Bounded and Almost Periodic Solvability of Nonautonomous Quasilinear Hyperbolic Systems

The paper concerns boundary value problems for general nonautonomous first order quasilinear hyperbolic systems in a strip. We construct small global classical solutions, assuming that the right hand sides are small. In the case that all data of the quasilinear problem are almost periodic, we prove that the bounded solution is also almost periodic. For the nonhomogeneous version of a linearized problem, we provide stable dissipativity conditions ensuring a unique bounded continuous solution for any smooth right-hand sides. In the autonomous case, this solution is two times continuously differentiable. In the nonautonomous case, the continuous solution is differentiable under additional dissipativity conditions, which are essential. A crucial ingredient of our approach is a perturbation theorem for general linear hyperbolic systems. One of the technical complications we overcome is the"loss of smoothness"property of hyperbolic PDEs.


Problem setting
We consider a first order quasilinear hyperbolic system where V = (V 1 , . . . , V n ) and f = (f 1 , . . . , f n ) are vectors of real-valued functions, and A = (A jk ) and B = (B jk ) are n × n-matrices of real-valued functions. The matrix A is supposed to have n real eigenvalues A j (x, t, V ) in a neighborhood of V = 0 in R n such that for some integer 0 ≤ m ≤ n. These assumptions imply that there exists a smooth and nondegenerate n × n-matrix Q(x, t, V ) = (Q jk (x, t, V )) such that We supplement the system (1.1) with the boundary conditions U j (0, t) = (RZ) j (t) + h j (t), 1 ≤ j ≤ m, t ∈ R, U j (1, t) = (RZ) j (t) + h j (t), m < j ≤ n, t ∈ R, (1.2) where R is a (time-dependent) bounded linear operator, (1, t), . . . , U m (1, t), U m+1 (0, t), . . . , U n (0, t)) , and U(x, t) = Q −1 (x, t, V )V (x, t). (1. 3) The purpose of the paper is to establish conditions on the coefficients A, B, f , and h and the boundary operator R ensuring that the problem (1.1)-(1.3) has a unique small global classical solution, which is two times continuously differentiable. If the data in (1.1) and (1.2) are almost periodic (respectively, periodic) in t, we prove that the bounded solution is almost periodic (respectively, periodic) in t also. Let and BC(Π; R n ) be the Banach space of all continuous and bounded maps u : Π → R n with the usual sup-norm u BC = sup {|u j (x, t)| : (x, t) ∈ Π, j ≤ n} .
Moreover, BC k (Π; R n ) denotes the space of k-times continuously differentiable and bounded maps u : Π → R n , with norm We also use the spaces BC k t (Π; R n ) of functions u ∈ BC(Π; R n ) such that ∂ t u, . . . , ∂ k t u ∈ BC(Π; R n ), with norm Similarly, BC k (R; R n ) denotes the space of k-times continuously differentiable and bounded maps u : R → R n . If n = 1, we will simply write BC k (R) for BC k (R; R), and likewise for all the spaces introduced above. Given two Banach spaces X and Y , the space of all bounded linear operators A : X → Y is denoted by L(X, Y ), with the operator norm A L(X,Y ) = sup{ Au Y : u ∈ X, u X ≤ 1}. We will use also the usual notation L(X) for L(X, X).
Let · denote the norm in R n defined by y = max j≤n |y j |. We suppose that the data of the problem (1.1)-(1.3) satisfy the following conditions.
(A1) There exists δ 0 > 0 such that the entries of the matrices A(x, t, V ), B(x, t, V ), and Q(x, t, V ) have bounded and continuous partial derivatives up to the second order in (x, t) ∈ Π and in V ∈ R n with V ≤ δ 0 , there exists Λ 0 > 0 such that (A2) f ∈ BC 2 t (Π; R n ), ∂ x f ∈ BC 1 t (Π; R n ), and h ∈ BC 2 (R; R n ).
(A3) R is a bounded linear operator on BC(R; R n ). The restriction of R to BC 1 (R; R n ) (respectively, to BC 2 (R; R n )) is a bounded linear operator on BC 1 (R; R n ) (respectively, on BC 2 (R; R n )). Moreover, for v ∈ BC 1 (R; R n ) it holds where v ′ (t) = d dt v(t) and R ′ , R, R ′ , R : BC(R; R n ) → BC(R; R n ) are some bounded linear operators.
We consider two sets of stable conditions on the data of the original problem.
(B1) For each j ≤ n, it holds (B2) For each j ≤ n, it holds inf x,t b jj > 0, e −γ j R j < 1, Moreover, in the particular case of periodic boundary conditions (Rz) j = z j or, the same, in the case we consider yet another set of conditions.
Note that, if inf x,t b jj > 0, then the conditions (B1) and (B2) differ at least in the restrictions imposed on the boundary operator R. More precisely, since the constants γ j are positive for all j ≤ n, the condition (B2) allows for R j ≥ 1, what is not allowed by (B1).

