This study is devoted to the problem of proving some analog of maximum principle and its further application to the questions of uniqueness, existence, and regularity for weak solutions of the Goursat, the Cauchy, and the Dirichlet problems for fourth-order linear hyperbolic equations with the constant coefficients and homogeneous non-degenerate symbol in a plane bounded domain Ω ∈ ℝ² convex with respect to characteristics:

Here, coefficients a_j, j = 0, 1, ..., 4 are constant, f(x) ∈ L²(Ω), ∂x=(∂∂x1,∂∂x2). We consider hyperbolic equations that means all roots of the characteristic equation

are prime and real and are not equal to ±i or the symbol of Equation 1 is non-degenerate (Equation 1 is a equation of principal type). If roots of characteristics equation of which are multiple and can take the values ±i we will call the equation with degenerate symbol (see Buryachenko [7]).

The main novelty of this study is to establish some analog of the maximum principle for fourth-order hyperbolic equations. This question is very important due to natural physical interpretation and helps to establish the qualitative properties for solutions (uniqueness and existence results for weak solutions). It is well known that even for the simple case of hyperbolic equation (one dimensional wave equation [23]), [1] the maximum principle is quite different from those for elliptic and parabolic cases, for which it is a natural fact. Such a way a role of characteristics curves and surfaces becomes evident for hyperbolic equations.

We call the angle of characteristics slope solution to the equation −tanφ_j = λ_j, and the angle between j− and k− characteristics: φ_k − φ_j ≠ πl, l ∈ ℤ, where λ_j ≠ ±i are real and prime roots of the characteristics equation, j, k = 1, 2, 3, 4.

Most of these equations serve as mathematical models of many physical processes and attract interest of researchers. The most famous of them are elasticity beam equations (Timoshenko beam equations with and without internal damping) [9], short laser pulse equation [12], equations which describe the structures are subjected to moving loads, and equation of Euler-Bernoulli beam resting on two-parameter Pasternak foundation and subjected to a moving load or mass [11, 24].

Due to evident practice application, these models require more precise tools for study, and as a result, attract fundamental knowledge. As usual, most of these models are studied by analytical-numerical methods (Galerkin's methods).

The range of problems studied in this study belongs to a class of quite actual problems of well-posedness of so-called general boundary value problems for higher-order differential equations. These problems originated from the studies of L. Hormander and M. Vishik, who used the theory of extensions to prove the existence of well-posed boundary value problems for linear differential equations of arbitrary order with constant complex coefficients in a bounded domain with smooth boundary. This theory got its present-day development in the studies of G. Grubb [13], Hörmander [14], and Posilicano [22] (see also [16]). Later, the problem of well-posedness of boundary value problems for various types of second-order differential equations was studied by Burskii [2], Burskii and Zhedanov [3], who developed a method of traces associated with a differential operator and applied this method for study the Poncelet, the Abel, and the Goursat problems. In the previous studies of Burskii and Buryachenko [6], there have been developed the qualitative methods for studying the Cauchy problem and non-standard for hyperbolic equations the Dirichlet and the Neumann problems. Moreover, for equation of any even order 2m, m ≥ 2, using operator methods (L-traces, theory of extension, moment problem, method of duality equation domain, and others), the existence and uniqueness results were proved, and the criteria of non-trivial solvability of the Dirichlet and the Neumann problems in a disk for the principal type equations and equations with degenerate symbol were obtained [4, 8]. In particular, the interrelations between multiplicity of roots of the characteristic equation were established, and the existence of a non-trivial solution of the corresponding problems was proved. As a consequence, the Fredholm property for the problems under consideration was established.

