Classical solution of the Cauchy problem for biwave equation: Application of Fourier transform

In this paper, we use some Fourier analysis techniques to find an exact solution to the Cauchy problem for the $n$-dimensional biwave equation in the upper half-space $\mathbb{R}^n\times [0,+\infty)$.

The biwave equation has been studied in some models related to the mathematical theory of elasticity. Let us consider the mathematical formulation for the displacement equation of a homogeneous isotropic elastic body. Remark that, the Newton's second law leads to the Cauchy's motion equation of an elastic body, which takes the form ∇ · σ + f = ρü, (1.3) where σ is the Cauchy stress tensor field, u is the displacement vector field, f is the vector field of body force per unit volume and ρ is the mass density. The infinitesimal strain tensor field is given by the equation Let us denote a 2 = (λ + 2µ)/ρ, b 2 = µ/ρ, then (1.6) can be rewritten as It is easy to show that the equation (1.7) has a solution in the following form where w is a solution to the biwave equation This formula is called as Cauchy-Kovalevski-Somigliana solution to the elastodynamic wave equation. Indeed, substituting (1.8)-(1.9) to the left-hand side of (1.7), we have Therefore, we get For more explanations about physical context, we refer the reader to [HI04,Mus10,Som50]. Actually, there are not many mathematical papers related to biwave equations because it gets more difficult when studying high-order PDEs. In some recent researches, the symmetry analysis of biwave equations is considered and exact solutions are obtained by Fushchych, Roman and Zhdanov [FRZ96]; the existence and uniqueness of the solution to Cauchy initial value problem, bounded valued problem are given by Korzyuk, Cheb and Konopelko [KKC10,KC07]; the finite element methods for approximations of biwave equation are developed by Feng and Neilan [FN10,FN11]. In our present work, the main result is to show the exact classical solution to the Cauchy initial value problem for the n-dimensional biwave equation by using some techniques of Fourier analysis.
Returning to the Cauchy problem for the biwave equation (1.1), we suppose that φ 0 , φ 1 , φ 2 , φ 3 , and f are elements in Schwartz space S(R n ) of rapidly decreasing functions on R n . Remark that, an indefinitely differentiable function φ is called rapidly decreasing when φ and all its derivatives are required to satisfy that for every multi-index α and β. The Fourier transform of Schwartz function φ ∈ S(R n ) is defined by The convolution of two integrable functions φ and ψ is written as φ * ψ. It is defined as the integral of the product of the two functions after one is reversed and shifted. As such, it is a particular kind of integral transform: In the Euclidean space R n , the spherical mean of an integrable function φ around a point x is the average of all values of that function on a sphere of radius R centered at that point, i.e. it is defined by the formula where ω n is the surface area of the n-dimensional unit ball and σ is the spherical measure area.

Main results
The Cauchy problem for the homogeneous biwave equation in R n × [0, +∞) that we will be studying in this section, reads as follows with the initial conditions The equation (2.1) can be rewritten as a fourth-order PDE, which has the following form Taking Fourier transform to the both sides of the equation (2.3), we obtain This fourth ODE has the general solution, which takes the form u (ξ, t) = C 1 cos (a|ξ|t) + C 2 sin (a|ξ|t) + C 3 cos (b|ξ|t) + C 4 sin (b|ξ|t) , where parameters C 1 , C 2 , C 3 , C 4 are determined from the initial conditions: Solving above system of equations, we easily get the image of solution u(x, t) via Fourier transform given by or by the rewritten form (2.4) In the next sequence, we will find the inverse formula of (2.4) and obtain an exact solution to the equation (2.1).
Theorem 2.1. The Cauchy problem for the homogeneous biwave equation in R × [0, +∞) has the following solution (2.5) Proof. We have some facts that Moreover, where δ (x) is the Dirac delta function. Hence, by the property of the Dirac's delta function, we note that Substituting the above identities into the formula (2.4), we obtain that Consequently, we get the inverse formula of u given by The last formula is equivalent to the one given at (2.5), so the theorem is proved.
For the generalized case, we will use the following result: For the proof, we refer the reader to Torchinsky's paper in [Tor09]. Note that, in the case k = m, it follows that sin(a|ξ|t)  Moreover, for each function θ ∈ S(R n ), we also have Therefore, we conclude that By the Fourier inversion and convolution formulas, we obtain the identity Applying the same way for the expressions (2.7)-(2.9) and substituting the obtained identities into the formula (2.4), we have found an exact solution to the n-dimensional biwave equation, where n ≥ 3 is an odd number: Theorem 2.3. The Cauchy initial value problem for the homogeneous n-dimensional biwave equation, where n ≥ 3 is an odd number, has the following solution Now we consider the case when n is an even number. The Hadamard's method of descent (see e.g. [Had53]) is useful to connect with the case in the odd dimensional space R n+1 . For fixed T > 0, we choose a Schwartz function η ∈ S(R), such that η(x n+1 ) = 1 for all |x n+1 | ≤ nT . Let us denote It is easy to see that φ i ∈ S(R n+1 ). For |x n+1 | ≤ T, t ≤ T , the solution u(x 1 , x 2 , ..., x n+1 , t) to the Cauchy problem for n + 1-dimensional biwave equation with initial valued functions φ i , i = 0, 1, 2, 3 does not depend on x n+1 . In particular, is the solution to the n-dimensional biwave equation for all |t| ≤ T . Since T is arbitrary, so u is the solution to the Cauchy problem in even dimensional space R n .
Finally, we obtain that So the lemma is proved.
We use the notation M t (f )(x) = 2 ω n+1 Bn(0,1) dz, which is called as modified spherical mean of f (see e.g. [SS97,SS03]). Applying the result of Lemma 2.4, we obtain the formula of the solution to the biwave equation in the even dimensional space R n : Theorem 2.5. The Cauchy initial value problem for the homogeneous n-dimensional biwave equation, where n ≥ 2 is an even number, has the following solution By a similar idea with the Duhamel principle for wave equations, the solution of the Cauchy problem for the nonhomogeneous biwave equation will be given at the next theorem Proof. We start with the observation that . Then, the above identity follows that ∂ 4 u ∂t 4 − a 2 + b 2 ∂ 2 ∂t 2 ∆u + a 2 b 2 ∆ 2 u = f (x, t) . Moreover, u| t=0 = u| t=0 + v| t=0 = φ 0 (x) + 0 = φ 0 (x) , ∂u ∂t t=0 = ∂ u ∂t t=0 + ∂v ∂t t=0 = φ 1 (x) + 0 = φ 1 (x) , So the theorem is proved.

Example
Let us give an example demonstrating Theorem 2.1.
Consider the equation