On the operator [[cross product].sup.k] related to nonlinear heat equation and its spectrum.

Introduction

It is well known that for the heat equation

[[partial derivative]/[partial derivative]t]u(x,t) = [c.sup.2][DELTA}u(x,t) (1.1)

with the initial condition

u(x,0 = f(x)

where [DELTA] = [n.summation over (i=1)][[partial derivative].sup.2]/[partial derivative][x.sup.2.sub.i] is the Laplace operator and (x,t) = ([x.sub.1],[x.sub.2],...,[x.sub.n],t) [member of] [R.sup.n] x (0,[infinity]), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

as the solution of (1.1). Now, (1.2) can be written u(x,t) = E(x,t) x f(x) where

E(x,t) = [1/[(4[c.sup.2][pi]t).sup.n/2]]exp(-[[absolute value of x].sup.2]/4[c.sup.2]t) (1.3)

E(x,t) is called the heat kernel, where [[absolute value of x].sup.2] = [x.sup.2.sub.1] + [x.sup.2.sub.2] + ... + [x.sup.2.sub.n] and t > 0 see [1, p208-209].

In 1996, A. Kananthai  has introduced the Diamond operator [??] defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [??] can be written as then product of the operators in the form [??] = [DELTA][] = [][DELTA] where [DELTA] = [n.summation over (i=1)][[partial derivative].sup.2]/[partial derivative][x.sup.2.sub.i] is the Laplacian and [] = [([p.summation over (i=1)][[partial derivative].sup.2]/[partial derivative][x.sup.2.sub.i]).sup.2] - [([p+q.summation over (j=p+1)][[partial derivative].sup.2]/[partial derivative][x.sup.2.sub.j]).sup.2] is the ultra-hyperbolic. The Fourier transform of the Diamond operator also has been studied and the elementary solution of such operator, see .

Next, K. Nonlaopon and A. Kananthai (see ) study the equation

[[[partial derivative].sup.2]/[partial derivative]t]u(x,t) = [c.sup.2][]u(x,t)

In this paper, we study the nonlinear equation

[[partial derivative]/[partial derivative]t]u(x,t) - [c.sup.2][(-[cross product]).sup.k]u(x,t) = f(x,t,u(x,t)) (1.4)

The operator [[cross product].sup.k] can be expressed in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is in the form of nonlinear heat equation. We consider the equation (1.4) with the following conditions on u and f as follows.

1. u(x,t) [member of] [C.sup.(6k)]([R.sup.n]) for any t > 0 where [C.sup.(6k)]([R.sup.n]) is the space of continuous function with 6k-derivatives.

2. f satisfies the Lipchitz condition,

[absolute value of f(x,t,u) - f(x,t,w)] [less than or equal to] A[absolute value of u - w]

where A is constant with 0 < A < 1.

3. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for x = ([x.sub.1],[x.sub.2],...,[x.sub.n]) [member of] [R.sup.n], 0 < 1 < [infinity] and u(x,t) is continuous function on [R.sup.n] x (0,[infinity]).

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

u(x,t) = E (x,t) x f(x,t,u(x,t))

as a unique solution of (1.4) where E(x,t) is an elementary solution of (1.4).

Preliminaries

Definition 2.1 Let f(x) [member of] [L.sub.1]([R.sup.n])--the space of integrable function in [R.sup.n]. The Fourier transform of f(x) is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

where [xi] = ([[xi].sub.1],[[xi].sub.2],...,[[xi].sub.n]), x = ([x.sub.1],[x.sub.2],...,[x.sub.n]) [member of] [R.sup.n],([xi],x) = [[xi].sub.1][x.sub.1] + [[xi].sub.2][x.sub.2] + ... + [[xi].sub.n][x.sub.n] is the usual inner product in [R.sup.n] and dx = d[x.sub.1]d[x.sub.2]...d[x.sub.n]. Also, the inverse of Fourier transform is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

If f is a distribution with compact supports by , Theorem 7.4-3, p. 187 Eq. (2.1) can be written as

[??]([xi]) = [1/[(2[pi]).sup.n/2]]<f(x),[e.sup.-i([xi],x)]>. (2.3)

Definition 2.2 The spectrum of the kernel E(x,t) defined by (2.6) is the bounded support of the Fourier transform [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any fixed t > 0.

