On the Operator ⨁𝑘𝐵 Related to Bessel Heat Equation

Abstract

We study the equation (𝜕/𝜕𝑡)𝑢(𝑥,𝑡)=𝑐²⊕^𝑘_𝐵𝑢(𝑥,𝑡) with the initial condition 𝑢(𝑥,0)=𝑓(𝑥) for 𝑥∈𝑅⁺_𝑛. The operator ⊕^𝑘_𝐵 is the operator iterated k-times and is defined by ⊕^𝑘_𝐵=((∑^𝑝_𝑖=1𝐵_𝑥𝑖)⁴−(∑^𝑝+𝑞_𝑗=𝑝+1𝐵_𝑥𝑖)⁴)^𝑘, where 𝑝+𝑞=𝑛 is the dimension of the ℝ⁺_𝑛, 𝐵_𝑥𝑖=𝜕²/𝜕𝑥²_𝑖+(2𝑣_𝑖/𝑥_𝑖)(𝜕/𝜕𝑥_𝑖), 2𝑣_𝑖=2𝛼_𝑖+1, 𝛼_𝑖>−1/2, 𝑖=1,2,3,…,𝑛, and 𝑘 is a nonnegative integer, 𝑢(𝑥,𝑡) is an unknown function for (𝑥,𝑡)=(𝑥₁,𝑥₂,…,𝑥_𝑛,𝑡)∈𝑅⁺_𝑛×(0,∞), 𝑓(𝑥) is a given generalized function, and 𝑐 is a positive constant. We obtain the solution of such equation, which is related to the spectrum and the kernel, which is so called Bessel heat kernel. Moreover, such Bessel heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.

1. Introduction

It is well known that for the heat equation 𝜕𝜕𝑡𝑢(𝑥,𝑡)=𝑐²Δ𝑢(𝑥,𝑡) (1.1) with the initial condition 𝑢(𝑥,0)=𝑓(𝑥), (1.2) where Δ=∑^𝑛_𝑖=1𝜕²/𝜕𝑥²_𝑖 is the Laplace operator and (𝑥,𝑡)=(𝑥₁,𝑥₂,…,𝑥_𝑛,𝑡)∈ℝ^𝑛×(0,∞), we obtain 1𝑢(𝑥,𝑡)=(4𝑐²𝜋𝑡)^{𝑛/2ℝ𝑛}exp(−|𝑥−𝑦|²4𝑐²𝑡)𝑓(𝑦)𝑑𝑦 (1.3) as the solution of (1.1). Equation (1.3) can be written as 𝑢(𝑥,𝑡)=𝐸(𝑥,𝑡)∗𝑓(𝑥), (1.4) where 1𝐸(𝑥,𝑡)=(4𝑐²𝜋𝑡)^𝑛/2exp(−|𝑥|²4𝑐²𝑡). (1.5)𝐸(𝑥,𝑡) is called the heat kernel, where |𝑥|²=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑛 and 𝑡>0 (see [1, pages 208-209]).

In 2004, Yildirim et al. [2, 3] first introduced the Bessel diamond operator ⋄^𝑘_𝐵 iterated 𝑘-times, defined by⋄^𝑘_𝐵=((^𝑝_𝑖=1𝐵_𝑥𝑖)²−(^𝑝+𝑞_𝑗=𝑝+1𝐵_𝑥𝑗)²)^𝑘, (1.6) where 𝐵_𝑥𝑖=𝜕²/𝜕𝑥²_𝑖+(2𝜐_𝑖/𝑥_𝑖)(𝜕/𝜕𝑥_𝑖), 2𝜐_𝑖=2𝛼_𝑖+1,𝛼_𝑖>−1/2,𝑥_𝑖>0. The operator ⋄^𝑘_𝐵 can be expressed by ⋄^𝑘_𝐵=Δ^𝑘_B□^𝑘_𝐵=□^𝑘_𝐵Δ^𝑘_𝐵, where Δ^𝑘_𝐵=(^𝑝_𝑖=1𝐵_𝑥𝑖)^𝑘,□^𝑘_𝐵=(^𝑝_𝑖=1𝐵_𝑥𝑖−^𝑝+𝑞_𝑗=𝑝+1𝐵_𝑥𝑗)^𝑘. (1.7)(1.8)

