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## Computational & Applied Mathematics

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*On-line version* ISSN 1807-0302

### Comput. Appl. Math. vol.28 no.2 São Carlos 2009

#### http://dx.doi.org/10.1590/S1807-03022009000200002

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

**Amnuay Kananthai ^{I}; Kamsing Nonlaopon^{II,}^{*}**

^{I}Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand E-mail: malamnka@science.cmu.ac.th

^{II}Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand E-mail: nkamsi@kku.ac.th

**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*) = (*x*1, *x*2,..., *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*) = (*x*1, *x*2,..., *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 }= *x*_{1}^{2 }+ *x*_{2}^{2 }+...+ *x _{n}*

^{2}and

*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 anyt> 0 where C^{(2k)}(^{n }) is the space of continuous function with 2k-derivatives.(2) ƒ satisfies the Lipchitz condition, that is

where

Ais constant and 0 <A< 1.(3)

for

x= (x1,x2,...,xn)^{n},t(0, ) andu(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 = (ξ_{1},ξ_{2},...,ξ* _{n}*)

*,x*= (

*x*

_{1},

*x*

_{2},...,

*x*)

_{n}^{n}

*,*(ξ,

*x*) = ξ

_{1}

*x*

_{1}+ ξ

_{2}

*x*

_{2}+ ...+ ξ

_{n}x_{n}is the usual inner product in^{n }

*and d*=

_{x}*dx*

_{1}

*dx*

_{2}...

*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 = (ξ_{1},ξ_{2},...,ξ* _{0}*)

*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}*, *(*x*_{1}, *x*_{2},..., *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 L*E*(*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 }× R*by 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) = 0for t> 0.(3)

,

for,

where M(t)is a function of t in the spectrumΩandΓdenote 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) Cforx^{n},t> 0.(2) By computing directly, we obtain

.

(3) We have

By changing to bipolar coordinates

and

where and . Thus 1 1

where *d*ξ = *r*^{p-1}*s*^{q-1 }*dr ds d*Ω* _{p }d*Ω

*Ω*

_{q }, d*and Ω*

_{p }*q*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, thenSee [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> 0whereC^{(2k)}(^{n})is the space of continuous function with2k-derivatives.(2) ƒ

satisfies the Lipchitz condition, that is

where A is constant and0 <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)}(Ω_{0}) C^{(2k)}(10) for 0 __<__ *t *__<__ *T *.

Now (C^{(2k)}(Ω_{0}), ) 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)}(Ω_{0}). Since (C^{(2k)}(10), )is a Banach space and L : C^{(2k)}(Ω_{0}) C^{(2k)}(Ω_{0}) is a contraction mapping on C^{(2k)}(Ω_{0}), 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*) Ω_{0} × (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.

**REFERENCES **

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

[2] R. Haberman, "Elementary Applied Partial Differential Equations", 2^{nd }Edition, Prentice-Hall International, Inc. (1983). [ Links ]

[3] E. Kreyszig, "Introductory Functional Analysis with Applications", John Wiley & Sons Inc. (1978). [ Links ]

[4] K. Nonlaopon and A. Kananthai, *On the Ultrahyperbolic Heat Kernel*, International Journal of Applied Mathematics, **13 **(2) (2003), 215-225. [ Links ]

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

#752/08.

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