Vol. 44 No. 8

SCIENCE IN CHINA

(Series A)

August2001

On a nonhomogeneous Burgers' equation DING Xiaqi (TJ~_~:) ~, JIU Quansen (~i~ r

2 & HE Cheng (~'~

1. Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China; 2. Department of Mathematics, Capital Normal University, Beijing 100037, China Correspondence should be addressed to Jiu Quansen (email: jiuqs@mail. cnu. edu. cn) Received September 14, 2000 In this paper the Cauchy problem for the following nonhomogeneous Burgers' equation is considered: ( 1 ) u, + uux = tzu= - k x , x E R, t > 0, where/~ and k are positive constants. Since the nonhomogeneous term kx does not belong to any Lp(R) space, this type of equation is beyond usual Sobolev framework in some sense. By Hopf-Cole transformation, (1) takes the form (2) 9, - cp= = - x2~. With the help of the Hermite polynomials and their properties, ( ] ) and (2) are solved exactly. Moreover, the large time behavior of the solutions is also considered, similar to the discussion in Hopf's paper. Especially, we observe that the nonhomogeneous Burgers' equation (1) is nonlinearly unstable.

Abstract

Keywords: nonhomogeneousBurgers' equation, explicit solution, large time behavior, Hermite polynomials. 1

Introduction and some related results

We consider the following nonhomogeneous Burgers' equation ut + uu~ = f z u = -

kx, x E R,

t > O,

(1.1)

with initial data u(x,O) = Uo(X), (1.2) where ,u and k are positive constants. When k = 0 , ( 1 . 1 ) is the well-known Burgers' equation ut + uu~ = ,uu~, x E R , t > 0. (1.3) It is well known that few nonlinear partial differential equation(s) can be solved exactly. Burgers' equation is an exception. In 1950, Hopf[11 obtained the explicit solution for Burgers' equation and systematically studied the properties of the solution such as the large time behavior ( ~ > 0) and the behavior of solution when/~ ---,- 0 . To obtain exact solution, Hopf-Cole transformation was used in ref. [ 1 ], which is ~(x,t)

= exp{- ~Iudx

},

(1.4)

2F ~o~

(1.5)

OF

u

2/z (log~)~

Substituting ( 1.5 ) into ( 1 . 3 ) , Burgers' equation ( 1 . 3 ) can be transformed to the standard heat equation 9t = / ~ x 9 (1.6)

No. 8

ON A NONHOMOGENEOUS BURGERS' EQUATION

985

Therefore, the explicit solution of Burgers' equation ( 1 . 3 ) is obtained through the solution fc~rmula of ( 1 . 6 ) and transformation ( 1 . 5 ) . In this paper, we first deduce the explicit solution of ( 1.1 ), similar to the discussion in ref. [ 1 ]. It should be noted that the nonhomogeneous function kx on the fight side of ( 1.1 ) does not belong to any Lp ( R ) space. So ( 1 . 1 ) is beyond usual Sobolev framework in some sense. Let t "-~

t,

tt --'~ - - ,

X .---~. X .

Nonhomogeneous Burgers' equation ( 1.1 ) becomes ut + uux =

,uuxx- 4x,

x E R, t > 0,

(1.7)

Without loss of generality, we mainly consider the following type of nonhomogeneous Burgers ' equation u, + uu~ = u ~ - 4 x . (1.8) By ( 1 . 4 ) or ( 1 . 5 ) , eq. ( 1 . 8 ) can be rewritten as opt- 5% = - x2~p9 (1.9) Generally speaking, we cannot hope to get the exact solution of a parabolic type equation like q~, = ~,~,, + a ( x ) ~ (1.10) even for a smooth function a ( x ) , which shows that ( 1 . 8 ) is different from Burgers' equation. While by Hermite polynomial, we can explicitly solve ( 1 . 9 ) . This approach is firstly motivated by Ding and Luo' s recent work on a class of generalized functions and weak convergence of some series[2] ~). The second purpose of this paper is to study the large time behavior of ( 1. I ) and ( I. 9) respectively.