And, Yildirim et al. [2, 3] have shown that the solution of the convolution form 𝑢(𝑥)=(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥) (1.9) is a unique elementary solution of ⋄^𝑘_𝐵 that is ⋄^𝑘_𝐵((−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥))=𝛿. (1.10)

Now, the purpose of this work is to study the following equation: 𝜕𝜕𝑡𝑢(𝑥,𝑡)=𝑐^2𝑘_𝐵𝑢(𝑥,𝑡) (1.11) with the initial condition 𝑢(𝑥,0)=𝑓(𝑥),for𝑥∈𝑅⁺_𝑛, (1.12) where the operator ⊕^𝑘_𝐵 is first introduced by Satsanit and Kananthai [4] and is defined by ^𝑘_𝐵=((^𝑝_𝑖=1𝐵_𝑥𝑖)⁴−(^𝑝+𝑞_𝑗=𝑝+1𝐵_𝑥𝑖)⁴)^𝑘=((^𝑝_𝑖=1𝐵_𝑥𝑖)²−(^𝑝+𝑞_𝑗=𝑝+1𝐵_𝑥𝑖)²)^𝑘((^𝑝_𝑖=1𝐵_𝑥𝑖)²+(^𝑝+𝑞_𝑗=𝑝+1𝐵_𝑥𝑖)²)^𝑘. (1.13) Let us denote the operator ⊚^𝑘_𝐵=((^𝑝_𝑖=1𝐵_𝑥𝑖)²+(^𝑝+𝑞_𝑗=𝑝+1𝐵_𝑥𝑗)²)^𝑘. (1.14) By (1.7) we obtain ⊚^𝑘_𝐵=[(^𝑝_𝑖=1𝐵_𝑥²_𝑖)²+(^𝑝+𝑞_𝑗=𝑝+1𝐵_𝑥²_𝑗)²]^𝑘Δ=[(^𝐵+□_𝐵2)²Δ+(^𝐵−□_𝐵2)²]^𝑘Δ=(^2𝐵+□²_𝐵2)^𝑘. (1.15) Thus, (1.13) can be written as ^𝑘_𝐵=⋄^𝑘_𝐵⊚^𝑘_𝐵, (1.16) where ⋄^𝑘_𝐵and⊚^𝑘_𝐵 are defined by (1.6) and (1.15), respectively, 𝑝+𝑞=𝑛 is the dimension of the 𝑅⁺_𝑛=𝑥∶𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛,𝑡),𝑥_𝑖>0,𝑖=1,2,3,…,𝑛, 𝑢(𝑥,𝑡) is an unknown function, 𝐵_𝑥𝑖=𝜕²/𝜕𝑥²_𝑖+(2𝜐_𝑖/𝑥_𝑖)(𝜕/𝜕𝑥_𝑖), 2𝜐_𝑖=2𝛼_𝑖+1,𝛼_𝑖>−1/2,𝑓(𝑥) is the given generalized function, 𝑘 is a positive integer, and 𝑐 is a constant.

Moreover, Bessel heat kernel has interesting properties and also related to the kernel of an extension of the heat equation. We obtain the solution in the classical convolution form 𝑢(𝑥,𝑡)=𝐸(𝑥,𝑡)∗𝑓(𝑥), (1.17) where the symbol ∗ is the 𝐵-convolution in (2.3), as a solution of (1.11), which satisfies (1.12), and 𝐸(𝑥,𝑡)=𝐶_𝑣Ω+𝑒^{𝑐2𝑡[(𝑦21+⋯+𝑦2𝑝)4−(𝑦2𝑝+1+⋯+𝑦2𝑝+𝑞)4]𝑘𝑛}_𝑖=1𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖)𝑦^2𝜐𝑖_𝑖𝑑𝑦, (1.18) and Ω⁺⊂𝑅⁺_𝑛 is the spectrum of 𝐸(𝑥,𝑡) for any fixed 𝑡>0 and 𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖) is the normalized Bessel function.

Before going into details, the following definitions and some important concepts are needed.