As the concern maximum principle, at the present time there are not any results for fourth-order equations even in linear case. As it was mentioned above, maximum principle even for the simplest case of one dimensional wave equation [23] and for second-order telegraph equation [18–21] is quite different from those for elliptic and parabolic cases. In the monograph of Protter and Weinberger [23], there was shown that solutions of hyperbolic equations and inequalities do not exhibit the classical formulation of maximum principle. Even in the simplest case of the wave equation u_tt − u_xx = 0, a maximum of a non-constant solution u = sinxsint in a rectangle domain {(x, t):x ∈ [0, π], t ∈ [0, π]} occurs at the interior point (π2,π2). In Chapter 4 [23], maximum principle for linear second hyperbolic equations of general type with variable coefficients has also been obtained for the Cauchy problems and boundary value problems on characteristics (the Goursat problem).

Following Ortega and Robles-Perez [21], we introduce the definition of the maximum principle for hyperbolic equations.

Definition 1. [21] Let L be linear differential operator, acting on functions u : D → ℝ in some domain D. These functions will belong to the certain family B, which includes boundary conditions or other requirements. It is said that L satisfies the maximum principle, if

implies u ≥ 0 in D.

In further studies of these authors (see Mawhin et al. [18–20]), the maximum principle for weak bounded twice periodical solutions from the space L^∞ for the telegraph equation with parameter λ in lower term, one-, two-, and -three dimensional spaces was studied. The precise condition for λ under which the maximum principle still valid was font. There was also introduced a method of upper and lower solutions associated with the non-linear equation, which allows to obtain the analogous results (uniqueness, existence, and regularity theorems) for the telegraph equations with external non-linear forcing.

Maximum principle for second-order quasilinear hyperbolic systems with dissipation was proved by De-Xing [17]. There were given two estimates for solution to the general quasilinear hyperbolic system and introduced the concept of dissipation (strong dissipation and weak dissipation); then, some maximum principles for secound-order quasilinear hyperbolic systems with dissipation were derived. As an application of maximum principle, the existence and uniqueness theorems of the global smooth solution to the Cauchy problem for considered quasilinear hyperbolic system were proved. In recent study by Yi and Ying [10], some analog of Equation 1 with lower order terms and non-linear external force was considered. Qualitative properties of solution of the Dirichlet problem with affine data for differential elasticity inclusion were proved by Ruland et al. [25].

The challenge to prove the maximum principle for weak solutions remains more complicated and at that time becomes more interesting in the case of fourth-order hyperbolic equations, especially, in the case of non-classical boundary value problems with data of weak regularity. Unlike second-order equations, qualitative analysis of solutions to fourth-order equations is not a trivial problem, since not only a solution is involved in boundary or initial conditions but also its high- order derivatives. Other difficulty concerns the concept of weak solution of the boundary value problems with L² − data. Such solutions do not have usual traces; thus, we have to use a special notion for traces to poss correctly the boundary value problems. This notion is traces associated with operator L or L− traces. We derive an example (see Remark 1), which shows that for every L²- solution to the Dirichlet problem for the wave equation, its value u|_∂K on the boundary ∂K does not exist, but its “improved” value −x₁x₂u|_∂K on boundary ∂K exists. It means that multiplying by some polynomial we “improve” a solution. This polynomial depends on the equation. In the case of the wave operator Lu=∂2u∂x1∂x2, this polynomial equals x₁x₂, what is the symbol L(x) = x₁x₂ of the wave operator. Therefore, such “improved” traces are called the traces associated with operator L or simply the L−traces.

At that moment, there are not any results on the maximum principle even for the model case of linear two- dimensional fourth-order hyperbolic equations with constant coefficients and homogeneous symbol (without lower terms), which are under consideration of the present study.

We also derive an interesting interpretation (as periodicity of characteristic billiard or the John's mapping) of the Fredholm's property violation. For second-order hyperbolic equations, the fact that periodicity of the John's algorithm is sufficient for violation of the Fredholm property for the Dirichlet problem was proved by John [15] (for the wave equation) and Burskii and Zhedanov [3] (for general second-order hyperbolic equations with constant complex coefficients). Analogous result is true for fourth-order hyperbolic equations and will be proved in the present study.

Therefore, obtaining such results as the maximum principle, uniqueness, existence and regularity, kernel dimension, the Fredholm property for weak solutions to fourth-order hyperbolic equations and boundary value problems for them is very important for the reason of their further applications and is the main goal of the study.