Definition 2.3 Let [xi] = ([[xi].sub.1],[[xi].sub.2],...,[[xi].sub.n]) be a point in [R.sup.n] and write

u = [[xi].sup.2.sub.1] + [[xi].sup.2.sub.2] + ... + [[xi].sup.2.sub.p] - [[xi].sup.2.sub.p+1] - [[xi].sup.2.sub.p+2] -... - [[xi].sup.2.sub.p+q], p + q = n.

Denote by [[GAMMA].sub.+] = {[xi] [member of] [R.sup.n]: [[xi].sub.1] > 0 and u > 0} the set of an interior of the forward cone and denote by [[bar.[GAMMA]].sub.+] the closure of [[bar.[GAMMA]].sub.+]. Let [OMEGA] be the spectrum of E(x,t) for any fixed t > 0 and [OMEGA] [subset] [[bar.[GAMMA]].sub.+]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the Fourier transform of E(x,t) and define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

Lemma 2.1 The Fourier transform of [(-[cross product]).sup.k][delta])

F[(-[cross product]).sup.k][delta] = [[(- 1).sup.4k]/[(2[pi]).sup.n/2]][[([[xi].sup.2.sub.1] + [[xi].sup.2.sub.2] +...+ [[xi].sup.2.sub.p]).sup.3] - [([[xi].sup.2.sub.p+1] + [[xi].sup.2.sub.p+2] +...+ [[xi].sup.2.sub.p+q]).sup.3]].sup.k]

where F is the Fourier transform defined by Eq. (2.1) and if the norm of [xi] is given by [parallel][xi][parallel] = [([[xi].sup.2.sub.1] + [[xi].sup.2.sub.2] + ... + [[xi].sup.2.sub.n]).sup.1/2] then

F[(-[cross product]).sup.k][delta] [less than or equal to] [3/[(2[pi]).sup.n/2]][[parallel][xi][parallel].sup.6k]

that is F[(-[cross product]).sup.k] is bounded and continuous on the space S' of the tempered distribution.

Moreover, by Eq. (2.2)

[(-[cross product]).sup.k][delta] = [F.sup.- 1][1/[(2[pi]).sup.n/2]][[([[xi].sup.2.sub.1] + [[xi].sup.2.sub.2] +...+ [[xi].sup.2.sub.p]).sup.3] - [([[xi].sup.2.sub.p+1] + [[xi].sup.2.sub.p+2] +...+ [[xi].sup.2.sub.p+q]).sup.3]].sup.k]

Proof. By Eq. (2.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By keeping on operator (-[cross product]) with k-1 times, we obtain

F[(-[cross product]).sup.k][delta] = [[(- 1).sup.4k]/[(2[pi]).sup.n/2]][[([[xi].sup.2.sub.1] + [[xi].sup.2.sub.2] +...+ [[xi].sup.2.sub.p]).sup.3] - [([[xi].sup.2.sub.p+1] + [[xi].sup.2.sub.p+2] +...+ [[xi].sup.2.sub.p+q]).sup.3]].sup.k]

Now,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [parallel][xi][parallel] = [([[xi].sup.2.sub.1] + [[xi].sup.2.sub.2] + ... + [[xi].sup.2.sub.n]).sup.1/2], [[xi].sub.i](i = 1,2,...,n) [member of] R. Hence we obtain F[cross product]S is bounded and continuous on the space S' of the tempered distribution.

Since F is 1-1 transformation from the space S' of the tempered distribution to the real space R, then by (2.2)

[cross product][delta] = [F.sup.-1][1/[(2[pi]).sup.n/2]][([[xi].sup.2.sub.1] + [[xi].sup.2.sub.2] +...+ [[xi].sup.2.sub.p]).sup.3] - [([[xi].sup.2.sub.p+1] + [[xi].sup.2.sub.p+2] +...+ [[xi].sup.2.sub.p+q]).sup.3]]

That completes the proof.

Lemma 2.2 Let L be the operator defined by

L = [[partial derivative]/[partial derivative]t] - [c.sup.2][(-[cross product]).sup.k] (2.5)

where [[cross product].sup.k] is the operator iterated k-times defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

p + q = n is the dimension of [R.sup.n],([x.sub.1],[x.sub.2],...,[x.sub.n]) [member of] [R.sup.n], t [member of] (0,[infinity]),k is a positive integer and c is the positive constant. Then we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

as the elementary solution of (1.4) in the spectrum [OMEGA] [subset] [R.sup.n] for t > 0, where

[p+q.summation over (j=p+1)][[xi].sup.2.sub.j] > [p.summation over (i=1)][[xi].sup.2.sub.i].