2. Preliminaries

The shift operator according to the law remarks that this shift operator connected to the Bessel differential operator (see [2, 3, 5]): 𝑇^𝑦_𝑥𝜑(𝑥)=𝐶^∗_𝑣^𝜋₀⋯^𝜋₀𝜑(𝑥²₁+𝑦²₁−2𝑥₁𝑦₁cos𝜃₁,…,𝑥²_𝑛+𝑦²_𝑛−2𝑥_𝑛𝑦_𝑛cos𝜃_𝑛)×(^𝑛_𝑖=1sin^{2𝑣𝑖−1}𝜃_𝑖)𝑑𝜃₁⋯𝑑𝜃_𝑛, (2.1) where 𝑥,𝑦∈𝑅⁺_𝑛,𝐶^∗_𝑣=∏^𝑛_𝑖=1Γ(𝑣_𝑖+1)/Γ(1/2)Γ(𝑣_𝑖). We remark that this shift operator is closely connected to the Bessel differential operator (see [4]): 𝑑²𝑈𝑑𝑥²+2𝑣𝑥𝑑𝑈=𝑑𝑑𝑥²𝑈𝑑𝑦²+2𝑣𝑦𝑑𝑈,𝑑𝑦𝑈(𝑥,0)=𝑓(𝑥),𝑈_𝑦(𝑥,0)=0. (2.2) The convolution operator determined by the 𝑇^𝑦_𝑥 is as follows: (𝑓∗𝜑)(𝑦)=_𝑅+𝑛𝑓(𝑦)𝑇^𝑦_𝑥𝜑(𝑥)(^𝑛_𝑖=1𝑦^2𝑣𝑖_𝑖)𝑑𝑦. (2.3) Convolution (2.3) is known as a B-convolution. We note the following properties of the 𝐵-convolution and the generalized shift operator. (1)𝑇^𝑦_𝑥⋅1=1. (2)𝑇⁰_𝑥⋅𝑓(𝑥)=𝑓(𝑥).(3)If 𝑓(𝑥),𝑔(𝑥)∈𝐶(𝑅⁺_𝑛),𝑔(𝑥) is a bounded function for all 𝑥>0, and ∫^∞₀|𝑓(𝑥)|(∏^𝑛_𝑖=1𝑥^2𝑣𝑖_𝑖)𝑑𝑥<∞, then ∫_𝑅+𝑛𝑇^𝑦_𝑥𝑓(𝑥)𝑔(𝑦)  (∏^𝑛_𝑖=1𝑦^2𝑣𝑖_𝑖)𝑑𝑦 = ∫_𝑅+𝑛𝑓(𝑦)𝑇^𝑦_𝑥𝑔(𝑥)  (∏^𝑛_𝑖=1𝑦^2𝑣𝑖_𝑖)𝑑𝑦.(4)From (3),we have the following equality fo𝑟𝑔(𝑥)=1∶∫_𝑅+𝑛𝑇^𝑦_𝑥𝑓(𝑥)(∏^𝑛_𝑖=1𝑦^2𝑣𝑖_𝑖)𝑑𝑦=∫_𝑅+𝑛𝑓(𝑦)(∏^𝑛_𝑖=1𝑦^2𝑣𝑖_𝑖)𝑑𝑦. (5)(𝑓∗𝑔)(𝑥)=(𝑔∗𝑓)(𝑥).

The Fourier-Bessel transformation and its inverse transformation are defined as follows: (ℱ_𝐵𝑓)(𝑥)=𝐶_{𝑣𝑅+𝑛}𝑓(𝑦)(^𝑛_𝑖=1𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖)𝑦^2𝜐𝑖_𝑖)𝑑𝑦,(ℱ_𝐵⁻¹𝑓)(𝑥)=(ℱ_𝐵𝑓)(−𝑥),𝐶_𝑣=(^𝑛_𝑖=12^{𝜐𝑖−1/2}Γ(𝜐_𝑖+12))⁻¹, (2.4)(2.5) where 𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖) is the normalized Bessel function which is the eigenfunction of the Bessel differential operator. The following equalities for Fourier-Bessel transformation are true (see [5–7]): ℱ_𝐵𝛿(𝑥)=1,ℱ_𝐵(𝑓∗𝑔)(𝑥)=ℱ_𝐵𝑓(𝑥)⋅ℱ_𝐵𝑔(𝑥). (2.6)(2.7)

Definition 2.1. The spectrum of the kernel 𝐸(𝑥,𝑡) of (1.18) is the bounded support of the Fourier Bessel transform ℱ_𝐵𝐸(𝑦,𝑡) for any fixed 𝑡>0.

