priori estimates for solutions to (1.1). Classical estimates are based on the usual Hamiltonian and therefore for every t ∈ [0, T ]. This is the classical way to obtain estimates both on |u ′ (t)| and on |A 1/2 u(t)|. Unfortunately, things are not so simple under our assumptions. A first problem comes from the lack of strict hyperbolicity when σ > 1/2. In that case there is no guarantee that M(x) → +∞ as x → +∞, and hence (6.3) does not provide a bound on |A 1/2 u(t)|, not even locally.
The second and even worse problem comes from regularity issues. Indeed, when computing the time-derivative of H(t), we have to deal with terms such as u ′ (t), Au(t) , which are quite delicate in the case of weak solutions. In the literature this issue has been addressed in two different ways. • In the strongly dissipative case with σ = 1, equality (6.3) seems to suggest that , then it makes sense (at least almost everywhere) also for solutions in the energy space. This is the key point in the paper [28], in which Nishihara's theory [31] is extended to initial data in the energy space. Unfortunately, this heuristic argument strongly suggests that this approach is hopeless when σ < 1.
Our assumptions are apparently too weak for both strategies. This leads us to follow a different path. As for the lack of coerciveness of M(x), we introduce a modified Hamiltonian. As for the lack of regularity of solutions, when σ ≥ 1 we exploit a strategy similar to the one described in the second point above. When σ ∈ (0, 1), we exploit the regularizing effect presented in section 3.2.
In the first a priori estimate we control |A 1/2 u(t)| in the supercritical case. We obtain an exponential bound, which is far from being optimal and could be improved to a polynomial bound with some additional effort. Nevertheless, any bound is enough in the sequel. Then there exists two constants K 3 (δ) and K 4 (δ), depending only on δ (an in particular independent of T and u), such that for every t ∈ [0, T ].
Case σ ≥ 1 Let us set c(t) := m |A 1/2 u(t)| 2 . It is easy to see that v n (t) is a solution to the linear problem v ′′ n (t) + 2δA σ v ′ n (t) + c(t)Av n (t) = 0. From the regularity assumption (3.2) and the convergence is uniform in [0, T ] in all three cases (but in the sequel we need just pointwise convergence and uniform boundedness). Let us consider the modified approximated Hamiltonian . Since σ ≥ 1, it turns out that Integrating this differential inequality we obtain that Passing to the limit as n → +∞ we conclude that which proves (6.4) in the case σ ≥ 1 with K 3 (δ) := 1 and K 4 (δ) := 4δ + (4δ) −1 .
Case 1/2 < σ < 1 Let us consider u(t) as a solution to the linear equation (1.10) with the coefficient c(t) defined as in (6.5). Let us consider the Hamiltonian H(t) defined in (6.1). Since σ ∈ (1/2, 1), from Theorem C we know that u ′ (t) is continuous with values in D(A 1/2 ) for every t ∈ (0, T ], or at least for every t ∈ (0, S 0 ) ∪ (S 0 , T ] for some S 0 ∈ (0, T ). This is enough to justify rigorously the calculation leading to (6.2). Indeed, the terms such as u ′ (t), Au(t) which appear during the computation of H ′ (t) can be interpreted in the form A 1/2 u ′ (t), A 1/2 u(t) , and hence they are well-defined when both u(t) and u ′ (t) are continuous with values in D(A 1/2 ). This proves that (6.2) holds true for every t ∈ (0, T ], or at least for every t ∈ (0, S 0 ) ∪ (S 0 , T ]. Since H(t) is continuous because of the regularity assumption (3.2), this is enough to conclude that (6.3) holds true for every t ∈ [0, T ].
Let us consider now the modified Hamiltonian which is well-defined in the range 1/2 ≤ σ ≤ 1. In this range it turns out that and therefore and Now we claim that H(t) is time-differentiable where H(t) is time-differentiable, namely in (0, T ] or in (0, S 0 ) ∪ (S 0 , T ], and its time-derivative is Indeed, in addition to the terms due to H ′ (t), now the most dangerous term is the last one. In order to show that this term is well-defined, we consider the same three cases as in the statement of Theorem C. In the first case, the term is well-defined because of the regularity of u(t). In the second case, it is well-defined because c(t) ≡ 0 in [0, T ]. In the third case, it is well-defined in [0, S 0 ] because c(t) ≡ 0 in that interval, and it is well-defined in (S 0 , T ] because of the regularity of u(t) in that interval.
In order to estimate H ′ (t), we observe that and, since σ ≥ 1/2, It follows that Now we estimate |u ′ (t)| by means of (6.3), and we estimate |u(t)| by observing that It follows that for every t ∈ (0, T ], or at least for every t ∈ (0, S 0 ) ∪ (S 0 , T ]. Since H(t) is continuous in [0, T ], a simple integration gives that Now we estimate H(0) as in (6.6) and powers of t with e t . Keeping (6.7) into account, we obtain an estimate of the form (6.4), with K 4 (δ) := 1 and a suitable K 3 (δ).