2

The Hermite polynomials

In this section, we recall some properties about the Hermite polynomials which will be used later (see ref. [3 ] for details). The Hermite polynomial of degree n is defined by 2 d+

+

= ( - l ) + e + -7---z, Ce-+ ) , clx

H,,Cx)

TM

and we write 1 2

n --lx2 d n

" - x 2"

~ ( x ) = e-~ xH~(x) = (- l) e2 ~x,(e

),

and r

Cn(x)

--

( 2 % !~-~ ) 89 Then, it is easy to verify that y = ~n ( x ) or y = #n ( x ) solves the following differential equation

d2y dx2

1 ) Ding, X. Q . ,

(2n + 1 ) y .

x2y -

_- _

I.~to, P. Z . , Weak convergence of some series, preprint.

(2.1)

986

SCIENCE IN CHINA (Series A)

Vol. 44

The following are some properties of the Hermite functions. Lelmma 2 . 1 . (see ref. [ 3 ] , p. 7 8 ) . The t r (- ~, ~),i.e.

f[ Lemma2.2 .=o

Let K ( x , y , t )

.r162

=

( s e e r e f . [ 3 ] , p. 7 9 ) .

tn~bn( X ) ~bn( y )

I

Ix2-y~

exp

~/~(1

-

t 2)

t

(x-yt)~l

-

2

1

t2

(2.2)

J"

Then

If f ( x ) is continuous in ( -

!imJ

ifm = n, if m # n

O,

Ifl t I < 1,then

denote the right term of ( 2 . 2 ) .

Lemma 2.3. large, then

{1,

form a normal orthogonal set over

zr

K(x,y,t)f(y)

+ m), andf(x)

= e~

for I x I

dy = f ( x ) .

(2.3)

- |

2xt

Proof.

Puttingy _ 1 + t 2 + u , then 1 l-t 2 2

f| -r

e 2 "--'Sx l§ f~

K(x,y,t)dy

- /ir(i__t2 ) ~l~t2e----

1+1 a

2

e-2(,_?)"du

_|

l - t 2 x~ 2 ( l + t 2) ~

1 ,

as t -~ 1. Therefore,

lim;"

K(x,y,t)f(y)dy

- f(x)

=

- |

;im;" -

K(x,y,t)[f(y)

- f(x)]dy.

oo

And after variable transformations, one has

J[: K ( x , y , t ) [ f ( y )

- f(x)]dy

_! l-? , e 21+,'~ f / ) . %/ -

2

l+/ 2f / 2xt ) 2c~-,) t S / l + t 2 + u - f ( x ) 1-t

2

'

a-(1 + t 2 ) e - ~ J - |

r

|

]

du

2[ ( 2xt / 2 ( - 1 - t2) / f ~ + t 2 + N 1 + t-s z ] - f ( x )

]

e-"

dz.

Noting t h a t f ( x ) = e ~ for large I x I , for any e > 0 a n d any fixed x E R , there exists a large enough number L ( E, x ) > 0 and a small number 81 ( e ) > 0 such that as I t - 1 I < 81,

,/

2

lxr

~r(1 + t2) e 2(1+'2) J,~,>Le-~

[ (f 2 ~x ,

+N

l + t2 z - f(x)

]1'

dz < - f .

(2.4) Moreover, using the fact that f ( x ) is uniformly continuous in [ - L, L ], one deduces that there exists another small number c72( r ) > 0 such that as I t - I I < c72, 2 )e-~'J 7r(1 + t 2

-

L e- IJ~ 1 - - ~ t 2 +

"-

+

t2

z -f(x)

dz

< ~-.(2.5)

No. 8

ON A NONHOMOGENEOUS BURGERS' EQUATION

987

For any r > 0, l e t 3 = mint31, 32t. As I t - 1 I < 3, from ( 2 . 4 ) and ( 2 . 5 ) one has

f'*

K(x,y,t)Ef(y)

- f(x)]dy

< ~.

Lemma 2 . 3 is then proved. Remark 1. In ref. [ 3 ] , ( 2 . 3 ) is established under the assumption that f ( x ) is continuous in ( - ~ , + ~ ) and f ( x ) vanishes outside a finite range. Lemma 2 . 4 . I f f ( x ) is any function of L 2 ( - ~ , ~ ) , and

am =

f(x)r

( m -- 0, 1, 2 , ' " ) ,

then

f(x)

!:f"

-

3

a,4bm(x)

•

--OD

m=0

' dx

= O.

Explicit solution and large time behavior of parabolic equation (1.9)

Firstly with the help of the Hermite polynomials, we solve the Cauchy problem for ( 1 . 9 ) with initial data

~(x,0)

~0(x).