Definition 2.2. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) be a point in ℝ⁺_𝑛 and denote by Γ₊={𝑥∈ℝ⁺_𝑛∶𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝−𝑥²_𝑝+1−𝑥²_𝑝+2−⋯−𝑥²_𝑝+𝑞>0,𝜉₁>0} (2.8) the set of an interior of the forward cone, and Γ₊ denotes the closure of Γ₊.

Let Ω⁺ be spectrum of 𝐸(𝑥,𝑡) defined by (1.18) for any fixed 𝑡>0 and Ω⊂Γ₊. Let 𝐹_𝐵𝐸(𝑦,𝑡) be the Fourier Bessel transform of 𝐸(𝑥,𝑡), which is defined by ℱ_𝐵𝐸(𝑦,𝑡)=𝑒^{𝑐2𝑡[(𝑦21+⋯+𝑦2𝑝)4−(𝑦2𝑝+1+⋯+𝑦2𝑝+𝑞)4]𝑘}for𝜉∈Ω₊,0for𝜉∉Ω₊. (2.9)

Lemma 2.3 (Fourier Bessel transform of □^𝑘_𝐵 operator). One has ℱ_𝐵□^𝑘_𝐵𝑢(𝑥)=(−1)^𝑘𝑉^𝑘₁(𝑥)ℱ_𝐵𝑢(𝑥), (2.10) where 𝑉^𝑘₁(𝑥)=(^𝑝_𝑖=1𝑥²_𝑖−^𝑝+𝑞_𝑗=𝑝+1𝑥²_𝑗)^𝑘. (2.11)

Proof. See [8].

Lemma 2.4 (Fourier Bessel transform of Δ^𝑘_𝐵 operator). One has ℱ_𝐵Δ^𝑘_𝐵𝑢(𝑥)=(−1)^𝑘|𝑥|^2𝑘ℱ_𝐵𝑢(𝑥), (2.12) where |𝑥|^2𝑘=(𝑥²₁+𝑥²₂+⋯+𝑥²_𝑛)^𝑘. (2.13)

Proof. See [8].

Lemma 2.5 (Fourier Bessel transform of ⊕^𝑘_𝐵 operator). One has ℱ_𝐵^𝑘_𝐵𝑢(𝑥)=𝑉^𝑘(𝑥)ℱ_𝐵𝑢(𝑥), (2.14) where 𝑉^𝑘(𝑥)=((^𝑝_𝑖=1𝑥²_𝑖)⁴−(^𝑝+𝑞_𝑗=𝑝+1𝑥²_𝑗)⁴)^𝑘. (2.15)

