Maximum principle for the weak solutions of the Cauchy problem for the fourth-order hyperbolic equations

We investigate the maximum principle for the weak solutions to the Cauchy problem for the hyperbolic fourth-order linear equations with constant complex coefficients in the plane bounded domain


Introduction
We concern here the problem of proving the analog of maximum principle for the weak solutions of the Cauchy problem for the fourth-order linear hyperbolic equations with the complex constant coefficients and homogeneous non-degenerate symbol in some plane bounded domain Ω ∈ R 2 convex with respect to characteristics: Here coefficients a j , j = 0, 1, ..., 4 are constant, f (x) ∈ L 2 (Ω), We assume, that Eq. (0.1) is hyperbolic, that means that all roots of characteristics equation L(1, λ) = a 0 λ 4 + a 1 λ 3 + a 2 λ 2 + a 3 λ + a 4 = 0 are prime, real and are not equal to ±i, that means that the symbol of Eq. (0.1) is nondegenerate or that the Eq.(0.1) is a principal-type equation.The equations for which the roots of the corresponding characteristic equation are multiple and can take the values ±i are called the equation with degenerate symbol (see [7]).
The main novelty of the paper is to prove the analog of maximum principle for the fourthorder hyperbolic equations.This question is very important due to usually a natural physical interpretation, and that it helps to establish the qualitative properties of the solutions (in our case the questions of uniqueness and existence of weak solution).But as it is well known, the maximum principle even for the simple case of hyperbolic equation (one dimensional wave equation [21]) are 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 in the situation of hyperbolic type PDEs.
We call the angle of characteristics slop the solution to the equation − tan ϕ j = λ j , and 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 the 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 [11], 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 [22] and others.
Due to evident practice application, these models need more exact tools for studying, and as consequence, to attract fundamental knowledge.As usual, most of these models are studied by the analytical-numerical methods (Galerkin's methods).
On the other hand, the maximum principle is an efficient tool for fundamental knowledge of PDEs: the range of problems, which maximum principle allows to study, belongs to a class of quite actual problems of well-posedness of so-called general boundary-value problems for higherorder differential equations originating from the works by L. Hörmander 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 works by G. Grubb [12], L.Hormander [13], and A. Posilicano [20].Later, the problem of well-posedness of boundary-value problems for various types of second order differential equations was studied by V. Burskii and A. Zhedanov [2], [3] which developed a method of traces associated with a differential operator and applied this method to establish the Poncelet, Abel and Goursat problems, and by I. Kmit [14].In the previous works of author (see [6]) there have been developed qualitative methods of studying Cauchy problems and nonstandard in the case of hyperbolic equations Dirichlet and Neumann problems for the linear fourth-order equations (moreover, for an equation of any even order 2m, m ≥ 2, ) with the help of operator methods (Ltraces, theory of extension, moment problem, method of duality equation-domain and others), [4].
As concern maximum principle, at the present time there are not any results for the fourth order equations even in linear case.As it was mentioned above, the maximum principle even for the simple case of one dimensional wave equation [21], and for the second-order telegraph equation [16]- [19] are quite different from those for elliptic and parabolic cases.In the monograph of Protter and Weinberger [21] 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 in two independent variables u tt − u xx = 0 the maximum of a nonconstant solution u = sin x sin t in a rectangle domain {(x, t) : x ∈ [0, π], t ∈ [0, π]} occurs at an interior point π 2 , π 2 .In Chapter 4 [21] the maximum principle for linear second hyperbolic equations of general type, with variable coefficients has also been obtained for Cauchy problems and boundary value problems on characteristics (Goursat problem).Following R.Ortega, A.Robles-Perez [19], we introduce the definition of a "weak form" of the maximum principle, which is used for the hyperbolic equations, which will be used later.Definition 1. [19] Let L = Lu 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 others requirements.It is said that L satisfies the maximum principle, if In further works of these authors (see [16], [17], [18]) there was studied the maximum principle for weak bounded twice periodical solutions from the space L ∞ of the telegraph equation with parameter λ in lower term, one-, two-, and -tree dimensional spaces, and which includes the cases of variables coefficients.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 nonlinear equation, which allows to obtain the analogous results (uniqueness, existence and regularity theorems) for the telegraph equations with external nonlinear forcing, applying maximum principle.There was considered also the case when the external forcing belongs to a certain space of measures.
The maximum principle for general quasilinear hyperbolic systems with dissipation was proved by Kong De Xing [15].There were given two estimates for the solution to the general quasilinear hyperbolic system and introduced the concept of dissipation (strong dissipation and weak dissipation), then state some maximum principles of quasilinear hyperbolic systems with dissipation.Using the maximum principle there were reproved the existence and uniqueness theorems of the global smooth solution to the Cauchy problem for considered quasilinear hyperbolic system.So, the problem to prove the maximum principle for the weak solutions stills 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 weak-regularity data.
There are no results on maximum principle even for model case of linear 2-dimensional fourthorder hyperbolic equations with the constant coefficients and without lower terms.Moreover, we can not use the term of usual traces in the cases of initial data of weak regularity, and we come to the notions of L−traces, the traces, which associated with differential operator.Let us remind (see, for example, [2]), that L− traces exist for the weak solutions from space L 2 even in the situations when classical notions of traces does not work for such solutions.
1 Statement of the problem and auxiliary definitions.
Let us start to establish the maximum principle for the weak solutions to the Cauchy problem for the Eq.(0.1) in some admissible planar domain.It is expected, that in hyperbolic case the characteristics of the equations play a crucial role.
Let C j , j = 1, 2, 3, 4 be characteristics, Γ 0 := {x 1 ∈ [a, b], x 2 = 0}, and define domain Ω as a domain which is restricted by the characteristics C j , j = 1, 2, 3, 4 and Γ 0 .Following [21] we will call below the domain Ω as characteristics domain.For the second order hyperbolic equations (see [21]), Ω is characteristics triangle, in the case of fourth-order equations with constant coefficients, that is existence of 4 different and real characteristics lines C j , j = 1, 2, 3, 4, Ω is a characteristic pentagon.
In the bounded domain Ω we consider the linear differential operation L of the order m, m ≥ 2, and formally adjoint 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: for any functions v ∈ H m (Ω).
As it has been mentioned above, some examples show (see [2]) that in the general case the solutions u ∈ D(L) do not exist ordinary traces in the sense of distributions even for the simplest hyperbolic equations.Indeed, for the wave equation Lu = ∂ 2 u ∂x 1 ∂x 2 = 0 in the unit disk such a way the trace u| ∂K does not exist even as a distribution.However, for every solution u ∈ L 2 (K) the − trace L (1) u exists for every u ∈ L 2 (K): where τ is the angular coordinate and u ′ τ is the tangential derivative, and 2 Maximum principle for the weak solutions of Cauchy problem.
We prove here the first simple case: the maximum principle for the weak solution of the Cauchy problem (0.1)-(1.1) in admissible plane domain Ω, restricted by the different and non-congruent characteristics C j , j = 1, 2, ..., 4 and initial line Γ 0 .
Theorem 1. Maximum principle.Let u ∈ D(L) satisfy the following inequalities: and then u ≤ 0 in D.
Proof. 1.First of all prove the statement for smooth solutions u ∈ C ∞ (Ω).Due to homogeneity of the symbol in Eq. (0.1), , we can rewrite this equation in the following form: 3) The vectors a j = (a j 1 , a j 2 ), j = 1, 2, 3, 4 are determined by the coefficients a i , i = 0, 1, 2, 3, 4, and < a, b >= a 1 b1 + a 2 b2 is a scalar product in C 2 .It is easy to see that vector a j is a tangent vector of j−th characteristic, slope ϕ j of which is determined by − tan ϕ j = λ j , j = 1, 2, 3, 4. In what follows, we also consider the vectors ãj = (−ā j 2 , āj 1 ), j = 1, 2, 3, 4. It is obvious that < ãj , a j >= 0, so ãj is a normal vector of j−th characteristic.
By analogous way we calculate others L− traces, L (0) u, L These conditions hold for operators with constant coefficients in domain convex with respect characteristics (see [13]).
The Theorem 1 is proved.
Remark 2. The weak form of the maximum principle for u ∈ L 2 (Ω) can be derived not only for the solutions of the Cauchy problem (1.1) but for all linear differential operator problems Lu = F ∈ L 2 (Ω), with condition Im L + = L 2 (Ω) and with the constant coefficients.Indeed, using conditions (2.1), (2.2), and the definition 9 we obtain Ω u • L + v dx ≤ 0, for all v ∈ H m (Ω).If Im L + = L 2 (Ω), then we have Ω u • w dx ≤ 0, for any w ∈ L 2 (Ω).The last inequality can serve as a weak maximum principle for solutions to the boundary value problems from L 2 (Ω).Remark 3. In the case of classical solution of the Cauchy problem for the second order hyperbolic equations of the general form with the constant coefficients the statement of the Theorem 1 coincides with the result of [21].In this case conditions (2.2) have usual form without using the notion of L−traces (see [21]):