=

(3.1)

Assume the solution has the following forms:

q~(x,t)

=

~a,(t)~b,(x),

(3.2)

n=0

and

go(x) = ~ aO.r nffi0

where an(0) = ao, and { r 2 . 1 , we have

are Hermite polynomials introduced in sec. 2. By Lemma

aOn =

I -**

q~o(y)r

(3.3)

Substituting ( 3 . 2 ) into ( 1 . 9 ) , it follows that

qh + ~,.~ = ~ a ( a ' , ( t )

+ (2n + 1 ) a , ( t ) ) ~ b , ( x ) - xe~-]~a,(t)r

nffiO

n=O

Let

a'n(t) + ( 2 n + 1 ) a n ( t ) = O, that is,

an(t) = a0ne -(2"+1)'. In view of ( 3 . 3 ) , we have ~O(x,t)

=

e-t ~-] CtOne-2ntcn ( X ) nffiO

n = 0

: .-':2. where

-

,, o-2,,,0<,,

(3.4)

988

SCIENCE IN CHINA (Series A)

K(x,y,t)

Vol. 44

= >', t " r 1 6 2

I t I < 1.

n=0

By Lemma 2 . 2 , 1

K(x,y,t)

- f i n ( 1 - t 2)

X2

exp

y2

2

(.~ -- ).t)2~

-

1

t2 J'

I t I < 1.

(3.5)

Therefore, the above formal deduction gives us Theorem 3.1. Suppose that ~P0( x ) is continuous in ( - ~ , ~ ) and (3.6)

Oo(X) = e ~

for large I x I. Then the function

~(x,t)

= e -t

K(x,y,

e-2')~0(y)

dy,

(3.7)

defines a smooth solution of ( 1 . 9 ) which satisfies lim ~ ( x , t ) = ~o(x).

(3.8)

t ~ O"

Here K ( x , y , e

-2') in ( 3 . 7 ) is defined in ( 3 . 5 ) .

Moreover, ifcpo(x) E Lx( - ~ ,

) , then

-4t

I ~O(X , t ) I ~

forx E Randt

e-t(1

e - 4 t ) - l / 2 e - 2 ( l~+ ex 2

--

11 r

It L~,

(3.9)

> 0, and

[I ~ ( x , t ) [ 1 L' <~ 2 e - ' ( 1 - e - " ) - ' If~po(X) E L |

)

[I g o ( X ) [ [ L', for t > 0.

~,~),then

(3.10)

r

I q~(x,t) I~< -,/-2e-'(1 + e -4t) 1/2e 2(11+ ') xz II (pO(X) I1 : , f o r x E R a n d t I> 0. Proof. Due to ( 3 . 6 ) ,

(3.11)

it is clear that the integral

~o(x,t) = e -t

I

K(x,y,e-2t)q~o(y)dy,

converges for all x E R and t > 0. Let G(x,y,t)

= e-'K(x,y,

e -2t) =_

e-~

H(x,y,

e-2t),

- e -4')

{x2where H ( x , y , t )

= exp

(x-

y2 2

-

yt) 2}

1 Gt-

t2

. We only need to verify that

G= = - x2 G.

(3.12)

Through some direct computation, one has -- -- e -4t 4 7 e - Z t ( x - ye-2t)(1 - e -4t) - 4e-4t(x - y e - 2t ~-~Gt : e - t H ( x , y , e - 2 t ) ( (1 1 e-4t) 3 / ~ (1 e -4' )2 / -

I - 1

-

e -4t

: e - ' H ( x , y ,e -2' )I (1- -~ e - f f ~ A -

4x2e -4t _ 4 x y e -2t - 4 x y e -6t + 4y2e-4t / (1 - e-4' )5/2 ]

ff-'~Gx = e - t H ( x , y , e -2t)

- x - e-4tx + 2 y e -2t (1 - e-4t) 3/2) "

( - X - e-4tx + 2 y e - 2 t ) 2 1

- e -4' Gxx = e - t H ( x , y , e - 2 ' ( ( l l e _ 4 - ~ - / 2

So

+

(1_

e-4t)5/2

]"

No. 8

ON A NONHOMOGENEOUS BURGERS' EQUA~ON

989

4'-~(G= - x2G) (-

- 1 - e -4'