Main result
n is a Bohr almost periodic in t uniformly in x and v (see [6, p. 55]) if for every µ > 0 there exists a relatively dense set of µ-almost periods of w, i.e., for every µ > 0 there exists a positive number l such that every interval of length l on R contains a number τ such that Let AP (R, R n ) be the space of Bohr almost periodic vector-functions. Analogously, let AP (Π, R n ) be the space of Bohr almost periodic vector-functions in t uniformly in x. By C T (R, R n ) and C T (Π, R n ) we denote the spaces of continuous and T -periodic in t vectorfunctions, defined on R and Π, respectively. The main result of the paper is stated in the following two theorems.
are satisfied for both i = 1 and i = 2, then the following is true: Suppose that A, B, Q, f , and h are Bohr almost periodic in t uniformly in x ∈ [0, 1] and V ∈ [−δ 0 , δ 0 ] n (resp., A, B, Q, f , and h are T -periodic in t). Moreover, suppose that the restriction of the boundary operator R to AP (R; R n ) (resp., to C T (R, R n )) is a bounded linear operator on AP (R; R n ) (resp., on C T (R, R n )). Then the bounded classical solution V * to the problem (1.1)-(1.3) belongs to AP (Π, R n ) (resp., to C T (Π, R n )). The paper is organized as follows. In Section 2 we formulate statements of independent interest for general linear first order nonautonomous boundary value problems related to solving the original quasilinear problem. In Section 3 we comment on the problem (1.1)-(1.3) and on our main assumptions. In particular, we give an example showing that in the nonautonomous setting the conditions (1.9) and (1.10) are essential for C 2 -regularity of the bounded continuous solutions. Section 4.1 is devoted to bounded continuous solvability of the linear boundary value problems (including the linearized version of the original problem). In Section 4.2 we prove C 2 -regularity of the bounded continuous solutions. A crucial point in our approach is a perturbation theorem for the general linear problem (2.1), (2.5), (2.7). This result, Theorem 2.4, is proved in Section 4.3. Our main result, Theorems 1.1 and 1.2, is proved in Section 5.

Relevant linear problems
Setting Our approach to the quasilinear problem (1.1)-(1.3) is based on a thorough analysis of a linearized problem. As we will see later, the main reason behind global classical solvability of the quasilinear problem (1.1)-(1.3) lies in the fact that the corresponding nonhomogeneous linear problem has a unique smooth bounded solution for any smooth right-hand side. We therefore first establish stable sufficient conditions ensuring the last property. To this end, consider the following general nonhomogeneous linear system where g = (g 1 , . . . , g n ) is a vector of real-valued functions, a * = (a * jk ) and b * = (b * jk ) are n×nmatrices of real-valued functions. Note that, if a * (x, t) = A(x, t, 0) and b * (x, t) = B(x, t, 0), then (2.1) is a nonhomogeneous version of the linearized system (1.1). This is a reason why we use the same notation for the general linear problem and for the linearized version of the original quasilinear problem.
Suppose that a * jk ∈ BC 1 (Π) and b * jk ∈ BC(Π) for all j, k ≤ n (2.2) and the matrix a * has n real eigenvalues a 1 (x, t), . . . , a n (x, t) such that a 1 (x, t) > . . . > a m (x, t) > 0 > a m+1 (x, t) > . . . > a n (x, t). Let q(x, t) = (q jk (x, t)) be a nondegenerate n × n-matrix such that q jk ∈ BC 1 (Π) and 3) The existence of such a matrix follows from the assumptions on a * . Note that, if (2.1) is a linearized version of (1.1), then the matrix q is defined by q(x, t) = Q(x, t, 0). Let λ 0 be a positive real such that We subject the system (2.1) to the boundary conditions where The system (2.1) with respect to u reads . It is evident that the problems (2.1), (2.5), (2.7) and (2.8), (2.5) are equivalent.

