Singular Cauchy problem for the general Euler-Poisson-Darboux equation

In this paper we obtain the solution of the singular Cauchy problem for the Euler-Poisson-Darboux equation when di erential Bessel operator acts by each variable.


Introduction
The classical Euler-Poisson-Darboux equation has the form The operator acting by t in (1) is called the Bessel operator. For the Bessel operator we use the notation (see. [1], p. 3) The Euler-Poisson-Darboux equation for n = appears in Euler's work (see [2], p. 227). Further Euler's case of (1) was studied by Poisson in [3], Riemann in [4] and Darboux in [5] (for the history of this issue see also in [6], p. 532 and [7], p. 527). The generalization of it was studied in [8]. When n ≥ the equation (1) was considered, for example, in [9,10]. The Euler-Poisson-Darboux equation appears in di erent physics and mechanics problems (see [11][12][13][14][15]). In [16] (see also [17], p. 243) and in [18] there were di erent approaches to the solution of the Cauchy problem for the general Euler-Poisson-Darboux equation with the initials conditions The Cauchy problem with the nonequal to zero rst derivative by t of u for the (2) (and for (1)) is incorrect. However, if we use the special type of the initial conditions containing the nonequal to zero rst derivative by t of u then such Cauchy problem for the (2) will by solvable. Following [17] and [19] we will use the term singular Cauchy problem in this case. The abstract Euler-Poisson-Darboux equation (when in the left hand of (2) an arbitrary closed linear operator is presented) was studied in [20][21][22]. In this article we consider the solution of the problem (2)-(3) when −∞ < k < +∞ and its properties. Besides this, we get the formula for the connection of solution of the problem (2)-(3) and solution of a simpler problem. Also using the solution of the problem (2)-(3) we obtain solution of the singular Cauchy problem for the equation (2) when k < with the conditions

Property of general Euler-Poisson-Darboux equations' solutions
In this section we give some necessary de nitions and obtain two fundamental recursion formulas for solution of (2). Let and Ω is open set in R n which is symmetric correspondingly to each hyperplane x i = , i= , ..., n, Ω + = Ω ∩R n We have Ω + ⊆ R n + and Ω + ⊆ R n + . Consider the set C m (Ω + ), m ≥ , consisting of di erentiable functions on Ω + by order m. Let where Equation (5) we will call the general Euler-Poisson-Darboux equation.
denote the solution of (5) when the next two fundamental recursion formulas hold Proof. Following [23] we prove (7).
If w = t k− v satis es the equation then using (9) we get which means that v satis es the equation Denoting w = u −k we obtain (7). Now we prove the (8).
We nd now k+ Then we get Recursion formulas (7) and (8) allow us to obtain, from a solution u k of equation (5), the solutions of the same equation with the parameter k+ and −k, respectively. Both formulas are proved for Euler-Poisson-Darboux

Weighted spherical mean and the rst Cauchy problem for the general Euler-Poisson-Darboux equation
Here we present the solutions of the problem (2)-(3) for di erent values of k for which we obtain solution of (2)-(4) in the next section, and get formula for the connection of solution of problem (2)-(3) and solution of simpler problem when k = in (2).
In R n + we will use multidimensional generalized translation corresponding to multi-index γ: where each γ i T τ i x i is de ned by the formula (see [24]) The below-considered weighted spherical mean generated by a multidimensional generalized translation γ T t has the form (see [25]) where θ γ = n ∏ i= θ γ i i , S + (n)={θ∶ θ = , θ∈R n + } and the coe cient S + (n) γ is computed by the formula (see [26], p. 20, formula (1.2.5) in which we should put N=n). Construction of a multidimensional generalized translation and the weighted spherical mean are transmutation operators (see [27]). Theorems 3.1-3.4 have been proved in [28]. We give formulations of these theorems here because they will be needed in the next section.
This theorem has been proved in [25]). We give theorems on the solution of the Cauchy problem for the general Euler-Poisson-Darboux equation for the remaining values of k.
Theorem 3.2. Let f ∈ C ev . Then for the case k > n + γ − the solution of (15)-(16) is unique and given by Using weighted spherical mean we can write where m is a minimum integer such that m ≥ n+ γ −k− and u k+ m (x, t) is the solution of the Cauchy problem The solution of (15)- (16) is unique for k ≥ and not unique for negative k.
The solution of (15)- (16) is not unique for negative k.
The theorem 3.5 contains the explicit form of the transmutation operator for the solution. De nition, methods of construction and applications of the transmutation operators can be found in [27,29,30].

Theorem 3.5. Let k > . The twice continuously di erentiable on R n+
.., n is connected with the twice continuously di erentiable on R n + × R solution w=w(x, t) of the Cauchy problem such that w x i (x , ..., x i− , , x i+ , ..., x n , t) = , i = , ..., n by formula where (P λ τ ) α is transmutation Poisson operator (see [24]) acting by α Proof. The fact that the function u k de ned by the equality (28) satis es the conditions (31) is obvious. Let us show that u k de ned by (28) satis es (24) where ξ = αt. Further integrating by parts we obtain Finally, Thus the function u k de ned by equality (28) satis es the problem (24)- (31). Let us prove that from the relation (28) we can uniquely obtain a solution of the problem (26)- (27). By introducing new variables αt = √ τ , t = √ y, we get Let k > then y k− u k (x, √ y) is the Riemann-Liouville left-sided fractional integral of the order k (see [31], p.

The second Cauchy problem for the general Euler-Poisson-Darboux equation
In this section we obtain solution of (2)-(4).
is given by if n + γ + k is an odd integer, where q ≥ is the smallest positive integer number such that − k + q ≥ n + γ − .
Proof. Let q ≥ be the smallest positive integer number such that − k + q ≥ n + γ − i.e. q = n+ γ +k− and let v −k+ q (x, t) be a solution of (29) when we take − k + q instead of k such that Then by property (7) we obtain that v k− q = t −k+ q v −k+ q is a solution of the equation Further, applying q-times the formula (8) We have shown that (32) satis es the equation (29). Now we will prove that v k satis es the conditions (31). For v k ∈ C q ev (Ω + ) we have the formula (see [19], p.9)