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

[[partial derivative]/[partial derivative]t]E(x,t) - [c.sup.2][(-[cross product]).sup.k]E(x,t) = [delta](x)[delta](t)

take the Fourier transform defined by (2.1) to both sides of the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By (2.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where [OMEGA] is the spectrum of E(x,t). Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for t > 0.

Definition 2.4 We can extend E(x,t) to [R.sup.n] x R by setting

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 2.3 (The properties of E(x,t)

The kernel E(x,t) defined by (2.6) have the following properties

1. E(x,t) [member of] [C.sup.[infinity]]--the space of continuous function for x [member of] [R.sup.n], t > 0 with infinitely detterentiable.

2. ([[partial derivative]/[partial derivative]t] - [c.sup.2][(-[cross product]).sup.k])E(x,t) = 0 for t > 0.

3. [absolute value of E(x,t)] [less than or equal to] [[2.sup.2- n]/[[pi].sup.n/2]][M(t)/[[pi].sup.1/2][GAMMA](p/2)[GAMMA](q/2)] for t > 0 where M(t) is a function of t in the spectrum and [GAMMA] denote the Gamma function. Thus E(x,t) is bounded for any fixed t > 0.

4. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. (1) From (2.6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus E(x,t) [member of] [C.sup.[infinity]] for x [member of] [R.sup.n], t > 0.

(2) By computing directly, we obtain

([[partial derivative]/[partial derivative]t] - [c.sup.2][(-[cross product]).sup.k])E(x,t) = 0

(3) We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By chaning to bipolar coordinates

[[xi].sub.1] = r[w.sub.1], [[xi].sub.2] = r[w.sub.2],...,[[xi].sub.p] = r[w.sub.p] and [[xi].sub.p+1] = s[w.sub.p+1], [[xi].sub.p+2] = s[w.sub.p+2],...,[[xi].sub.p+q] = s[w.sub.p+q]

where [p.summation over (i=1)][w.sup.2.sub.i] = 1 and [p+q.summation over (j=p+1)][w.sup.2.sub.j] = 1

Thus

[p.summation over (i=1)][w.sup.2.sub.i] = 1 and [p+q.summation over (j=p+1)][w.sup.2.sub.j] = 1

where d[xi] = [r.sup.p-1][s.sup.q-1]drdsd[[OMEGA].sub.p]d[[OMEGA].sub.p] and d[[OMEGA].sub.q] are the elements of surface area of the unit sphere in [R.sup.p] and [R.sup.q] respectively. Since [OMEGA] [subset] [R.sup.n] is the spectrum of E(x,t) and suppose 0 [less than or equal to] r [less than or equal to] R and 0 [less than or equal to] s [less than or equal to] L where R and L are constants. Thus we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7)

where M(t) = [[integral].sup.R.sub.0][[integral].sup.L.sub.0]exp[[c.sup.2]t[([r.sup.6] - [s.sup.6]).sup.6]][r.sup.p-1][s.sup.q-1]drds is a function for t > 0, [[OMEGA].sub.p] = [2[pi].sup.p/2]/[GAMMA](p/2) and [[OMEGA].sub.q] = [2[pi].sup.q/2]/[GAMMA](q/2). Thus for any fixed t > 0, E(x,t) is bounded.

(4) From (2.5),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for x [member of] [R.sup.n], see[4, p. 396, Eq. (10.2.19b)].

Main Results

Theorem 3.1 Given the nonlinear equation

[[partial derivative]/[partial derivative]t]u(x,t) - [c.sup.2][(-[cross product]).sup.k]u(x,t) = f(x,t,u(x,t)) (3.1)

for (x,t) [member of] [R.sup.n] x (0,[infinity]), k is a positive number and with the following conditions on u and f as follows

1. u(x,t) [member of] [C.sup.(6k)]([R.sup.n]) for any t > 0 where [C.sup.(6k)]([R.sup.n]) is the space of continuous function with 6k-derivative.

2. f satisfies the Lipchitz condition,

[absolute value of f(x,t,u) - f(x,t,w)] [less than or equal to] A[absolute value of u - w]

where A is constant with 0 < A < 1.

3. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for x = ([x.sub.1],[x.sub.2],...,[x.sub.n]) [member of] [R.sup.n], 0 < 1 < [infinity] and u(x,t) is continuous function on [R.sup.n] x (0,[infinity]).