2 y e - 2 ' ) 2 - x2(1 - e-4') 2 (1 - e -4t)Sn ]

X -

e-4tx +

-2t) (]- - e_4t)3 n -

= e-'H(x,y,e

4x2e -4' _ 4xye -2t _ 4xye -6t + 4y2e-4t I

_ e -#t

= e - ' H ( x , y , e -2')

(l le_~

2-

(1

-

e -4t)5/2

-

]

It is obvious that ( 3 . 1 2 ) is valid. And ( 3 . 8 ) is a direct consequence of Lemma 2.3 and ( 3 . 7 ) . In order to prove ( 3 . 9 ) and (3.11 ) we rewrite the integral ( 3 . 7 ) as =e-'

9(x,t)

f-

K(x,y,e-2t)fo(y)dy

e -t

~/~(1

_

e-")

Therefore if 90( x) E L ~ ( - ~ ,

-"~. x 2 ~

e

~),

l + e -4' x2

- 2(1-~ . ~ - ~ , -)

_|

q~o 1 + e -4t + z dz "

one has -4t

19(x,t )

I<~ e-t(1 -

e-4t)

-1/2

~ x 2 ) e - 2(l+e

II 9 0 ( 2 7 )

It L t , for x E R,

t > 0,

in particular, 1[ ~ ( x , t ) [ I

Ifcp0(x) 6 L | I ~(x,t)

L' <~ 2e-t(1 - e-4') -' 11 9o(X)11L', for t > 0.

~,~),wehold 1-e

2

1~<42e-t(1 + e-4t)-l/Ze-2(,+e-") x I[ 9o(X) tl L', for x 6 R, t > 0.

The proof of the theorem is finished. Remark 2. By the maximum principle, we know that the solution of ( 1 . 9 ) with initial condition ( 3 . 1 ) is unique in the function class { u ( x , t ) 11 u ( x , t ) I <_ C e c~2 , x E R , 0 < t ~< T}, where C is some positive constant [43 . 4

Explicit solution of nonhomogeneous

Burgers'

equation

{ 1.8 )

Let F be a function set defined by F = {u(x,t) I I u(x,t) I<~ C ( I x I+ 1), x E R, 0 < t ~< T t , where C is a constant. Our main result of this section can be stated as follows. Theorem 4 . 1 .

L~oc(R )

Suppose that uo( x ) E

and

~0 Uo(~)d~ = o(x 2)

(4.1)

for large I x I. Then |

u(x,t)

1

:|

=

{ K(x,y,e-2t)exp

lfy -

r

} oUO($)d$ dy

,

(4.2)

_0e

define a regular solution of ( 1 . 8 ) in the half plane t > 0 which satisfies the initial condition

? o

u(~,t)d~--~

?

Uo(~)d~ as t ~ 0.

(4.3)

o

In addition, if Uo( x ) is continuous, then u(x,t) --~ U o ( X ) as t ---," 0.

(4.4)

990

SCIENCE IN CHINA (Series A)

Vol. 44

Moreover, the regular solution u ( x, t ) of ( 1 . 8 ) , which belongs to F in some strip 0 < t ~< T and satisfies ( 4 . 4 ) , must coincide with one defined in ( 4 . 2 ) in the strip. Proof. The continuous function go(Y) = e x p { - l f ~ u o ( ~ ) d ~ }

(4.5)

is, in virtue of ( 4 . 1 ) , of the order g0(Y) = e ~ for large I y I. Then the integral

~(x,t)

=e-'

f-

= ~/n(1

K(x,y,e-2t)~o(y)dy e t - e -4')

e

-2(1+e-4') x J _ * *

e 2CI-,-")"

9o

1 + e -4t + z

dz,

converges for all x E R and t > 0. Furthermore for t > 0, ~ ( x , t) has continuous partial derivatives of any order, which are obtained by differentiating the Kernel K( x, y ,e-2t). Since ( x, t ) solves ( 1 . 9 ) and satisfies the conditions !imo~(x,t) = ~p0(x), the function

u(x,t)

=-2

~,

defines a solution to ( 1 . 8 ) for t > 0, and has continuous derivatives of all orders. Moreover, f" u($,t)d$

