# 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 [2] 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 [3].

Next, K. Nonlaopon and A. Kananthai (see [5]) 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 [6], 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

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

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

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

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

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

[6] 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