in the half-plane x 2 >
0 an admissible domain if it has the property that for each point C ∈ D the corresponding characteristics domain Ω is also in D.More generally, D is admissible, if it is the finite or countable union of characteristics domains.We choose some arbitrary point C ∈ D in admissible plane domain D, draw through this point two arbitrary characteristics, C 1 and C 2 .Another two characteristics (C 3 and C 4 ) we draw through the ends a and b of initial line Γ 0 .We determine the points O 1 and O 2 as intersections of C 1 , C 3 and C 2 , C 4 correspondingly: O 1 = C 1 ∩ C 3 , O 2 = C 2 ∩ C 4 .Such a way, domain Ω is a pentagon aO 1 CO 2 b.Establishment of the maximum principle in this situation allows us to obtain a local properties of the solution to Cauchy problem (0.1)-(1.1) on the arbitrary interior point C ∈ D. We will consider the weak solution to the problem (0.1)-(1.1) from the D(L), domain of definition of maximal operator, associated with the differential operation L in Eq.(0.1).

For L 2 −Definition 8 .
) for any functions v ∈ H m (Ω).The functionals L (p) u is called the L (p) − traces of the function u ∈ D( L).Here (•, •) L 2 (Ω) is a scalar product in Hilbert space L 2 (Ω).solutions the notion of L (p) − traces can be realized by the following way.The distributions L (p) u ∈ H −p− 1 1 (∂Ω), p = 0, ..., m − 1, are called the p−th L−traces of the function u ∈ D(L) on ∂Ω, if the following identity is true

Finally, we areDefinition 9 .
going to the definition of the weak solution to the problem (0.1)-(1.1):We will call the function u ∈ D(L) a weak solution to the Cauchy problem (0.1)-(1.1), if it satisfies to the following integral identity

( 1 ) 2 .
u and L (2) u.The value of the function u at the point C ∈ D, u(C) we estimate from the last equality, integrating by the characteristics C 1 and C 2 and using conditions (1.1), (1.6)-(2.2).Since, the chosen point C ∈ D is arbitrary, we arrive at u ≤ 0 in D. For solutions u ∈ D(L) the statement of the Theorem 1 follows from the conditions: C ∞ (Ω) = D(L), and C ∞ (Ω) = D(L + ).