An operator representation Let
Suppose that g and h are sufficiently smooth and bounded functions. As usual, a function u ∈ BC 1 (Π; R n ) is called a bounded classical solution to (  Let us introduce operators C, D ∈ L(BC(Π; R n )) and an operator F ∈ L (BC(Π; R 2n ); BC(Π; R n )) by (2.11) Then the system (2.10) can be written in the operator form (2.12) BC-solutions Theorems 2.1 and 2.2 below give stable sufficient conditions for BC-solvability of the linear problem (2.1), (2.5), (2.7). If the data of the problem are sufficiently smooth, in the autonomous case these conditions even ensure BC 2 -regularity. In the nonautonomous case, we need an additional condition to ensure BC 1 -regularity and yet another condition to ensure BC 2 -regularity. These additional conditions, which are stated in Theorem 2.3, turn out to be essential; see Subsection 3.6. This seems to be a new interesting phenomenon for nonautonomous hyperbolic PDEs.
Moreover, the apriori estimate is fulfilled for a constant K > 0 not depending on g and h. Higher regularity of bounded continuous solutions is the subject of the next theorem.
, and h j ∈ BC 1 (R) for all j, k ≤ n. Suppose that the restriction of R to BC 1 (R; R n ) is a bounded linear operator on BC 1 (R; R n ) satisfying (1.4). If the inequality (1.9) for i = 1 (resp., the inequality (1.10) for i = 1) is true, then the bounded continuous solution v to the problem (2.1), (2.5), (2.7) (resp., to the problem (2.1), (1.8), (2.7)) belongs to BC 1 (Π, R n ). Moreover, the apriori estimate is fulfilled for a constant K 1 > 0 not depending on g and h.
2. Let, additionally, b * jk , g j ∈ BC 2 t (Π) and h j ∈ BC 2 (R) for all j, k ≤ n and the restriction of R to BC 2 (R; R n ) be a bounded linear operator on BC 2 (R; R n ). If the inequality (1.9) for i = 2 (resp., the inequality (1.10) for i = 2) is true, then v ∈ BC 2 t (Π, R n ) and ∂ x v ∈ BC 1 t (Π, R n ). Moreover, the apriori estimate is fulfilled for a constant K 2 > 0 not depending on g and h.
2. If the assumptions of Part 2 of Theorem 2.3 are fulfilled, then there exists ε 1 ≤ ε 0 such that, for allã * jk andb * jk satisfying the conditions (2.18) and the stronger conditions Moreover,ṽ satisfies the apriori estimate (2.15) withṽ in place of v for a constant K 2 not depending onã * ,b * ,q, g, and h.
3 Comments on the problem and the assumptions 3.1 About the quasilinear system (1.1) It is well known that quasilinear hyperbolic PDEs are accompanied by various singularities as shocks and blow-ups. Since the characteristic curves are controlled by unknown functions, the characteristics of the same family intersect each other in general and, therefore, bring different values of the corresponding unknown functions into the intersection points (appearance of shocks). The nonlinearities in B(x, t, u) often lead to infinite increase of solutions in a finite time (appearance of blow-ups). When speaking about global classical solutions, one needs to provide conditions preventing the singular behavior.
Certain classes of nonlinearities ensuring a non-singular behavior for autonomous quasilinear systems are described in [13,22]. Some monotonicity and sign preserving conditions on the coefficients of the nonautonomous quasilinear hyperbolic systems are imposed in [1,25]. In the present paper, we study nonautonomous quasilinear hyperbolic systems with lower order terms and use a different approach focusing on small solutions only. We do not need any of the above constraints. Instead, we assume a regular behavior of the linearized system and smallness of the right hand sides. Small periodic classical solutions for autonomous quasilinear hyperbolic systems without lower order terms and with small nondiagonal elements of the matrix A = A(V ) for V ≈ 0 were investigated in [27]. In our setting, the nondiagonal entries of the matrix A = A(x, t, V ) are not necessarily small and the lower order coefficients B(x, t, V ) are not necessarily zero. Our dissipativity conditions depend both on the boundary operator and on the coefficients of the hyperbolic system.
In Section 3.6 we show that the additional dissipativity conditions (1.9) and (1.10) are in general necessary for C 2 -regularity of continuous solutions, which is a notable fact in the context of nonautonomous hyperbolic problems.

About the boundary conditions (1.2)
The boundary operator R covers different kinds of reflections and delays, in particular, where r jk , p jk , θ jk , and ϑ jk are known BC 1 -functions. The boundary operators R ′ and R introduced in (1.4) are in this case computed by the formulas Boundary conditions of the reflection type appear, in particular, in semiconductor laser modeling [21,29] and in boundary feedback control problems [2,7,10,26], while integral boundary conditions (with delays [23]) appear, for instance, in hyperbolic age-structured models [5,14].

3.3
Nilpotency of the operator C Theorem 2.1 can be extended if the operator C is nilpotent. This is the case of the so-called smoothing boundary conditions, see e.g. [16]. The smoothing property allowed us in [20] to solve the problem (1.1)-(1.3) where the boundary conditions (1.2) are specified to be of the reflection type, without the requirement of the smallness of D L(BC(Π,R n )) . In [20] we used the assumption that the evolution family generated by a linearized problem has exponential dichotomy on R and proved that the dichotomy survives under small perturbations in the coefficients of the hyperbolic system. For more general boundary conditions (in particular, for (1.2)) when the operator C is not nilpotent, the issue of the robustness of exponential dichotomy for hyperbolic PDEs remains a challenging open problem.

Space-periodic problems and exponential dichotomy
In the case of space-periodic boundary conditions (1.8), our assumption (B3) implies, according to [15], that the evolution family generated by the linearized problem has the exponential dichotomy on R. For more general boundary conditions (2.5) one can expect the same dichotomy behavior of the evolution family whenever one of the two assumptions (B1) and (B2) is fulfilled, but this still remains a subject of future work.

