On the generalized nonlinear ultra-hyperbolic heatequation related to the spectrum

Abstract

In this paper, we study the nonlinear equation of the form <img border=0 src="../../../../../../img/revistas/cam/v28n2/a02ent02.gif"> where is <img border=0 src="../../../../../../img/revistas/cam/v28n2/a02tex03.gif" align=absmiddle>the ultra-hyperbolic operator iterated k-times, defined by <img border=0 src="../../../../../../img/revistas/cam/v28n2/a02ent03.gif" align=absmiddle>, p + q = n is the dimension of the Euclidean space <img border=0 src="../../../../../../img/revistas/cam/v28n2/a02tex02.gif" align=absmiddle>n, (x, t) = (x1, x2,..., xn, t) <img border=0 src="../../../../../../img/revistas/cam/v28n2/a01ent09.gif" align=absmiddle><img border=0 src="../../../../../../img/revistas/cam/v28n2/a02tex02.gif" align=absmiddle>n× (0, <img border=0 src="../../../../../../img/revistas/cam/v28n2/a06tex01.gif">), k is a positive integer and c is a positive constant. On the suitable conditions for f , u and for the spectrum of the heat kernel, we can find the unique solution in the compact subset of <img border=0 src="../../../../../../img/revistas/cam/v28n2/a02tex02.gif" align=absmiddle>n × (0, <img border=0 src="../../../../../../img/revistas/cam/v28n2/a06tex01.gif">). Moreover, if we put k = 1 and q = 0 we obtain the solution of nonlinear equation related to the heat equation. Mathematical subject classification: 35L30, 46F12, 32W30.

ultra-hyperbolic heat equation; the Dirac delta distribution; the spectrum; Fourier transform

On the generalized nonlinear ultra-hyperbolic heatequation related to the spectrum

Amnuay Kananthai^I; Kamsing Nonlaopon^II,*

^IDepartment of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand E-mail: [e-mail]

^IIDepartment of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand E-mail: [e-mail]

ABSTRACT

In this paper, we study the nonlinear equation of the form

where is the ultra-hyperbolic operator iterated k-times, defined by

,

p + q = n is the dimension of the Euclidean space ⁿ, (x, t) = (x1, x2,..., xn, t)

ⁿ× (0, ), k is a positive integer and c is a positive constant.

On the suitable conditions for f , u and for the spectrum of the heat kernel, we can find the unique solution in the compact subset of ⁿ × (0, ). Moreover, if we put k = 1 and q = 0 we obtain the solution of nonlinear equation related to the heat equation.

Mathematical subject classification: 35L30, 46F12, 32W30.

Key words: ultra-hyperbolic heat equation, the Dirac delta distribution, the spectrum, Fourier transform.

1 Introduction

It is well known that for the heat equation

with the initial condition

where Δ = is the Laplace operator and (x, t) = (x1, x2,..., xn, t) n × (0, ), and ƒ is a continuous function, we obtain the solution

as the solution of (1.1).

Now, (1.2) can be written as u(x, t) = E(x, t) _*ƒ(x) where

E(x, t) is called the heat kernel, where |x|² = x₁² + x₂² +...+ x_n²and t > 0, see [1, p. 208-209].

Moreover, we obtain E(x, t) δ as t 0, where δ is the Dirac-delta distribution. We also have extended (1.1) to be the equation

where is the ultra-hyperbolic operator, defined by

.

We obtain the ultra-hyperbolic heat kernel

where p + q = n is the dimension of the Euclidean space ⁿ and i =. For finding the kernel E(x, t) see [4].

In this paper, we extend (1.4) to be the general of the nonlinear form

for (x, t)

ⁿ ×(0,) and with the following conditions on u and ƒ as follows,

(1) u(x, t) C^(2k)(ⁿ) for any t > 0 where C^(2k) (ⁿ ) is the space of continuous function with 2k-derivatives.

(2) ƒ satisfies the Lipchitz condition, that is

where A is constant and 0 < A < 1.

(3)

for x = (x1, x2,..., xn)

ⁿ, t (0, ) and u(x, t) is continuous function on ⁿ × (0, ).

Under such conditions of ƒ, u and for the spectrum of E(x, t), we obtain the convolution

as a unique solution in the compact subset of

ⁿ × (0, ) and E(x, t) is an elementary solution defined by (2.5).

2 Preliminaries

Definition 2.1.Let ƒ(x) -the space of integrable function in ⁿ. The Fourier transform of ƒ(x) is defined by

where ξ = (ξ₁,ξ₂,...,ξ_n),x = (x₁, x₂,..., x_n)

ⁿ, (ξ, x) = ξ₁x₁ + ξ₂x₂ + ...+ ξ_nx_n is the usual inner product in

ⁿ and d_x= dx₁dx₂ ... dx_n.

Also, the inverse of Fourier transform is defined by

Definition 2.2. The spectrum of the kernel E(x, t) defined by (2.5) is the bounded support of the Fourier transform for any fixed t > 0.

Definition 2.3. Let ξ = (ξ₁,ξ₂,...,ξ₀) be a point in

ⁿ and we write

.

Denote by

and

the set of an interior of the forward cone, and + denotes the closure of Γ+.

Let Ω be spectrum of E(x, t) defined by Definition 2.2 for any fixed t > 0 and Ω +. Let

be the Fourier transform of E(x, t) and define

Lemma 2.1. Let L be the operator defined by

where

is the ultra-hyperbolic operator iterated k-times defined by k

,

p + q = n is the dimension of

ⁿ, (x₁, x₂,..., x_n)

ⁿ, t (0, ), k is a positive integer and c is a positive constant. Then we obtain

as a elementary solution of (2.4) in the spectrum Ω

ⁿ for t > 0.