Then obtain the convolution

u(x,t) = E(x,t) x f(x,t,u(x,t)) (3.2)

as a unique solution of (3.1) for x [member of] [OMEGA] where [OMEGA] is a compact subset of [R.sup.n] and 0 [less than or equal to] t [less than or equal to] T with T is constant and E(x,t) is an elementary solution defined by (2.6) and also u(x,t) is bounded for any fixed t > 0. In particular, if we put k = 1 and p = 0 in (3.1), then (3.1) reduces to the nonlinear hear equation

[[partial derivative]/[partial derivative]t]u(x,t) + [c.sup.2][[DELTA].sup.3]u(x,t) = f(x,t,u(x,t))

which is relate to the heat equation.

Proof. Convolving both sides of (3.1) with E(x,t), that is

E(x,t) x [[[partial derivative]/[partial derivative]t]u(x,t) - [c.sup.2](- [[cross product]).sup.k]u(x,t)] = E(x,t) x f(x,t,u(x,t))

or

[[[partial derivative]/[partial derivative]t]E(x,t) - [c.sup.2](-[[cross product]).sup.k]E(x,t)] x u(x,t) = E(x,t) x f(x,t,u(x,t)),

so

[delta](x,t) x u(x,t) = E(x,t) x f(x,t,u(x,t))

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where E(r,s) is given by definition (2.5). We next show that u(x,t) is bounded on [R.sup.n] x (0,[infinity]). We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus u(x,t) is bounded on [R.sup.n] x (0,[infinity]). To show that u(x,t) is unique. Now, we next to show that u(x,t) is unique. Let w(x,t) be another solution of (3.1), then

w(x,t) = E(x,t) x f(x,t,w(x,t))

for (x,t) [member of] [[OMEGA].sub.0] x (0,T] the compact subset of [R.sup.n] x [0,[infinity]) and E(x,t) is defined by (2.6).

Now, define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by (2.6) and the condition (2) of the theorem. Now, for (x,t) [member of] [[OMEGA].sub.0] x (0,T] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

where V([[OMEGA].sub.0]) is the volume of the surface on [[OMEGA].sub.0].

Choose A[absolute value of E(r,s)]TV([[OMEGA].sub.0]) [less than or equal to] 1 or A [less than or equal to] 1/[absolute value of E(r,s)]TV([[OMEGA].sub.0]).

Thus from (3.3),

[parallel]u - w[parallel] [less than or equal to] [alpha][parallel]u - w[parallel] where [alpha] = A[absolute value of E(r,s)]TV([[OMEGA].sub.0]) [less than or equal to] 1.

It follows that [parallel]u - w[parallel] = 0, thus u = w.

That is the solution u of (3.1) is unique.

In particular, if we put k = 1 and q = 0 in (3.1), then (3.1) reduces to the nonlinear heat equation

[[partial derivative]/[partial derivative]t]u(x,t) + [c.sup.2][[DELTA].sup.3]u(x,t) = f(x,t,u(x,t))

which has solution

u(x,t) = E(x,t) x f(x,t,u(x,t))

where E(x,t) is defined by (2.6) with k = 1 and q = 0.

Acknowledgement

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

References

 F. John,\Partial Differential Equations", 4th Edition, Springer-Verlag, New York, (1982).

 Kananthai, On the Solution of the n-Dimensional Diamond Operator, Applied Mathematics and Computational 88:27-37(1997).

 Kananthai, On the Fourier Transform of the Diamond Kernel of Marcel Riesz, Applied Mathematics and Computation 101:151-158(1999)..

 R. Haberman,\Elementary Applied Partial Differential Equations", 2nd Edition, Prentice-Hall International, Inc. (1983).

 K. Nonlaopon, A. Kananthai, On the Ultra-hyperbolic heat kernel, Applied Mathematics Vol.13 No.2 2003, 215-225.

 H. Zemanian, Distribution Theory and Transform Analysis, McGraw-Hill, New York, 1965.

Wanchak Satsanit and Amnuay Kananthai

Department of Mathematics, Chiang Mai University, Chiang Mai, 50200 Thailand. E-mail: aunphue@live.com
Author: Printer friendly Cite/link Email Feedback Satsanit, Wanchak; Kananthai, Amnuay Global Journal of Pure and Applied Mathematics Report 9THAI Dec 1, 2009 3024 Polya theory for orbiquotient sets. Some of properties of intuitionistic fuzzy normal subrings. Differential equations, Nonlinear Euclidean geometry Geometry, Plane Geometry, Solid Heat equation Nonlinear differential equations Operator theory Space Space and time