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interaction, the change in the momentum p of the colliding photon is not small. But the

change in its energy \p\ is small, (due to the exponential decay of the term S(p,p')e~l

when

|p|

—

|p'

is large). Therefore, the main contribution in the integral comes now from

the region ||p|

—

|p'|| small. Therefore, in order to obtain the Kompaneets equation, the

Taylor's expansion is performed in terms of |p| instead of

This introduces a boundary

at the point \p\ = 0. This boundary arises from the very nature of the photon electron

interaction and plays an important role in all the following (cf. in particular Remark 3

below).

Finally, let us emphasize that, in the Kompaneets equation the nonlinearity comes from

the terms (1 + /) i.e. quantum effects.

Remark 1. If we do not assume that the particles of one of the two species (the light

particles in the first example or the electrons in the second) are at equilibrium, the

approximation may still be performed but it gives rise to a more complicated integro

differential equation. In the simplest case, we may consider just one specie of interacting

classical particles. The system reduces then to one single equation

QiAfJ)= [ I f w[f(

/.>W,t)-f(p.,t)f(p,t)]dp

dpfdj/

For some particular types of collisions, the measure W is still concentrated in the region

where p

—

p' is small. This allows to perform again formally a Taylor's expansion in the

integral collision.

A simple example for classical particles undergoing grazing collisions with Coulomb

potential and with radially symetric distribution is the following:

W£{/

<X,fc

(

')[|f(i.0/(i'.*)-/(a:.*)£(i'.0]«&'}

k(x,x') = mm{x

3/2

,x'

3/2

}

which can also be written as:

/•oo

A(f,x) = a(x) k(x,x')f(x')dx',

B(f,x) = ^M fmin^/yi/J}/^,^

^ Jo

c{x)

M^l

(c.f. [V2]

and, in

particular

the

ocurrence

or not of

finite time blow

up ).

Notice that

Kompaneets equation could

written

as:

fc +

fe2

2fe

V)g

f).

The study

Boltzmann Compton

and

Kompaneets equations,

has

deserved

large

amount

interest

physical litterature from

the

late

50's to the mid 70's: [Dr],

[ZL],

[C],

[CCS], [CL]. They have been considered either

formal asymptotics

or by

numerical

and

computational methods.

The

Kompaneets equation

currently used

describe

the

distorsions

in the

cosmic background radiation

(cf. [P] and [S]).

Some properties

Boltzmann Compton

and

Kompaneets

equations

The Boltzmann Compton and the Kompaneets equation present some joint features

and some differences that

briefly describe below.

First

all,

we consider

the

entropy

r 5(/)=y°°

(/(fe),fc)/c

dfc,

s(x,

fc)

+ x)

ln(l

+ x)

—

x hix

—

then, along

the

trajectories

the

two

equations

have:

|W))>0.

the

other hand,

very important property

the Boltzmann Compton equation

(1)

is that

the

total number

particles

constant,

i.e.:

«>

f(k,t)dk

= 0.

If we want that property

to be

true also

for the

solutions

to the

Kompaneets equation,

which

is a

second order P.D.E

in the

domain

k > 0, we

need

flux condition

at fc = 0

and

—>

+oo. Indeed, integrating

parts

least formally gives:

|°° /(fc,

t)k

J"±{k^

f)}dk.

We have then

impose

the

flux condition:

(5)

(% + f

+ f)-¥0 as

fc->-0

and fe -> oo,

ofe

get (4). The

Kompaneets equation

has

been considered with condition

(5) in [CL]

from

numerical

and

formal point

vue.

Finally, the functions

are steady states of the two equations. Notice nevertheless that for every // > 0 :

/ f„(k)k

dk< /

(k)k

This shows that there are no such steady states for a total mass greater than N

. We

have then to consider also the distributions F

such that

{k) = -

-_- + a8

, a>0

—

which are also stationary solutions of the kinetic Boltzmann Compton equation. Notice

that the ocurrence of Dirac measure is for the function k

f(k) which is the density

function in the radial variable k since J f(p)dp =

u>„

j f(k)k

dk.

It was already observed by Bose and Einstein (cf. [B], [El], [E2]) that for systems

of bosons in thermal equilibrium a careful analysis of the statistical physics of the

problem leads to enlarge the class of steady distributions and to include also the solutions

containing a Dirac mass at the origin.

The Cauchy Problem and asymptotic behaviour of solutions

for (1) and (2)

We expose in this section our main results, which show first that the Cauchy problem for

the two equations (1) and (2) are globally well posed. We then consider the asymptotic

behaviour of the solutions as t

—>

+00. We prove that the solutions of the Boltzmann

Compton equation undergo Bose condensation in infinite time if the density of particles

is larger that N

. Finally we show that for some initial data, the Kompaneets equation

is a good approximation of the Boltzmann Compton equation in the whole space k €

(0,

+00) and for all t > 0.

Let us define

f°° k

and consider the set:

poo

= {heL

(0,oo),h>0, / (1 + k)h(k)dk < 00}.