Time-periodic problems and small divisors
Analysis of time-periodic solutions to hyperbolic PDEs usually encounters a complication known as the problem of small divisors. However, this obstacle does not appear in our setting due to the non-resonance assumptions (B1), (B2), and (B3). Similar conditions were discussed in [17,18]. The completely resonant case (closely related to small divisors) is qualitatively different. This case is discussed in a series of papers by Temple and Young (see, e.g., [30,31]) about time-periodic solutions to one-dimensional linear Euler equations with the periodic boundary conditions (1.8). In this case one cannot expect any stable non-resonant conditions of our type. More precisely, in the setting of [30,31] it holds b jj = 0 for all j and, hence, our condition (B3) is not satisfied. Therefore, the operator I − C is not bijective, while the bijectivity property is a crucial point in our Theorems 2.1 and 2.2.
3.6 Conditions (1.9) and (1.10) are essential for higher regularity of bounded continuous solutions, in general In the autonomous case, when the operator R and the coefficients in the hyperbolic system (2.8) do not depend on t, we have R ′ = 0, R = R, and c l j ≡ c j for all j ≤ n and l = 1, 2. Then the bounds (1.9) and (1.10) straightforwardly follow from the assumptions of any of Theorems 2.1 and 2.2. The higher regularity of solutions follows automatically. This means that we have to explicitly impose the conditions (1.9) and (1.10) only in the nonautonomous case.
We now show that in the nonautonomous case, if the estimate (1.9) is not fulfilled for i = 1, then Part 1 of Theorem 2.3 is not true in general. Similarly, if (1.9) is not fulfilled for i = 2, then one can show that Part 2 of Theorem 2.3 is not necessarily true.
Consider the following example, satisfying all the assumptions in Part 1 of Theorem 2.3 except (1.9) for i = 1: where r 1 is 2π-periodic and positive C 1 -function r 1 (t) and a constant r 2 are such that In this case, all assumptions of Theorem 2.1 are true since R 1 = sup t∈R r 1 (t) < 1, R 2 = r 2 < 1, and b jk = 0 for all j, k ≤ 2. The system (3.1) has a unique bounded continuous solution u = (u 1 , u 2 ) ∈ BC(Π, R 2 ). Since all the coefficients of the problem are 2π-periodic in t, it is a simple matter to show that the solution u is 2π-periodic in t as well (sf. Section 5.3).

Quasilinear hyperbolic systems in applications
Quasilinear systems of the type (1.1) cover, in particular, the one-dimensional version of the classical Saint-Venant system for shallow water [28] and its generalizations (see, e.g. [3]), the water flow in open-channels [12], and one-dimensional Euler equations [11,30,31,33]. They are also used to describe rate-type materials in viscoelasticity [8,9,24] and the interactions between heterogeneous cancer cell [4].
The behavior of unsteady flows in open horizontal and frictionless channel is described in [32] by the Saint-Venant system of the type where L is the length of the channel, A = A(t, x) is the area of the cross section occupied by the water at position x and time t, V = V (t, x) is the average velocity over the cross section. Furthermore, is the depth of the water. This system is subjected to flux boundary conditions. Note that in the smooth setting the system (3.10) is of our type (2.1). As described in [32], in a neighborhood of an equilibrium point the system (3.10) can be written in Riemann invariants in the diagonal form (2.8). The flux boundary conditions are then transformed into boundary conditions of the type (2.5).
The nonautonomous first order quasilinear system is used to model the stress-strain law for metals [8,9,24].
Here v and u denote the stress and the Lagrangian velocity, while the functions φ and ψ measure, respectively, the noninstantaneous and the instantaneous response of the metal to an increment of the stress. We first give the proof under the assumption (B1). We have to prove that I − C − D is a bijective operator from BC(Π; R n ) to itself. It suffices to establish the estimate Using (2.9), we have If inf x,t b jj ≥ 0, then γ j ≥ 0 and, if inf By the definition (2.11) of the operator D, for all u ∈ BC(Π; R n ) with u BC = 1 and all (x, t) ∈ Π we have Note that γ j = 0 iff inf x,t b jj = 0. Using (B1), we immediately get the inequality (4.1). This implies that, for given g ∈ BC(Π; R n ) and h ∈ BC(R; R n ), the equation ( Now we assume that the assumption (B2) is fulfilled. Our aim is to prove that the operator I − C ∈ L (BC(Π; R n )) is bijective and that the following estimate is fulfilled: To prove the bijectivity of I − C ∈ L (BC(Π; R n )), we consider the equation with respect to u ∈ BC(Π; R n ), where r belongs to BC(Π; R n ) and z is given by (2.6). The operator I − C ∈ L (BC(Π; R n )) is bijective iff the equation (4.4) is uniquely solvable for any r ∈ BC(Π; R n ). Putting x = 0 for m < j ≤ n and x = 1 for 1 ≤ j ≤ m, the system (4.4) reads as follows with respect to z(t): Introduce operator G 0 ∈ L(BC(R, R n )) by This implies that for all u ∈ BC(Π; R n ) with u BC = 1 it holds Hence, the operator I − G 0 is bijective due to the assumption (B2). It should be noted that C L(BC(Π;R n )) = 1, while G 0 L(BC(R;R n )) < 1. We, therefore, can rewrite the system (4.5) in the form wherer(t) = (r 1 (1, t), . . . , r m (1, t), r m+1 (0, t), . . . , r n (0, t)). Substituting (4.8) into (4.4), we obtain The assumption that inf x,t b jj > 0 entails that c j (x j , x, t) ≤ 1 for all (x, t) ∈ Π and all j ≤ n. Therefore,

Proof of Theorem 2.2
We follow the proof of Theorem 2.1 under the assumption (B2), with the following changes.
Since in the periodic case one can integrate in both forward and backward time directions, we use an appropriate integral analog of the problem (2.8), (1.8), namely Note that in the case of general boundary conditions (2.5) we could integrate only in the backward time direction where the boundary conditions are given. Now, instead of the system (4.5), we have the following decoupled system: The analog of the operator G 0 introduced in (4.6), which we denote by H 0 , reads One can easily see that C L(BC(Π;R n )) = 1, while H 0 L(BC(R;R n )) < 1. It follows that the operator I − H 0 and, hence, the operator I − C is bijective, as desired. The rest of the proof goes similarly to the second part of the proof of Theorem 2.1.