Proof. We can use the mathematical induction method; for 𝑘=1, we have ℱ_𝐵(_𝐵𝑢)(𝑥)=𝐶_{𝑣𝑅+𝑛}(_𝐵𝑢(𝑦))(^𝑛_𝑖=1𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖)𝑦^2𝜐𝑖_𝑖)𝑑𝑦=𝐶_{𝑣𝑅+𝑛}(⋄_𝐵⊚_𝐵𝑢(𝑦))(^𝑛_𝑖=1𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖)𝑦^2𝜐𝑖_𝑖)𝑑𝑦=𝐶_{𝑣𝑅+𝑛}(Δ²_𝐵+□²_𝐵)2𝑔(𝑦)(^𝑛_𝑖=1𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖)𝑦^2𝜐𝑖_𝑖)𝑑𝑦,𝑔(𝑦)=⋄_𝐵𝑢(𝑦)=ℱ_𝐵(Δ²_𝐵𝑔+□²_𝐵𝑔)2(𝑥)=((−1)²(𝑥²₁+⋯+𝑥²_𝑛)²+(−1)²(𝑥²₁+⋯+𝑥²_𝑝−𝑥²_𝑝+1−⋯−𝑥²_𝑝+𝑞)²)2ℱ_𝐵𝑔(𝑥)=((𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝)²+(𝑥²_𝑝+1+⋯+𝑥²_𝑝+𝑞)²)ℱ_𝐵⋄_𝐵𝑢(𝑥)=((^𝑝_𝑖=1𝑥²_𝑖)²+(^𝑝+𝑞_𝑗=𝑝+1𝑥²_𝑗)²)⋅((^𝑝_𝑖=1𝑥²_𝑖)²−(^𝑝+𝑞_𝑗=𝑝+1𝑥²_𝑗)²)ℱ_𝐵𝑢(𝑥)=((𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝)⁴−(𝑥²_𝑝+1+⋯+𝑥²_𝑝+𝑞)⁴)ℱ_𝐵𝑢(𝑥)=𝑉(𝑥)ℱ_𝐵𝑢(𝑥), (2.16)(1)(2)(3)(4)(5)(6)(7)(8) where 𝑉(𝑥)=(𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝)⁴−(𝑥²_𝑝+1+⋯+𝑥²_𝑝+𝑞)⁴. Then, from inverse Fourier transform we obtain _𝐵𝑢(𝑥)=ℱ_𝐵⁻¹𝑉(𝑥)ℱ_𝐵𝑢(𝑥). (2.17) Assume that the statement is true for 𝑘−1, that is, _𝐵^𝑘−1𝑢(𝑥)=ℱ_𝐵⁻¹𝑉^𝑘−1(𝑥)ℱ_𝐵𝑢(𝑥). (2.18) Then, we must prove that is also true for 𝑘∈𝑁. So we have ^𝑘_𝐵𝑢(𝑥)=_𝐵(_𝐵^𝑘−1𝑢(𝑥))=ℱ_𝐵⁻¹𝑉(𝑥)ℱ_𝐵ℱ_𝐵⁻¹𝑉^𝑘−1(𝑥)ℱ_𝐵𝑢(𝑥)=ℱ_𝐵⁻¹𝑉^𝑘(𝑥)ℱ_𝐵𝑢(𝑥). (2.19) This completes the proof.

Lemma 2.6. For 𝑡,𝑣>0 and 𝑥,𝑦∈ℝ^𝑛, one has ^∞₀𝑒^{−𝑐2𝑥2𝑡}𝑥^2𝜐𝑑𝑥=Γ(𝜐)2𝑐^2𝜐+1t^𝜐+1/2_∞0,𝑒^{−𝑐2𝑥2𝑡}𝐽_𝜐−1/2(𝑥𝑦)𝑥^2𝜐𝑑𝑥=Γ(𝜐+1/2)2(𝑐²𝑡)^𝜐+1/2𝑒^{−𝑦2/4𝑐2𝑡}, (2.20) where 𝑐 is a positive constant.

Proof. See [9].

Lemma 2.7. Let the operator 𝐿 be defined by 𝜕𝐿=𝜕𝑡−𝑐^2𝑘_𝐵, (2.21) where ⊕^𝑘_𝐵 is the operator iterated 𝑘-times and is given by ^𝑘_𝐵=((^𝑝_𝑖=1𝐵_𝑥𝑖)⁴−(^𝑝+𝑞_𝑗=𝑝+1𝐵_𝑥𝑖)⁴)^𝑘,𝐵_𝑥𝑖=𝜕²𝜕𝑥²_𝑖+2𝑣_𝑖𝑥_𝑖𝜕𝜕𝑥_𝑖, (2.22)𝑝+𝑞=𝑛 is the dimension ℝ⁺_𝑛, 𝑘 is a positive integer, (𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ⁺_𝑛, and 𝑐 is a positive constant. Then 𝐸(𝑥,𝑡)=𝐶_𝑣Ω+𝑒^{𝑐2𝑡[(𝑦21+⋯+𝑦2𝑝)4−(𝑦2𝑝+1+⋯+𝑦2𝑝+𝑞)4]𝑘𝑛}_𝑖=1𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖)𝑦^2𝜐𝑖_𝑖𝑑𝑦 (2.23) is the elementary solution of (2.21) in the spectrum Ω⁺⊂𝑅⁺_𝑛 for 𝑡>0.