Theorem 1. [EM2] Iffi

£ £0 there exists a unique global solution f 6 C([0, 00), £

)

to the Boltzmann Compton equation, such that

/ f(k,t)k

dk=

(k)k

=: m Vt > 0.

Moreover

rod

(i) ifm = Na : lim / \f(k, t) - fjk)\k

dk = 0

(ii)ifm > N

f{-,

t) -+

+ {N

- m)5 and

lim f \f(k,t)-f

(k)\k

dk =

0,\/k

>0.

'-*

Jk>k

Remark 2. Theorem 1 shows that the Cauchy problem for Boltzmann Compton

is stable: we have existence of global solutions which converge asymptotically to the

equilibrium states.

Theorem 2 . [EHV] Assume that fi

is a bounded and continuous function over

(0,

ex)),

such that lim k

(k) = 0. Then there exists a unique global classical solution

k—*oo

to the Kompaneets equation (2) such that f(k, 0) = fi

(k) such that, for some constant

/(*,*)<§ Vfce(0,l),Vi>0,

< df

imfe

(

+ / +

)=0

fc->oo Ok

Moreover, there are fi

, bounded and continuous over (0,+oo), such that for some

T* > 0 the unique solutions satisfies:

lim fc

(|( + / +/

) = 0 Vt€(0,r*)

k-tO OK

lim k

(% + f + f

) > 0

\ft>T*.

fc-s-o ok

For some f^, steady state of the Kompaneets equation:

f(k, t) —^f/j,(k) uniformly on compacts sets of R

Remark 3. The Cauchy problem for Kompaneets equation is well posed with the

conditions imposed in (6) only. In particular, no boundary condition is needed at

k = 0. This is due to the degeneracy of the elliptic operator in Kompaneets equation

at k = 0.

On the other hand, the Cauchy problem for the Kompaneets equation with condition

(5),

which ensures the conservation law (4) to hold, is unstable for a large class of initial

data. This is not so surprising since, as it was said above, the Kompaneets equation is an

approximation of the Boltzmann Compton but only in the region k/m

—>

0. Therefore,

in principle, the solutions of the Kompaneets equation describe the photon density only

in that region, and there is no particular reason for these solutions to satisfy the global

condition (4). It is actually shown in [EHV] that the solutions constructed in the second

part of Theorem 2 are driven by the following Burgers-type equation:

They behave like smoothed shock waves moving to the left as t increases. More precisely,

there exists some positive constants C\, C2 and C3, depending on the initial data f

such that:

-+ if z>C

0 if 0<z<C

uniformly on compact subsets of (C2,oo) and (0, C2)

\imx

f(k,T*) = C

>0.

fc->0

This makes that the boundary condition (3) at the origin, is violated at the origin in

finite

time.

This in turn, implies that condition (4) is not fulfilled anymore.

Theorem 1 and Theorem 2 show that, for some initial data, the Kompaneets equation

can not be a good approximation of Boltzmann Compton globally in the whole region

k e (0, +00) and for all time t > 0.

The next result shows, on the other hand, that this may be true for some initial data.

Theorem 3. [EM2] Assume that

b(k, k') = e

and take a e £>(R) even, suppa C [—2,2], a > 0 on

[—1,1]

with J

°° a{z) dz = 1,

/ z

a(z) dz = 2. Define

(k,k') = b(k,k')

ae[z) =

(£).

For 0 < fi

< /o, let f

e C([0,00), L

(0,00)) be the solution to the Boltzmann equation

(1) corresponding to the cross-section b

and the initial datum fi

which is given by

Theorem 1. Then, for all T > 0,

lim

||fe

- fc

/*:||c([o,T],£2(o,oo)) = °.

fx is the unique solution to the Cauchy problem

dfic d ,

dfK \

-df= dk^ (fK

-8k)}>

/*(*,<>) = /«.(*) for k>0.

Remark 4. Since f

< f

, the function f

is the unique solution of the Kompaneets

equation with initial data fi

satisfying

(k,t)<^

Vke(0,l),Vf>0,

given by Theorem 2.

In that case, we also have that

and therefore

JJtW

—>

fu uniformly on compacts of

[0,

oo)

t—¥00

with JV

= /

dk.

Remark 5. Theorem 3 is not true for all initial data such that J fi

dk < N

Notice indeed that, by Theorem 2 we know some of these data, for which the solution

/if is such that for some T* > 0:

\imk

+ f +

)>0,

Vt>T*

fc-»o

and

J /-OO

-J

(k,t)k

dk<0

and

/if

(*)

—

^ /i/ uniformly on compact subsets of (0, oo)

for some v such that

/•OO

W„ < / fin{k)k

dk.

Since on the other hand, for every e > 0:

lim ||/

(i)-/

||i = 0

t—*oo

with p, < v we may not have /

—>•

JK for

large.