Let us start to establish the maximum principle for weak solutions to the Cauchy problem for Equation 1 in some admissible planar domain. It is expected that in the hyperbolic case, characteristics of the equations play a crucial role.

Let C_j, j = 1, 2, 3, 4 be characteristics, Γ₀: = {x₁ ∈ [a, b], x₂ = 0} is initial line, and define Ω as a domain which is restricted by the characteristics C_j, j = 1, 2, 3, 4 and Γ₀ by the following way. We choose some arbitrary point C and draw through this point two characteristics, C₁ and C₂, for instance. Another two characteristics (C₃ and C₄) we draw through the ends a and b of initial line Γ₀. We determine a points O₁ and O₂ as intersections of C₁, C₃ and C₂, C₄ correspondingly: O₁ = C₁ ∩ C₃, O₂ = C₂ ∩ C₄. Such a way, domain Ω is a pentagon aO₁CO₂b. Consider also the Cauchy problem for Equation 1 on Γ₀:

where φ, ψ, σ, and χ are given weak regular functions on Γ₀, in general case φ,ψ,σ,χ∈L2(Γ0),ν- is outer normal of Γ₀.

Definition 2. We call a domain D: = {(x₁, x₂): x₁ ∈ (−∞, +∞), x₂ > 0} in the half-plane x₂ > 0 an admissible domain if it has the property that for each point C ∈ D the corresponding characteristic domain Ω is also in D. More generally, D is a admissible if it is the finite or countable union of characteristics 5 angles (in the case of fourth-order equations with constant coefficients, there exist four different and real characteristics lines).

Establishment of the maximum principle allows us to obtain a local properties of solution to the Cauchy problem (Equations 1, 2) on a arbitrary interior point C ∈ D.

We will consider a weak solution to the problem (Equations 1, 2) from the domain of definition D(L) of maximal operator associated with the differential operation L in Equation 1. Following Burskii and Buryachenko [6], Grubb [13], and Hörmander [14], we remind the corresponding definitions.

In a bounded domain Ω, we consider linear differential operation L of m−th order, m ≥ 2, and formally adjoint L+:

where α = (α₁, α₂, ...α_n), |α| = α₁ + α₂ + ... + α_n is multi-index. Note, that for Equation 1 n = 2, m = 4.

Definition 3. Minimum operator. [6]. Let us consider differential operation L (Equation 3) on functions from the space C0∞(Ω). The minimum operator L₀ is called extension of operation L from C0∞(Ω) to the set D(L0): =C0∞(Ω)¯. The closure is realized in the norm of graph of operator L: ||u||L2: =||u||L2(Ω)2+||Lu||L2(Ω)2.

Definition 4. Maximum operator. [6]. The maximum operator L is defined as the restriction of differential operation L(Dx) to the set D(L): = {u ∈ L²(Ω):Lu ∈ L²(Ω)}.

Definition 5. [6]. The operator L~ is defined as the extension of minimum operator L₀, to the set D(L~): =C∞(Ω̄)¯.

Definition 6. Regular operator. [6]. The maximum operator is called regular if D(L)=D(L~).

It is easy to see that D(L~)=H4(Ω), D(L0)=H40(Ω), the Hilbert Sobolev space of fourthly weak differentiable functions from L²(Ω).

Analogously, we introduce operators L+,L~+, and L0+ associated with the formally adjoint operation L+.

Definition of a weak solution to problem (Equations 1, 2) from the space D(L) is closely connected with the notion of L−traces, traces associated with the differential operator L.

Definition 7. L-traces. [5]. Assume, that for a function u∈D(L~), there exist linear continuous functionals L_(p)u over the space H^m−p−1/2(∂Ω), p = 0, 1, 2..., m − 1, such that the following equality is satisfied:

Functionals L_(p)u are called L_(p) − traces of function u∈D(L~). Here, (·,·)L2(Ω) is a scalar product in the Hilbert space L²(Ω).