= - 21og~p(x,t) + 2 1 o g ~ ( 0 , t ) --~

o

- 21og~0(x) + 21ogq~(0,0) =

u0(~)d~:, as t ~ 0,

o

then ( 4 . 3 ) is proved. Now, in addition to ( 4 . 1 ) , if Uo(X) is continuous, we intend to prove ( 4 . 4 ) . First, we need

If f ( x ) and f ' ( x ) are both continuous in ( - o, , |

Lemma4.1. large I x I, then

ffxx

-

r

K(x,y,t)f(y)dy

=

K(x,y,t)f(y)

dy "-~ " ~ x f ( X ) ,

= e~

t--~ l, (4.6)

where K( x , y , t ) is defined in ( 3 . 5 ) . Proof.

Since f ( x )

Tx

--OD

= e~

it is obvious that the following equality is valid.

K(x,y,t)f(y)dr =

g ( x , y , t ) f ( y ) dr.

--W

'

Through direct computation, one has

a~-~K(x,y,t) = - x a~-~K(x,y,t) = - ~ K ( x , y , t )

t2x + 2 y t K ( x

1-

t2

,y,t),

- 11 -+ t t ( y + x ) K ( x , y , t ) .

(4.7)

No. 8

ON A NONHOMOGENEOUSBURGERS' EQUATION

991

So K(x,y,t)f(y)dy

=-

K(x,y,t)f(y)dy

1 tf | - I + t _| K ( x , y , t )

Notice that the integral

I

K(x,y,t)

(y + x)f(y)dy

(y + x)f(y)

dy.

is finite, one can deduce, with the

_at~

help of the integration by parts, that _

,y,t)f(y)dy--~f'(x),

t --~ 1,

where Lemma 2.3 has also been used. Now we can continue to prove Theorem 4 . 1 . If u0 ( x ) is continuous then 9o ( x ) is also continuous. Hence, by Lemma 4 . 1 , we have limg(x,t) = 9o(x). t~O

Note that

9~( x, t) =

K( x , y ,e -2') 9o( y ) d y ,

is valid for all x and all t > 0. Then one has u(x,t)

= - 2 9~(x,t) 9'o(x) 9 ( x , t ) ---~- 2 9o(X ) - u o ( x ) ,

as t --~ 0. When u ( x, t ) is regular, the uniqueness results from the equivalence of ( 1 . 8 ) and ( 1 . 9 ) and the uniqueness of solution to the Cauchy problem of ( 1 . 9 ) . If u E M , then we deduce that 2

9(x,t)

E M = {u(x,t)

I I u(x,t)

I<<. Ce c~xj , x E R , 0 < t <~ T}. By Remark2,

1 ~x0 Uo(~) d~ } has a unique s~176 we know that eq. ( 1 . 9 ) with initial data 9o(X ) = exp { - ~in the function class M for0 <~ t ~< T, where Uo( x ) is the initial data given in the theorem. Thus correspondingly, there is a unique solution to eq. ( 1 . 8 ) in F for 0 ~< t ~< T with initial data u0 ( x ). Moreover, we have (see Theorem 5.2 below) I u(x,t)-2x I~< C(I x I + l ) e -2'. Therefore, a regular solution of ( 1 . 8 ) , which belongs to M and satisfies ( 4 . 4 ) , coincides with u ( x , t ) defined in ( 4 . 2 ) . Remark 3. We conjecture that there would be no limitation on the uniqueness of solution of ( 1 . 8 ) ; that is, any regular solution of ( 1.8 ), which satisfies ( 4 . 4 ) , necessarily coincides with u ( x, t ) in ( 4 . 2 ) . This could result from the following conjecture: regular solution of ( 1.9) with positive initial data 9o( x ) is unique and coincides with 9 ( x , t ) in ( 3 . 7 ) .

5

Large time behavior of nonhomogeneous Burgers' equation ( 1.8) Denote the initial moment by uo(x)dx

From ( 1 . 5 ) , we deduce

992

SCIENCE IN CHINA (Series A)

(2log ~ o~o( == ))

=

Vol. 44

M.