Finally, if we want K 3 (δ) and K 4 (δ) to be the same in the case σ ≥ 1 and in the case 1/2 ≤ σ ≤ 1, it is enough to take for each of them the maximum of the values obtained in the two ranges.
Our second a priori estimate concerns the subcritical case σ ∈ (0, 1/2]. We state the result under the assumptions we need in the sequel, even if slightly different from those of Theorem 2.2. • σ ∈ (0, 1/2] and δ > 0 are two real numbers. Let us assume that there exists a local solution u(t), defined in the time-interval [0, T ] for some T > 0, with the regularity prescribed by (3.2). Let us assume in addition that the continuity modulus of the function t → m(|A 1/2 u(t)| 2 ) satisfies the assumptions of Theorem B, and in particular inequality (3.7) with Λ ∞ defined by (3.8).
Then it turns out that Proof Let us consider u(t) as a solution to the linear equation (1.10) with the coefficient c(t) defined as in (6.5). This coefficient is bounded from below by µ 1 because of the strict hyperbolicity assumption, and bounded from above by some µ 2 because of its continuity. Moreover, we assumed that the continuity modulus of c(t) satisfies (3.7). As a consequence, we are in the assumptions of Theorem B and Theorem D, from which we deduce further regularity of u ′ (t) in (0, T ]. This is enough to justify rigorously the computation of the time-derivative of H(t) leading to (6.2), which therefore holds true for every t ∈ (0, T ]. Since H(t) is continuous, equality (6.3) holds true for every t ∈ [0, T ].

Proof of main results
We are now ready to prove Theorem 2.1 and Theorem 2.2. Actually, we prove two stronger results, which we state below.
The first one is a stronger version of Theorem 2.1 in interpolation spaces. Let us assume in addition that (u 0 , u 1 ) ∈ V ω d for some continuity modulus ω d (x). Then the problem admits at least one global solution From Proposition 5.6, applied with α = 0, we know that every pair of initial conditions (u 0 , u 1 ) ∈ D(A 1/2 )×H lies in V ω d for a suitable continuity modulus ω d (x) depending on (u 0 , u 1 ). As a consequence, Theorem 7.1 above implies Theorem 2.1.
Then the following conclusions hold true.
From Proposition 5.6, applied with the same value of α for which (2.5) holds true, we know that every pair of initial conditions (u 0 , u 1 ) ∈ D(A α+1/2 )×D(A α ) lies in V ω d for a suitable continuity modulus ω d (x) satisfying (5.9) through (5.12). As a consequence, the local existence statement of Theorem 7.2 above implies the local existence statement of Theorem 2.2. Up to now, we did not exploit (5.12), which only comes into play in the global existence statement.
If in addition the pair (u 0 , u 1 ) is small in the sense of (2.6), then from (5.12) it follows that (u 0 , u 1 ) is small also in the sense of (7.6) with ε 1 = 2ε 0 . Since ε 1 does not depend on ω d (x), the global existence statement of Theorem 7.2 above implies the global existence statement of Theorem 2.2.
In order to prove the claim, we apply Proposition 5.5 in every closed interval [0, T ] ⊆ [0, T * ). We deduce that the function t → |A 1/2 u(t)| 2 is continuous in [0, T ] with continuity modulus equal to 3L 0 ω d (x). Since the continuity modulus does not depend on T , the function is continuous with the same continuity modulus in [0, T * ), and in particular it is bounded in the same interval. As a consequence, the function t → c(t) := m(|A 1/2 u(t)| 2 ) is uniformly continuous in [0, T * ), and hence it can be extended to a continuous function defined in the closed interval [0, T * ].
The function c(t), in the closed interval [0, T * ], is nonnegative and bounded from above. Therefore, from Theorem A and Proposition 5.2 it follows that the linear problem (1.10)-(1.2) has a unique solution in the closed interval [0, T * ] with the regularity (7.15) and (7.16), which is the required extension of u.
Passing to the limit as t → T − * we contradict (7.13).
Alternative We argue as in the corresponding paragraph of the proof of Theorem 7.1. Assuming by contradiction that T * < +∞ and (7.14) holds true, the key point is extending c(t) to the closed interval [0, T * ] in such a way that the assumptions of Theorem B and Proposition 5.3 are satisfied, so that also the solution u(t) can be continued to [0, T * ] with the required regularity.
In analogy with the proof of Theorem 7.1, from Proposition 5.5 we deduce that the function t → |A 1/2 u(t)| 2 is uniformly continuous in [0, T * ), with continuity modulus equal to 3L 0 ω d (x). As a consequence, the function t → c(t) is uniformly continuous in [0, T * ), with continuity modulus ω m (3L 0 ω d (x)), and hence it can be extended to the closed interval [0, T * ] with the same continuity modulus. On the other hand, this continuity modulus satisfies (3.7) with Λ ∞ = 0 (same proof as in the case of ω c (x), just with a different constant). Therefore, the assumptions of Theorem B are satisfied.