For L² − solutions, the notion of L_(p) − traces can be realized by the following way.

Definition 8. Distributions L(p)u∈H-p-11(∂Ω),p=0,...,m-1 are called the p−th L−traces of a function u ∈ D(L) on ∂Ω, if the following identity is true

for any functions v ∈ H^m(Ω).

For example, for some solution u ∈ D(L), L−traces have the form:

for all v ∈ KerL⁺ ∩ H^m(Ω).

Finally, we present the definition of a weak solution to problem (Equations 1, 2):

Definition 9. We will call a function u ∈ D(L) a weak solution to the Cauchy problem (Equations 1, 2), if it satisfies to the following integral identity

for any functions v∈C0∞(Ω). The functionals L_(p)u are called L_(p) − traces of function u, p = 0, 1, 2, 3, and completely determined by the initial data φ, ψ, σ, χ by the following way:

Here, α_i, i = 1, 2, β_j, j = 1, 2, ..., 5, and δ_k, k = 1, ..., 9 are smooth functions, completely determined by coefficients a_i, i = 0, 1, ..., 4.

We can use a general form of operators γ_j in left-hand side of identity (Equation 6) instead of operators of differentiation ∂ν(j)v. Indeed, we define γ_j = p_jγ, where

and pj:H(m)→Hm-j-1/2(∂Ω)- projection.

Remark 1. As it has been mentioned above, some examples show (see Burskii [2]) that for solutions u ∈ D(L) ordinary traces do not exist even in the sense of distributions. Indeed, let Lu=∂2u∂x1∂x2=0 in the unit disk K:|x| = 1, the solution u(x)=(1-x12)-52 belongs to L²(K), but < u|_∂K, 1 > _∂K = ∞, that means limr→1-0∫|x|=ru(x)dsx=∞. The trace u|_∂K does not exist even as a distribution. However, for every solution u ∈ L²(K) L₍₀₎ − trace L(0)u: =-L(x)u(x)||x|=1=-x1x2u(x)||x|=1∈L2(∂K). Likewise, L₍₁₎ − trace, L₍₁₎u, exists for every u ∈ L²(K):

Here, τ is the angular coordinate and uτ′ is the tangential derivative, and L(x) = x₁x₂− symbol of the wave operator L=∂2∂x1∂x2.

We prove the maximum principle for weak solutions of the Cauchy problem (Equations 1, 2) in an admissible plane domain Ω restricted by different and non-congruent characteristics C_j, j = 1, 2, ..., 4 and initial line Γ₀.

Theorem 1. Maximum principle. Let u ∈ D(L) satisfies the following inequalities:

and

then, u ≤ 0 in D.

Proof. 1. First of all, we prove the statement for smooth solutions u∈C∞(Ω̄).

Due to the homogeneity of the symbol in Equation 1, L(ξ)=a0ξ14+a1ξ13ξ2+a2ξ12ξ22+a3ξ1ξ23+a4ξ24=(ξ,a1)(ξ,a2)(ξ,a3)(ξ,a4),ξ=(ξ1,ξ2)∈ℝ2, we can rewrite this equation in the following form:

The vectors aj=(a1j,a2j),j=1,2,3,4 are determined by the coefficients a_i, i = 0, 1, 2, 3, 4, and (a,b)=a1b̄1+a2b̄2 is a scalar product in ℂ². It is easy to see that vector a^j is the tangent vector of the j−th characteristic, slope φ_j of which is determined by −tanφ_j = λ_j, j = 1, 2, 3, 4. In what follows, we also consider the vectors ãj=(-ā2j,ā1j),j=1,2,3,4. It is obvious that (ã^j, a^j) = 0, so ã^j is a normal vector of the j−th characteristic.