Proof. Let 𝐿𝐸(𝑥,𝑡)=𝛿(𝑥,𝑡), where 𝐸(𝑥,𝑡) is the elementary solution of 𝐿 and 𝛿 is the Dirac-delta distribution. Thus 𝜕𝜕𝑡𝐸(𝑥,𝑡)−c^2𝑘_𝐵𝐸(𝑥,𝑡)=𝛿(𝑥)𝛿(𝑡). (2.24) Applying the Fourier Bessel transform, which is defined by (2.4) to the both sides of the above equation and using Lemma 2.5 by considering ℱ_𝐵𝛿(𝑥)=1, we obtain 𝜕ℱ𝜕𝑡^𝐵𝐸(𝑥,𝑡)−𝑐²[(𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝)⁴−(𝑥²_𝑝+1+⋯+𝑥²_𝑝+𝑞)⁴]^𝑘ℱ_𝐵𝐸(𝑥,𝑡)=𝛿(𝑡). (2.25) Thus, we get ℱ_𝐵𝐸(𝑥,𝑡)=𝐻(𝑡)𝑒^{𝑐2𝑡[(𝑥21+𝑥22+⋯+𝑥2𝑝)4−(𝑥2𝑝+1+⋯+𝑥2𝑝+𝑞)4]𝑘}, (2.26) where 𝐻(𝑡) is the Heaviside function, because 𝐻(𝑡)=1 holds for 𝑡≥0.
Therefore, ℱ_𝐵𝐸(𝑥,𝑡)=𝑒^{𝑐2𝑡[(𝑥21+𝑥22+⋯+𝑥2𝑝)4−(𝑥2𝑝+1+⋯+𝑥2𝑝+𝑞)4]𝑘}, (2.27) which has been already defined by (2.7). Thus from (2.5), we have 𝐸(𝑥,𝑡)=𝐶_𝑣ℝ+𝑛𝑒^{𝑐2𝑡[(𝑦21+⋯+𝑦2𝑝)4−(𝑦2𝑝+1+⋯+𝑦2𝑝+𝑞)4]𝑘𝑛}_𝑖=1𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖)𝑦^2𝜐𝑖_𝑖𝑑𝑦, (2.28) where Ω⁺ is the spectrum of 𝐸(𝑥,𝑡). Thus, we obtain 𝐸(𝑥,𝑡)=𝐶_𝑣Ω+𝑒^{𝑐2𝑡[(𝑦21+⋯+𝑦2𝑝)4−(𝑦2𝑝+1+⋯+𝑦2𝑝+𝑞)4]𝑘𝑛}_𝑖=1𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖)𝑦^2𝜐𝑖_𝑖𝑑𝑦 (2.29) as an elementary solution of (2.21) in the spectrum Ω⁺⊂𝑅⁺_𝑛 for 𝑡>0.

3. Main Results

Theorem 3.1. Let us consider the equation 𝜕𝜕𝑡𝑢(𝑥,𝑡)−𝑐^2𝑘_𝐵𝑢(𝑥,𝑡)=0 (3.1) with the initial condition 𝑢(𝑥,0)=𝑓(𝑥), (3.2) where ⊕^𝑘_𝐵 is the operator iterated 𝑘-times and is defined by ^𝑘_𝐵=((^𝑝_𝑖=1𝐵_𝑥𝑖)⁴−(^𝑝+𝑞_𝑗=𝑝+1𝐵_𝑥𝑖)⁴)^𝑘,𝐵_𝑥𝑖=𝜕²𝜕𝑥²_𝑖+2𝑣_𝑖𝑥_𝑖𝜕𝜕𝑥_𝑖, (3.3)𝑝+𝑞=𝑛 is the dimension ℝ⁺_𝑛,𝑘 is a positive integer, 𝑢(𝑥,𝑡) is an unknown function for (𝑥,𝑡)=(𝑥₁,𝑥₂,…,𝑥_𝑛,𝑡)∈ℝ⁺_𝑛×(0,∞),𝑓(𝑥) is the given generalized function, and 𝑐 is a positive constant. Then 𝑢(𝑥,𝑡)=𝐸(𝑥,𝑡)∗𝑓(𝑥) (3.4) is a solution of (3.1), which satisfies (3.2), where 𝐸(𝑥,𝑡) is given by (2.23). In particular, if one puts 𝑘=1 and 𝑞=0 in (3.1), then (3.1) reduces to the equation 𝜕𝜕𝑡𝑢(𝑥,𝑡)−𝑐²Δ⁴_𝐵𝑢(𝑥,𝑡)=0, (3.5) which is related to the Bessel heat equation.