Higher regularity of the bounded continuous solutions: Proof of Theorem 2.3
We divide the proof into a number of claims. Part 1 of the theorem follows from Claims 1-4, while Part 2 follows from Claims 5-6. We give a proof under the assumptions of Theorem 2.1. The proof under the assumptions of Theorem 2.2 follows the same line, and we will point out only the differences.
We begin with Part 1. Let u ∈ BC(Π, R n ) be the bounded continuous solution to the problem (2.8), (2.5).
Claim 1. The generalized directional derivatives (∂ t + a j ∂ x )u j are continuous functions.
where the functions are uniformly bounded and have continuous and uniformly bounded first order derivatives in t. An upper bound as in (4.13) for the first sum in (4.14) follows directly from the regularity and the boundedness assumptions on the coefficients of the original problem. The strict hyperbolicity assumption (2.4) admits the following representation formula: , ω j (ξ))) ∂ t ω j (ξ)a j (ξ, ω j (ξ))a k (ξ, ω j (ξ)) a k (ξ, ω j (ξ)) − a j (ξ, ω j (ξ)) .
Combining (4.14) with (4.15), we conclude that ∂ t (DC) is bounded as stated in (4.13). Similarly, A desired estimate for the first summand is obvious, while for the second summand follows from the following transformations. For definiteness, assume that j, k ≤ m (the cases j > m or k > m are similar). Taking into account the identity The right hand side of (4.16) can be written as which implies an upper bound as in (4.13).
Claim 3. I − C is a bijective operator from BC 1 t (Π, R n ) to itself.