Using Definitions 7 and 9 (m = 4), we assume that domain Ω is restricted by the characteristics C_j, j = 1, 2, 3, 4 and Γ₀:

Using representation (Equation 10), we have

Integrating by parts, we obtain:

Since (∇, a⁴)(∇, a³)(∇, a²)(∇, a¹)v = L⁺v and

we have

Difference between Equations 11, 12 is that natural traces in Equation 11 L_(3−k) are multiplied by the k−th derivative of truncated function v:∂ν(k)v by outer normal ν. On the other hand, we determined by L~(3-k) in Equation 12 some expressions multiplied by differential operators Lk+v, which can serve as analogous of natural L_(3−k) traces, k = 0, 1, 2, 3. So, in Equation 12:

Let v ∈ KerL⁺ in Equation 12, and calculate L− traces on ∂Ω = C₁ ∪ C₂ ∪ C₃ ∪ C₄ ∪ Γ₀. For instance, for L₍₃₎u we obtain: L(3)u=(ν,a1)(∇,a2)(∇,a3)(∇,a4)u. We use (∇,aj)u=(ν,aj)uν′+(τ,aj)uτ′,j=1,2,3,4, where ν− normal vector and τ− tangent vector. It is easy to see that L₍₃₎u = 0 (due to presence the product (ν, a¹)) on characteristic C₁, normal vector ã¹ of which is orthogonal to the vector a¹. On the other parts of ∂Ω, there will be vanish terms containing (ν, a^j) on C_j. After that

Here, the coefficients α_i,j are numerated as follows: the first index i indicates the derivative of u: 1)u_νντ, 2)u_νττ, 3)u_τττ, 4)u_νν, 5)u_ντ, 6)u_ττ, 7)u_ν 8)u_τ, the second index j indicates the j + 1−th characteristic, j = 1, 2, 3. Such a way, Equation 11 has the form:

Coefficients α_i,j are constant and depend on only coefficients a₀, a₁, a₂, a₃, a₄. By analogous way, we calculate others L− traces: L₍₀₎u, L₍₁₎u and L₍₂₎u.

To obtain the statement of Theorem 1, we choose some arbitrary point C ∈ D in admissible plane domain D and draw through this point two arbitrary characteristics, C₁ and C₂. Another two characteristics (C₃ and C₄) we draw through the ends a and b of initial line Γ₀. We determine some points O₁ and O₂ as intersections of C₁, C₃ and C₂, C₄ correspondingly: O₁ = C₁ ∩ C₃, O₂ = C₂ ∩ C₄. Such a way, domain Ω is a pentagon aO₁CO₂b. The value of a function u at the point C ∈ D, u(C) we estimate from the last equality, integrating by the characteristics C₁ and C₂ and using conditions (Equations 2, 7–9). Since a chosen point C ∈ D is arbitrary, we arrive at u ≤ 0 in D.

2. For solutions u ∈ D(L), the statement of the theorem follows from the conditions:

and

These conditions hold true for operators with constant coefficients in domains convex with respect to characteristics (see Hörmander [14]).

Theorem 1 is proved.

Remark 2. The weak form of the maximum principle for u ∈ L²(Ω) can be derived not only for solutions of the Cauchy problem (Equation 2) but also for all linear problems with constant coefficients Lu = F ∈ L²(Ω) under condition ImL+¯=L2(Ω).

Indeed, using conditions (Equations 8, 9) and definition 9, we obtain

for all v ∈ H^m(Ω). If ImL+¯=L2(Ω), then

for any w ∈ L²(Ω). The last inequality serves as a weak maximum principle for L² − solutions.

Remark 3. In the case of classical solutions of the Cauchy problem for second-order hyperbolic equations of general form with constant coefficients, the statement of Theorem 1 coincides with the result of Protter and Weinberger [23]. In this case, conditions (Equation 9) have usual form without using the notion of L−traces (see Protter and Weinberger [23]):

We develop the method of equation-domain duality (see also Burskii and Buryachenko [6] and Burskii [2]) for study of the Goursat problem. This method allows us to reduce the Cauchy problem (Equation 1, 2) in bounded domain Ω to the equivalent Goursat boundary value problem. We will show that the method of equation-domain duality can be applied also to boundary value problems in the generalized statement. First of all, we consider the method of equation-domain duality for the case of classical (smooth) solutions.