Proof. Taking the Fourier Bessel transform, which is defined by (2.4), of both sides of (3.1) for 𝑥∈ℝ⁺_𝑛 and using Lemma 2.5, we obtain 𝜕ℱ𝜕𝑡^𝐵𝑢(𝑥,𝑡)=𝑐²((𝑥²₁+⋯+𝑥²_𝑝)⁴−(𝑥²_𝑝+1+⋯+𝑥²_𝑝+𝑞)⁴)^𝑘ℱ_𝐵𝑢(𝑥,𝑡). (3.6) Thus, we consider the initial condition (3.2); then we have the following equality for (3.6): 𝑢(𝑥,𝑡)=𝑓(𝑥)∗ℱ_𝐵⁻¹𝑒^{𝑐2𝑡[(𝑥21+⋯+𝑥2𝑝)4−(𝑥2𝑝+1+⋯+𝑥2𝑝+𝑞)4]𝑘}. (3.7) Here, if we use (2.4) and (2.5), then we have 𝑢(𝑥,𝑡)=𝑓(𝑥)∗ℱ_𝐵⁻¹𝑒^{𝑐2𝑡[(𝑦21+⋯+𝑦2𝑝)4−(𝑦2𝑝+1+⋯+𝑦2𝑝+𝑞)4]𝑘}=_ℝ+𝑛ℱ_𝐵⁻¹𝑒^{𝑐2𝑡[(𝑦21+⋯+𝑦2𝑝)4−(𝑦2𝑝+1+⋯+𝑦2𝑝+𝑞)4]𝑘}𝑇^𝑦_𝑥𝑓(𝑥)(^𝑛_𝑖=1𝑦^2𝜐𝑖_𝑖=)𝑑𝑦_ℝ+𝑛(𝐶_𝑣ℝ+𝑛𝑒^{𝑐2𝑡𝑉𝑘𝑛(𝑧)}_𝑖=1𝐽_{𝜐𝑖−1/2}(𝑦_𝑖,𝑧_𝑖)𝑧^2𝜐𝑖_𝑖𝑑𝑧)𝑇^𝑦_𝑥𝑓(𝑥)(^𝑛_𝑖=1𝑦^2𝜐𝑖_𝑖)𝑑𝑦, (3.8) where 𝑉(𝑧)=(𝑧²₁+𝑧²₂+⋯+𝑧²_𝑝)⁴−(𝑧²_𝑝+1+𝑧²_𝑝+2+⋯+𝑧²_𝑝+𝑞)⁴. Set 𝐸(𝑥,𝑡)=𝐶_𝑣ℝ+𝑛𝑒^{𝑐2𝑡[(𝑦21+⋯+𝑦2𝑝)4−(𝑦2𝑝+1+⋯+𝑦2𝑝+𝑞)4]𝑘𝑛}_𝑖=1𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖)𝑦^2𝜐𝑖_𝑖𝑑𝑦. (3.9) Since the integral in (3.9) is divergent, therefore we choose Ω⁺⊂ℝ⁺_𝑛 to be the spectrum of 𝐸(𝑥,𝑡), and by (2.21), we have 𝐸(𝑥,𝑡)=𝐶_𝑣ℝ+𝑛𝑒^{𝑐2𝑡[(𝑦21+⋯+𝑦2𝑝)4−(𝑦2𝑝+1+⋯+𝑦2𝑝+𝑞)4]𝑘𝑛}_𝑖=1𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖)𝑦^2𝜐𝑖_𝑖𝑑𝑦=𝐶_𝑣Ω+𝑒^{𝑐2𝑡[(𝑦21+⋯+𝑦2𝑝)4−(𝑦2𝑝+1+⋯+𝑦2𝑝+𝑞)4]𝑘𝑛}_𝑖=1𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖)𝑦^2𝜐𝑖_𝑖𝑑𝑦. (3.10) Thus (3.8) can be written in the convolution form 𝑢(𝑥,𝑡)=𝐸(𝑥,𝑡)∗𝑓(𝑥). (3.11) Moreover, since 𝐸(𝑥,𝑡) exists, we can see that lim_𝑡→0𝐸(𝑥,𝑡)=𝐶_𝑣Ω+^𝑛_𝑖=1𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖)𝑦^2𝑣𝑖_𝑖𝑑𝑦=𝐶_𝑣ℝ+𝑛^𝑛_𝑖=1𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖)𝑦^2𝑣𝑖_𝑖𝑑𝑦=𝛿(𝑥),for𝑥∈ℝ⁺_𝑛, (3.12) hold (see [8]). Thus for the solution 𝑢(𝑥,𝑡)=𝐸(𝑥,𝑡)∗𝑓(𝑥) of (3.1), then we have lim_𝑡→0𝑢(𝑥,𝑡)=𝑢(𝑥,0)=𝛿∗𝑓(𝑥)=𝑓(𝑥), (3.13) which satisfies (3.2). This completes the proof.