Remark 6. The long time behaviour of the "singular solutions" (i.e. such that M{

)

to the Boltzmann Compton and Kompaneets equation can be precisely described.

The strong differences in their behaviour is then easily observed.

For Kompaneets equation it is proved in [EHV] that for a large set of integrable initial

data, with Mi„ > N

, the solution satisfies

C(i)

(k,t)

-^ as t ->

oo,

->•

where C(i) is a bounded function of t.

On the other hand, it is shown in [EMV] that the solution of the Boltzmann Compton,

with the same initial data behaves like:

1 i

/(M)~

_ + —e~

mkt

, as i->oo, 0< kt < 1.

In the second case one can see that

1.2

/(£> t) ~ -

—- + kt

mkt

, as t -s- oo, 0 < fci < 1

and the formation of the Dirac mass clearly appears in the second term of the right

hand side.

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Phys. Fluids 29, 748-752 (1986)

[CCS] G. Chapline, G. Cooper and S. Slutz: Condensation of photons in a hot plasma;

Phys. Rev. A, 9, 1273-1277 (1974).

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243-251.

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Boundedness of global solutions

of nonlinear parabolic equations

Marek Fila*

Institute of Applied Mathematics

Comenius University

842 48 Bratislava, Slovakia

Email : fila@pcl.iam.fmph.uniba.sk

Abstract

We consider superlinear parabolic problems for which blow-up in finite time

occurs for some initial data but other initial data may yield global solutions. We

review results concerning the question whether or not global solutions are necessarily

bounded.

1 Introduction

In some classes of nonlinear parabolic problems, solutions may cease to exist in finite

time by becoming unbounded. Such a phenomenon is called blow-up in finite time and it

has been a subject of intensive mathematical studies in connection with various fields of

science such as plasma physics, combustion theory and population dynamics, for instance.

In this survey we consider problems for which blow-up in finite time occurs for some

initial data but other initial data may yield global solutions. We discuss results which say

that global solutions are necessarily bounded or, in other words, blow-up in infinite time

does not occur.

The problems that we review include semilinear (Section 2) and quasilinear (Section 3)

equations with Dirichlet boundary conditions, the Cauchy (Section 4) and Stefan (Section

5) problems for semilinear equations, the Cauchy problem for a semilinear equation in sim-

ilarity variables (Section 6), problems with nonlinear Neumann (Section 7) and nonlinear

dynamical (Section 8) boundary conditions, and also systems (Section 9). The last two

sections are devoted to a brief description of the methods (Section 10) and applications

(Section 11).

"The author was partially supported by VEGA grant 1/7677/20.

2 Semilinear equations with the Dirichlet boundary

condition

2.1 Power nonlinearity

Consider the problem

= Au+ |u|

p_1

u x e D, t > 0,

u = 0 x e 5D,

> 0,

u(x,0) = u

(x) i£D,

where p > 1, u

e L

°(D) and D is a bounded domain in R

. For this problem it is

well known that there exist choices of u

for which the corresponding solutions tend to

zero as t

—>

oo and other choices for which the solutions blow up in finite time. If we

are interested in other types of behavior we may consider the possibility that the solution

u(-,t; u

) exists globally but is not uniformly bounded.

The study of the question whether the last possibility may occur was initiated by Ni,

Sacks and Tavantzis [28]. The results of [28] can be expressed roughly in the following

way. If p < (N + 2)/N, D is convex, u

> 0 and u is a global (classical) solution then

u(x,t) < C(uo,D,p), xeD,t>0,

where the constant C(uo,D,p) > 0 depends on the shape of u

near dD. If p > (N +

2)/(N

—

2), JV > 2, then there are global unbounded L

-solutions.

The very interesting question whether these unbounded global weak solutions are

actually classical was left open for a long time. Recently, Galaktionov and Vazquez [17]

proved that for p = (N + 2)/(JV

—

2) this is true provided D is a ball and w

is radially

symmetric. On the other hand, if p > (N

2)/(N-2) andp < l + 6/(JV-10) for N > 10

then any unbounded global L

-solution blows up in finite time in the L°°-norm provided

D is a ball and u

is radially symmetric and decreasing in r.

An improvement of the result from [28] on the boundedness of global solutions was

given by Cazenave and Lions [2]. They removed the assumptions on convexity of D

and nonnegativity of

and showed that global solutions are uniformly bounded if p <

(N + 2)/(iV

—

2) (without giving any explicit dependence of the bound on the data). An

a priori bound of the form

\u(x,t)\<C{\\uo\\

iD)

,D,p), xeD,t>0 (1)

for any global solution u was established in [2] for p(3N

—

4) < 3N +

Later, Giga ([19])

derived an a priori bound of the form

u{x,t) < C(||uo||i~p),£>,p), xeD, t > 0

for any nonnegative global solution u when (N

—

2)p < N + 2.

The results from [2] and [19] were improved by Quittner ( [31]) who proved that (1)

holds for the whole subcritical range (N

—

2)p < N + 2.