Proof of Claim.
We are done if we show that the system (4.4) is uniquely solvable in BC 1 t (Π, R n ) for any r ∈ BC 1 t (Π, R n ). Obviously, this is true if and only if where the operator G 0 ∈ L(BC(R, R n )) is given by (4.6). To prove (4.17), let us norm the space BC 1 (R, R n ) with where a positive constant σ will be defined later on. Note that the norms (4.18) are equivalent for all σ > 0. We therefore have to prove that there exist constants σ < 1 and γ < 1 such that Combining (2.9) with the formula Define an operator W ∈ L(BC(R, R n )) by (4.20) On the account of (4.6) and (4.7), both assumptions (B1) and (B2) implies that G 0 L(BC(R,R n )) < 1. Moreover, the assumption (1.9) for i = 1 of Theorem 2.3 yields G 1 L(BC(R,R n )) < 1. Fix σ < 1 such that G 0 L(BC(R,R n )) + σ W L(BC(R,R n )) < 1. Set γ = max G 0 L(BC(R,R n )) + σ W L(BC(R,R n )) , G 1 L(BC(R,R n )) .
The proof under the assumptions of Theorem 2.2 follows the same line with this changes: We specify (Rz) j ≡ z j for all j ≤ n and replace the operator G 0 by the operator H 0 (see the formula (4.10)). Hence, (R ′ z) j ≡ 0 and ( Rz) j ≡ z j for all j ≤ n and all z ∈ BC 1 (R, R n ).
where u is the bounded classical solution to (2.8), (2.5) and K 12 is a positive constant not depending on g and h. Now, from (2.1) we get t + h BC 1 for some K 13 > 0 not depending on g and h. The estimate (2.14) follows.
The proof of Part 1 of the theorem is complete. Now we prove Part 2. Formal differentiation of the system (2.1) in the distributional sense with respect to t and the boundary conditions (2.5) pointwise, we get, respectively, and Introduce a new variable w = q −1 ∂ t v = ∂ t u + q −1 ∂ t qu and rewrite the problem (4.23)-(4.24) with respect to w as follows: (1, t), . . . , w m (1, t), w m+1 (0, t), . . . , w n (0, t)) , Claim 5. The function w ∈ BC(Π, R n ) satisfies both (4.25) in the distributional sense and (4.26) pointwise if and only if w satisfies the following system pointwise for all j ≤ n: , v(ξ, ω j (ξ))) dξ.
To prove the sufficiency, take an arbitrary sequence w l ∈ BC 1 (Π; R n ) approaching w in BC(Π; R n ). Take an arbitrary smooth function ϕ : (0, 1) × R → R with compact support. On the account of (4.27), we have where y l = w l 1 (1, t), . . . , w l m (1, t), w l m+1 (0, t), . . . , w l n (0, t) . It remains to note that for all what easily follows from the diagonality of the matrix a and the identity Moreover, putting x = x j in (4.27), we immediately get (4.26). The proof of the sufficiency is complete.
To prove the necessity, assume that the function w satisfies (4.25) in the distributional sense and (4.26) pointwise. On the account of (4.11), we rewrite the system (4.25) in the form without destroying the equalities in the sense of distributions. To prove that w satisfies (4.27) pointwise, we use the constancy theorem of distribution theory claiming that any distribution on an open set with zero generalized derivatives is a constant on any connected component of the set. By (4.29), this theorem implies that for each j ≤ n the expression is constant along the characteristic curve ω j (ξ, x, t). In other words, the distributional directional derivative (∂ t + a j (x, t)∂ x ) of the function (4.30) is equal to zero. Since (4.30) is a continuous function, c 1 j (x j , x j , t) = 1, and the trace w j (x j , t) is given by (4.26), it follows that w satisfies the system (4.27) pointwise, as desired. Proof of Claim. We rewrite the system (4.27) in the operator form where C 1 , D 1 ∈ L(BC(Π; R n )) and F 1 ∈ L(BC(Π; R 2n ), BC(Π; R n )) are operators defined by , v(ξ, ω j (ξ))) dξ (4.32) Iterating (4.31), we obtain Using the same argument as in Claim 2, we conclude that the operators D 1 C 1 and D 2 1 map continuously BC(Π, R n ) into BC 1 t (Π, R n ). Moreover, the following smoothing estimate is true: ( for some K 21 > 0 not depending on w ∈ BC(Π, R n ). Next, we prove that I − C 1 is a bijective operator from BC 1 t (Π, R n ) to itself. The proof is similar to the proof of Claim 3. We have to show that the system where the operator G 1 ∈ L(BC(R, R n )) is given by (1.7). To prove (4.35), we use the space BC 1 (R, R n ) normed by (4.18). We are done if we prove that there exist constants σ 1 < 1 and γ 1 < 1 such that for all ψ ∈ BC 1 (R, R n ). As it follows from (1.6) and (4.19), c 2 j (ξ, x, t) = c 1 j (ξ, x, t)∂ t ω j (ξ, x, t). Define operator W 1 ∈ L(BC(R, R n )) by (4.36) Taking into account (1.4) and (1.7), for given ψ ∈ BC 1 (R, R n ), it holds By the assumption (1.9), G 1 L(BC(R,R n )) < 1 and G 2 L(BC(R,R n )) < 1. Fix σ 1 < 1 such that G 1 L(BC(R,R n )) + σ 1 W 1 L(BC(R,R n )) < 1. Set It follows that Similarly to (4.22), the inverse to I − C 1 can be estimated from above as follows: Combining this estimate with (2.14), (4.33), and (4.34), we get , the constants K 22 and K 23 being independent of g and h. By (4.23), there exists a constant K 24 not depending on g and h such that , which implies the estimate (2.15), as desired.