Our first result of this section is Theorem 5 . 1 . Under the assumption of Theorem 4 . 1 , the solution u ( x, t) of ( 1 . 8 ) with the initial value Uo( x ) possesses a moment in the following sense:

f:

P. 7.

u ( x , t ) dx = |

Uo(x)dx,

;" -|

where P . V. means Cauchy integral principal value. Proof. The solution of ( 1 . 9 ) with initial value ~Po(x) has the representation (p(x,t)

= e -t

f-

K(x,y,e-2')~o(y)

-|

-t

-

e

~/rt(X - e -4t)

1-e

.

dr

x2~"

l+e

e 2(l+e-4') J _ |

,

:

2(l_e -4') " ~ 0

(

2 x e -2t

1 + e_4t 4- g

)

dz.

Then for any N > 0 and fixed t, one has ~ ( - N) IN u ( x , t ) d x =2log -N ~(N)

f

| _|

,+r

2

/_2Ne-2t

z 9 ~90[ "~ e_4"--+ ~

4-

z)dz

Let N --~ + ~ , it follows that

P.V.

q~o(- o* = 21Og~o(+ g)) -

f: |

;"-|

uo( x )dx

which implies our resuh. At last, we have Theorem S . 2 . Under the assumption of Theorem 4 . 1 , the solution u ( x, t) of ( 1 . 8 ) with the initial values Uo(X) has a limit state as t ---~+ ~ ; that is,

~,(x,t)

u(x,t)

= - 2 -~(;.t)

--- 2 x ,

(5.I)

t ~4- |

More precisely,

u(x,t)

-

2(11 -+ ec-4t - " ) x I <~ Ce-2',

z E

R,

t > 0,

(5.2)

and

I u(x,t)-2xl<~ Proof.

C ( I x I+ 1)e -zt, x E R , t > 0.

Applying the representation ( 3 . 7 ) , it is easily concluded that

4"-ne'q~(x,t) --* e-~ |

f

e-~0(z)

dz,t--~+ ~ .

_|

Noticing that

a-~K(x,y,t) = - x we have

(5.3)

t2x + 2ytK( x

1-

t2

,y,t),

(5.4)

No. 8

ON A NONHOMOGENEOUS BURGERS' EQUATION

l-~ -"' ~ f |

(1 + e -4' )

~/-Tre'gx ( X , t ) = _ (1 - e-4t) 3/2e-2(l+d'')'~ x j_| 1 - e-4Q-|

~

993

t+e*' , ( 2 x e -2t ) e 2(1_,-")" 9o 1 + e -4t + z dz

~1 + e -4t + z 90 1 + e -4t + z dz.

It is easy to see that the second term of the last equality is well defined. As t --~ + ~ , we deduce X2f ~ ..2 ~/-~e~gx(x,t)--~- xe-~! e-~-90(z)dz. (5.5) From ( 5 . 4 ) and ( 5 . 5 ) ,

we have

u(x,t)

which showed ( 5 . 1 ) .

=-

2 9~--"2x, t--~+ ~, 9 Moreover, ( 4 . 7 ) and direct calculation give one that p==

u(x,t)

2(1 + e -4') 1 - e -4t X

4e -2' --

1

-

e -4t

]_ f~

J

K( x , y ,e-2t ) 9o( y ) y d y K(x,y,e-Et)9o(Y)

dy

_at:

with 90 ( x ) defined in ( 4 . 5 ) . Then ( 5 . 2 ) and ( 5 . 3 ) can be obtained. Thus we complete the proof of Theorem 5 . 2 . R e m a r k 4. It is obvious that the limit state 2x does not satisfy the steady equation of ( 1 . 8 ) . Thus, the nonhomogeneous Burgers' equation ( 1 . 8 ) is nonlinearly unstable. Acknowledgements The authors would like to thank Profs. Peizhu Luo and Huijiang Zhao for their valuable discussions. This work was partially supported by Beijing Natural Sciences Foundation (Grant Nos. 1992002 and 1002004) and Beijing Education Committee Foundation, and partially supported by the National Youth Foundation of China.

References 1. 2. 3. 4.

Hopf, E . , The partial differential equation ut+ u u , = p u = , Comm. Pure Appl. Math., 1950, 3: 201--230. Ding, X. Q . , Luo, P. Z . , Generalized expansions in Hilbert space, Acta Mathematica Scientia, 1999, 19(3): 241--250. Titchmarsh, E . , Introduction to the Theory of Fourier Integrals, 2rid e d . , Oxford: Oxford University Press, 1948. Ladyzhenskaya, O. A . , Sclonnikov, V. A . , Ural' ceva, N. N . , Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, 1968.