Theorem 3.2. The kernel 𝐸(𝑥,𝑡) defined by (3.10) has the following properties. (1)𝐸(𝑥,𝑡)∈𝒞^∞-the space of continuous function for 𝑥∈ℝ^𝑛, 𝑡>0 with infinitely differentiable. (2)(𝜕/𝜕𝑡−𝑐²⊕^𝑘_𝐵)𝐸(𝑥,𝑡)=0 for all 𝑥∈ℝ⁺_𝑛,𝑡>0.(3)lim_𝑡→0𝐸(𝑥,𝑡)=𝛿 for all 𝑥∈ℝ⁺_𝑛.

Proof. (1) From (3.10) and 𝜕^𝑛𝜕𝑡^𝑛𝐸(𝑥,𝑡)=𝐶_𝑣ℝ+𝑛𝜕^𝑛𝜕𝑡^𝑛𝑒^{𝑐2𝑡[(𝑦21+⋯+𝑦2𝑝)4−(𝑦2𝑝+1+⋯+𝑦2𝑝+𝑞)4]𝑘𝑛}_𝑖=1𝐽_{𝑣𝑖−(1/2)}(𝑥_𝑖,𝑦_𝑖)𝑦^2𝜐𝑖_𝑖𝑑𝑦. (3.14) we have 𝐸(𝑥,𝑡)∈𝐶^∞ for 𝑥∈ℝ⁺_𝑛,𝑡>0.
(2) We have 𝑢(𝑥,𝑡)=𝐸(𝑥,𝑡) since 𝑢(𝑥,𝑡)=𝐸(𝑥,𝑡)∗𝑓(𝑥) holds. Note here that we use the fact 𝑓(𝑥)=𝛿(𝑥) by the Fourier Bessel transformation. Then, we obtain (𝜕𝜕𝑡−𝑐^2𝑘_𝐵)𝐸(𝑥,𝑡)=0 (3.15) by direct computation.
(3) This case is obvious by (3.12).
In particular, if we put 𝑘=1 and 𝑞=0 in (3.1), then (3.1) reduces to the equation 𝜕𝜕𝑡𝑢(𝑥,𝑡)−𝑐²Δ⁴_𝐵𝑢(𝑥,𝑡)=0, (3.16) and we obtain the solution of (3.16) in the convolution form 𝑢(𝑥,𝑡)=𝐸(𝑥,𝑡)∗𝑓(𝑥), (3.17) where 𝐸(𝑥,𝑡) is defined by (2.23) with 𝑘=1 which is related to Bessel heat equation. This completes the proof.

Acknowledgments

The authors would like to thank The Thailand Research Fund and Graduate School, Maejo University, Thailand for financial support.