Assume that the condition (B1) is fulfilled. Similar argument works in the case of (B2) or (B3).
Proof of Part 1. Note that the assumptions (B1) and (1.9) of Theorem 2.3 are stable with respect to small perturbations of a and b. Since small perturbations of a * , b * , and q imply small perturbations of a and b, there exists ε 11 ≤ ε 0 such that, for allã * andb * varying in the range the conditions (B1) and (1.9) for i = 1 remain to be true withã andb in place of a and b, respectively. Due to Part 1 of Theorem 2.3, the system (4.38), (2.5), (4.37) has a unique bounded classical solutionũ ∈ BC 1 (Π; R n ) for each fixedã * ,b * , andq.
To derive the apriori estimate (2.14) withṽ in place of v, note that the value of ε 11 > 0 can be chosen so small that there exists a positive real ν 1 < 1 such that the left hand sides of (B1) and (1.9) for i = 1, calculated for the perturbed problem (4.38), (4.37), (2.5), are bounded from above by 1 − ν 1 . Due to the proof of Theorem 2.1, this implies the inequality C L(BC(Π;R n )) + D L(BC(Π;R n )) ≤ 1 − ν 1 , which is uniform inã * ,b * , andq. Combining this inequality with (4.39), we conclude that there exists a constant K > 0 not depending onã * ,b * ,q, g, and h such that (4.42) We immediately see from (B1), (1.9) for i = 1, (4.7), and (4.20) that there exist constants K 1 > 0 and ν 2 < 1 such that uniformly inã * ,b * , andq fulfilling (4.41) with ε 12 in place of ε 11 . Put γ = 1 − ν 2 + σ K 1 and fix σ < 1 such that γ < 1. Now we apply the argument used to prove the estimate (4.22) and get Similarly to the proof of Claim 2 in Section 4.2, we show that the operators D C and D 2 are smoothing and map BC(Π, R n ) into BC 1 t (Π, R n ). Moreover, there exists a constant K 2 such that, for allã * ,b * , andq fulfilling the inequalities (4.41) with ε 12 in place of ε 11 , it holds for all u ∈ BC(Π, R n ). Now we combine the estimates (4.42)-(4.44) with the equations (4.40). We conclude that there exist constants ε 1 ≤ ε 12 and K 1 > 0 such that, for allã * , b * ,q, g, and h varying in the range (4.41) with ε 1 in place of ε 11 , the estimate (2.14) is true withṽ in place of v. Proof of Part 2. Let ε 1 be a constant satisfying Part 1 of Theorem 2.4. Consider a perturbed version of the equation (4.33) where C 1 , D 1 , and F 1 are replaced by C 1 , D 1 , and F 1 , respectively. Proceeding similarly to Part 1, we use (4.36) and (1.9) for i = 2 and conclude that the constant ε 1 can be chosen so small that there exist positive reals ν 3 < 1 and K 3 fulfilling the bounds uniformly inã * ,b * , andq satisfying the estimates (4.41) with ε 1 in place of ε 11 as well as the The desired apriori estimate (2.15) for the ε 1 -perturbed problem then easily follows from the perturbed versions of (4.33) and (4.34).
The proof of Theorem 2.4 is complete.
Put V 0 (x, t) = 0. For a given nonnegative integer number k, construct the iteration V k+1 (x, t) as the unique bounded classical solution to the linear system subjected to the boundary conditions and The function U k+1 then satisfies the system Here and below in this proof we also use the short notation A k , B k , Q k , a k , and b k for We divide the proof into a number of claims.
where K 2 is the constant as in Part 2 of Theorem 2.4. Then there exists a sequence V k of bounded classical solutions to (5.4)-(5.6) belonging to BC 2 (Π; R n ) such that Proof of Claim. Note that the first iteration V 1 satisfies the system (2.1) with g = f and the boundary conditions (2.5), (2.7). Due to Theorem 2.3, there exists a unique bounded classical solution V 1 such that V 1 ∈ BC 2 t (Π, R n ) and ∂ x V 1 ∈ BC 1 t (Π, R n ). Since A 0 , Q 0 , and B 0 are continuously differentiable in x, from the system (5.4) differentiated in x it follows that V 1 ∈ BC 2 (Π, R n ). Moreover, V 1 satisfies the bound (2.15) with v and g replaced by V 1 and f , respectively. Since f and h obey (5.8), the estimate (5.9) with k = 1 follows. Due to (5.1)-(5.2), we then have Similarly, V 2 (x, t) fulfills the bound (5.9) with k = 2 and, due to (5.4), belongs to BC 2 (Π, R n ). On the account of (5.1)-(5.2), we also have the estimates (5.10) with A 1 , B 1 , and Q 1 replaced by A 2 , B 2 , and Q 2 , respectively.
Proceeding by induction, assume that the problem (5.4)-(5.6) has a unique bounded classical solution V k belonging to BC 2 (Π, R n ) and satisfying the estimate (5.9) and, hence the estimates Now, using Theorem 2.4 and the system (5.4) differentiated in x, we conclude that the problem (5.4)-(5.6) has a unique bounded classical solution V k+1 ∈ BC 2 (Π, R n ). Moreover, this solution fulfills the inequalities (5.9) and (5.11) with k + 1 in place of k.