Proof. Let LE(x, t) = δ(x, t) where E(x, t) is the kernel or the elementary solution of operator L and δ is the Dirac-delta distribution. Thus

Take the Fourier transform defined by (2.1) to both sides of the equation, weobtain

.

Thus

where H(t) is the Heaviside function. Since H(t) = 1 for t > 0. Therefore,

which has been already defined by (2.3). Thus

where Ω is the spectrum of E(x, t). Thus from (2.3)

Definition 2.4. Let us extend E(x, t) to

ⁿ × Rby setting

3 Main Results

Theorem 3.1.The kernel E(x, t) defined by (2.5) have the following properties:

(1) E(x, t) C-the space infinitely differentiable.

(2)

E(x, t) = 0 for t > 0.

(3)

, for ,

where M(t) is a function of t in the spectrum Ω and Γ denote the Gamma function. Thus E(x, t) is bounded for any fixed t > 0.

(4)

Proof.

(1) From (2.5), since

.

Thus E(x, t) Cfor x

ⁿ, t > 0.

(2) By computing directly, we obtain

.

(3) We have

By changing to bipolar coordinates

and

where and . Thus 1 1

where dξ = r^p-1s^q-1 dr ds dΩ_p dΩ_q , dΩ_p and Ωq are the elements of surface area of the unit sphere in ^p and ^q respectively. Since 1

ⁿ is the spectrum of E(x, t) and we suppose 0 < r < R and 0 < s < L where R and L are constants. Thus we obtain

where

is a function of

, and .

Thus, for any fixed t > 0, E(x, t) is bounded.

(4) By (2.5), we have

Since E(x, t) exists, then

See [3, p. 396, Eq. (10.2.19b)].

Theorem 3.2. Given the nonlinear equation

for (x, t)

ⁿ × (0, ), k is positive number and with the following conditions on u and f as follows,

(1) u(x, t) C^(2k)(ⁿ) for any t > 0 where C^(2k)(ⁿ) is the space of continuous function with 2k-derivatives.

(2) ƒ satisfies the Lipchitz condition, that is

where A is constant and 0 < A < 1.

(3)

for x = (x1, x2,..., xn)

ⁿ, t (0, ) and u(x, t) is continuous function on

ⁿ × (0, ).

Then, for the spectrum of E(x, t) we obtain the convolution

as a unique solution of (3.3) for x 10 where 10 is an compact subset of

ⁿ, 0 < t < T with T is constant and E(x, t) is an elementary solution defined by (2.5) and also u(x, t) is bounded.

In particular, if we put k = 1 and q = 0 in (3.3) then (3.3) reduces to the nonlinear heat equation.

Proof. Convolving both sides of (3.3) with E(x, t) and then we obtain the solution

or

where E(r, s) is given by Definition 2.4.

We next show that u(x, t) is bounded on ⁿ × (0, ). We have

by the condition (3) and (3.1) where

Thus u(x, t) is bounded on ⁿ × (0, ).

To show that u(x, t) is unique, suppose there is another solution w(x, t) of equation (3.3). Let the operator

then (3.3) can be written in the form

.

Thus

.

By the condition (2) of the Theorem,

Let Ω0 × (0, T ] be compact subset of ⁿ × (0, ) and L : C^(2k)(Ω₀) C^(2k)(10) for 0 < t < T .

Now (C^(2k)(Ω₀), ) is a Banach space where u(x, t) C^(2k)(10) for 0 < t < T , given by

Then, from (3.5) with 0 < A < 1, the operator l is a contraction mapping on C^(2k)(Ω₀). Since (C^(2k)(10), )is a Banach space and L : C^(2k)(Ω₀) C^(2k)(Ω₀) is a contraction mapping on C^(2k)(Ω₀), by Contraction Theorem, see [3, p. 300], we obtain the operator L has a fixed point and has uniqueness property. Thus u(x, t) = w(x, t). It follows that the solution u(x, t) of (3.3) is unique for (x, t) Ω₀ × (0, T ] where u(x, t) is defined by (3.4).

In particular, if we put k = 1 and q = 0 in (3.3) then (3.3) reduces to the nonlinear heat equation

which has solution

where E(x, t) is defined by (2.5) with k = 1 and q = 0. That is complete of proof.

Acknowledgement. The authors would like to thank The Thailand ResearchFund for financial support.

Received: 07/III/08. Accepted: 08/III/09.

#752/08.

• [1] F. John, "Partial Differential Equations", 4^th Edition, Springer-Verlag, New York (1982).
• [2] R. Haberman, "Elementary Applied Partial Differential Equations", 2^nd Edition, Prentice-Hall International, Inc. (1983).
• [3] E. Kreyszig, "Introductory Functional Analysis with Applications", John Wiley & Sons Inc. (1978).
• [4] K. Nonlaopon and A. Kananthai, On the Ultrahyperbolic Heat Kernel, International Journal of Applied Mathematics, 13 (2) (2003), 215-225.

*

Supported by The Commission on Higher Education and the Thailand Research Fund(MRG5180058).

Publication Dates

• Publication in this collection
  14 July 2009
• Date of issue
  2009

History

• Received
  07 Mar 2008
• Accepted
  08 Mar 2009