Proof of Claim. Set
where First we derive a boundary value problem for w k+1 . To this end, introduce the following notation:χ k (t) = χ k 1 (1, t), . . . , χ k m (1, t), χ k m+1 (0, t), . . . , χ k n (0, t) , On the account of (5.13) and (5.14), the boundary conditions (5.5) with respect to Y k+1 can be written as follows: or, in the above notation, as Therefore, the function w k+1 is the classical BC 2 -solution to the system with the boundary conditions (5.13), (5.15), where Now we show that the sequence w k+1 converges to zero in BC 1 By the inequality (5.9) and the assumptions of Theorem 2.4, the inverse (A k ) −1 exists for every k and, moreover, is bounded in BC(Π; R n ) uniformly in k. Now, the equation (5.16) yields which together with (5.19) gives the convergence w k+1 BC 1 → 0 as k → ∞. Finally, because of (5.12), the sequence V k converges to some function V * in BC 1 (Π; R n ). It is a simple matter to show that V * is a classical solution to the problem (1.1)-(1.3). The proof of the claim is complete.
Claim 3. There exist positive constants ε ≤ ε 2 and δ ≤ δ 1 such that, if f BC 2 t + h BC 2 ≤ ε, then the classical solution V * belongs to BC 2 (Π; R n ) and satisfies the estimate Proof of Claim. we start with proving that the sequence V k converges in BC 2 (Π; R n ). First show that the sequence To this end, we differentiate the problem (5.4)-(5.6) with respect to t and, similarly to (4.25) and (4.26), write down the resulting problem in the diagonal form with respect to W k+1 , as follows: (1, t), . . . , W k+1 m (1, t), W k+1 m+1 (0, t), . . . , W k+1 n (0, t) , ρ k (t) = ̺ k 1 (1, t), . . . , ̺ k m (1, t), ̺ k m+1 (0, t), . . . , ̺ k n (0, t) , Now we intend to show that there exists δ 2 ≤ δ 1 such that, given a nonnegative integer k and V k satisfying the estimate (5.9) with δ 2 in place of δ 1 , the formula (5.28) is equivalent to the following one: where A(k) ∈ L(BC(Π; R n )) and X k ∈ BC(Π; R n ) are given by It suffices to show that, for every nonnegative integer k, the operator I − C(k) − (I + D(k))F (k) (G 1 (k), 0) is invertible and has a bounded inverse. Even more, we will show that the inverse is bounded uniformly in k. With this aim, denote by G 0 (k) operator defined by the right hand sides of (4.6), where a j , b jj , and ω j are replaced, respectively, by a k j , b 2k jj , and ω k j . Moreover, denote by G 2 (k) operator defined by the right hand sides of the second formula in (1.7), where a j , b jj , and ω j are replaced, respectively, by a k j , b k jj , and ω k j . Note that, similarly to (4.28), we have b 1k jj = b k jj − (a k j ) −1 ∂ t a k j . Then, accordingly to the notation introduced above, the function b 2k jj is given by the formula b 2k jj = b k jj − 2(a k j ) −1 ∂ t a k j . This means that the operators G 0 (k) and G 2 (k) coincide.
Finally, similarly to the proof of the invertibility of I − C in Subsection 4.1.1, the invertibility of I − C(k) follows from the invertibility of I − G 0 (k) (see the inequality (4.9)). Furthermore, the following estimate is true for all k ∈ N: (I − C(k)) −1 L(BC(Π;R n )) ≤ 1 + ν −1 3 C(k) L(BC(Π;R n )) .
As the operators C(k) are bounded uniformly in k, the inverse operators (I − C(k)) −1 are bounded uniformly in k also. Taking into account that the set of all invertible operators whose inverses are bounded is open, our task is, therefore, reduced to show that the operator (I + D(k))F (k) (G 1 (k), 0) is sufficiently small whenever δ 1 is sufficiently small. Note that Claim 1 is true with δ 2 in place of δ 1 for any δ 2 ≤ δ 1 . This implies that for any σ > 0 there is δ 2 such that for all V k fulfilling (5.9) with δ 2 in place of δ 1 , we have G 1 (k) L(BC(Π;R n )) = g 1 (x, t, W k (x, t)) BC ≤ σ for all k. Moreover, the operators D(k) and F (k) are bounded, uniformly in k. Consequently, if δ 2 is sufficiently small, then for all k ∈ N and all f and h satisfying (5.8) with δ 2 in place of δ 1 , the operator I − C(k) − (I + D(k))F (k) (G 1 (k), 0) is invertible and the inverse is bounded by a constant not depending on k. Fix δ 2 satisfying the last property. The equivalence of (5.28) and (5.29) is, therefore, proved. Now, to prove that the sequence W k+1 the estimate being uniform in ξ, x, t, and j. Due to the mean value theorem, a k j (ξ, ω k j (ξ, x, t + τ )) − a k j (ξ, ω k j (ξ, x, t) + τ ) = (ω k j (ξ, x, t + τ ) − ω k j (ξ, x, t) − τ ) × 1 0 ∂ 2 a k j ξ, αω k j (ξ, x, t + τ ) + (1 − α)(ω k j (ξ, x, t) + τ dα.
Applying the Gronwall's inequality to the identity (5.35), we derive the estimate the constant L 1 being independent of µ, η, x, t, and j.
The estimates (5.37) and (5.38) imply that the functions in the right hand sides of the equalities in (2.9) with a k j , b k jj , and ω k j in place of a j , b jj , and ω j , respectively, are almost periodic for all j ≤ n, uniformly in ξ, x ∈ [0, 1]. Let C(k), D(k), and F (k) be defined by the right hand side of (2.11) with a j , b jj , and ω j replaced by a k j , b k jj , and ω k j , respectively. Taking into account (5.39), we conclude that the operators C(k), D(k), and F (k) map the space AP (Π, R n ) ∩ BC 1 t (Π, R n ) into itself. Let the condition (B1) be fulfilled. Due to the proof of Theorem 2.1, this yields C(k) L(BC(Π;R n )) + D(k) L(BC(Π;R n )) < 1.
Hence, the operator I − C(k) − D(k) is bijective from BC(Π; R n )) into itself. As a consequence, the solution U k+1 ∈ BC(Π; R n ) to the equation U k+1 = ( C(k) + D(k))U k+1 + F (k)(f, h) is given by the (uniformly convergent) Neumann series Since the functions f and h are continuously differentiable in t, the function F (k)(f, h) belongs to BC 1 t (Π, R n ). Moreover, C(k) + D(k) j maps AP (Π, R n ) ∩ BC 1 t (Π, R n ) to AP (Π, R n ) for each j. Therefore, the right hand side and, hence, the left hand side of (5.40) belongs to AP (Π, R n ). This means that that the function V k+1 = Q k U k+1 belongs to AP (Π, R n ), as desired.

Proof of Part 2 of Theorem 1.1: Periodic solutions
We follow the proof of the almost periodic case in Section 5.2, on each step referring to the periodicity instead of the almost periodicity. Obvious simplifications in the proof are caused by the identity ω k j (η, x, t) + T = ω k j (η, x, t + T ).

Proof of Theorem 1.2: Bounded solutions for space-periodic problems
The proof of Theorem 1.2 repeats the proof of Theorem 1.1, with the only difference being that we need to refer to Theorem 2.2 instead of Theorem 2.1.