Bemelmans J. et. al. (editor) Elliptic and Parabolic Problems: Rolduc and Gaeta 2001

Подождите немного. Документ загружается.

The first universal bound (independent of initial data) has been established recently

in [16]. More precisely, if (N - l)p < N + 1, r > 0 and u is a global nonnegative solution

then

u{x,t)<C{T,D,p), xeD,t>r. (2)

Even more recently, Quittner has derived a universal bound of this kind in [32] for

(N - 2)p < N + 2 and N < 3.

2.2 Exponential nonlinearity

Consider next the problem:

= Au + Ae

i€Bi(0),<>0,

u = 0 x 6 dfli(O), i>0,

u(x,0) = uo(x) i £ Bi(0),

where Bi(0) = {x e R

: |x| < 1}, u

is a radial continuous function on B

(0) vanishing

on dBi(0) and

is a positive parameter.

For JV = 1,2 it was shown in [5] that every global classical solution is uniformly

bounded even for general domains and initial data. On the other hand, for N > 10 and

= 2(iV-2), global unbounded classical solutions do exist (cf. [22]). For 3 < N < 9 it was

known that global unbounded L

-solutions exist for certain

> 0 (see [23]), which makes

the problem of existence of unbounded global classical solutions even more interesting. It

was conjectured (see [23] and [42]) that such solutions might exist for some dimensions

3 < N < 9 and A > 0. A result from [12] rules out this conjecture, as far as radial

solutions are concerned. Namely, the main aim of [12] was to prove that every global

classical solution is uniformly bounded if 3 < N < 9 and

> 0.

2.3 Chipot-Weissler equation

In this subsection we discuss the problem

ut = Au - \Wu\

+ Xu" xeD,t>0,

u = 0 x € 3D, t > 0,

u(x,0) = u

(x) > 0 ieB.

Here p, q > 1 and

> 0. Under the assumptions

3<-7T, A>0 or q = ^r,

> 2"(p+ l)(p -

l)-*"

p+ 1

and (N

—

2)p < N + 2it was shown in [6] that global increasing solutions are bounded if

D is a bounded domain in R

If q > p and A is small enough then all solutions are global, bounded and decay

exponentially to zero provided D C R

is a domain (possibly unbounded) where the

Poincare inequality is valid (see [40]). Conversely, if D is such that the Poincare inequality

is not valid then there always exist unbounded solutions (cf. [40]). These results of

[40] were slightly improved and given a more complete form in [38]. In particular, the

restriction on smallness of

was removed for global existence.

Studies of global solutions of similar problems with more general nonlinearities /

(u,

Vu)

were performed in [24] and [35].

2.4 Nonlocal nonlinearity

The following nonlocal problem is considered in [7]:

x e D, t > 0,

xeT-L,

t > 0,

e r

, t > o,

x el>,

where D is a bounded domain in R

, 3D =

L)T

I\ DF

= 0,

f e C\R), / > 0, J f(u) du < oo, F(u) = J f(v) dv,

— oo —oo

g : (0, oo)

—>

(0, oo) is a locally Lipschitz function.

The assumptions on / and g imposed in [7] guarantee blow-up in finite time if

large. Sufficient conditions for boundedness of global solutions are given in [7] and their

sharpness is discussed there. A particular case of the main result from [7] was proved

independently in [1] where the method from [5] was also used. A priori bounds for global

solutions of nonlocal equations of this type and also of the form

= Au +

f(x,u)-

—J f(x,u)dx

were established in [33].

2.5 Equation with a localized reaction term

In [36], the following problem was studied:

= &u + u

(t),t) xeD,t>0,

u = 0 x

dD, t > 0,

u(x,0) = u

(x) > 0 ieB.

where p > 1, D is a bounded domain in R

and x

[0,

oo)

—>

D is a Holder continuous

function. The results in [36] say that if either i

= 0 6 D or J\f = 1 then every global

solution is uniformly bounded. The former result is in contrast with the fact that for

the problem discussed in Subsection 2.1, boundedness of global solutions holds only for p

which is Sobolev subcritical ((TV - 2)p <

+ 2).

u(x,Q)

Au + g(J

F(u)dx)f(u)

(x)

2.6 Periodic forcing term

Consider the problem

ih = Au + m(t)f(u) x £ D, t > 0,

u = 0 x e dD, t > 0,

u(x,0) = u

(x) x e D,

where D is a bounded domain in Ft", / is a superlinear function and m is positive and

periodic. An a priori bound for global solutions of this problem was derived in [34].

2.7 Abstract equation

In [9] we studied the initial value problem for evolution equations of the form

^+Au(t)=F(u(t)),

in a Banach space where A is the infinitesimal generator of an analytic semi-group and F

is a nonlinear function such that the initial value problem possesses non-global solutions

for some initial data. (See [9] for details of the setting which are omitted here). We gave

sufficient conditions (on A, F and the underlying spaces) to ensure that global solutions

are necessarily bounded in a suitable norm. We also gave several examples of problems

to which the abstract result applies. They include some nonlocal equations and equations

of order higher than two.

3 Quasilinear equations with the Dirichlet boundary

condition

Several works have been devoted to the study of boundedness of global solutions of the

degenerate problem

= A(|u|

u) + /(w) xeD, t>0,

u = 0 xedD, t >0,

u(x,0) = u

(x) x € D,

where m > 0. It was shown in [28] that for m > 1, D convex, uo > 0 and f(u) = u

, all

global solutions are bounded if m < p < (N + 2)/N, while global unbounded solutions

exist iip/m > (N + 2)/(N

—

2),N> 2. The results from [17] that were mentioned above

also apply to this problem if f(u) = u

, D is a ball and u

is radial (cf. [17] for details).

In [5], boundedness of global solutions is proved for general D and

and nonlinearities

/ that include

/ s , ,„-i r ^ f iV-21 p N + 2

/(«) = H

»: p>max{l,m}, m > max JO, -^—- j , — < ^3^-

This result was improved in [25] in several ways. Instead of |u|

m_1

u a large class of

continuous increasing functions $(u) was taken there. Non-power growth of $ was allowed

for

< 2 as well as more general /'s. For

N =

1, Suzuki ([41]) proved boundedness of

global solutions for

> 1 for some nonlinearities

that are not allowed in [5] and [25].

Arguments and results from [5] were generalized to parabolic p-Laplace equations in [26].

For any global solution, an a priori bound of the form

Hx,t)\

< C(\\u

«,

(D)

,D,p,rn) ,

x €

i >

has been recently established in [39] for f(u)

|u|

p_1

w under the assumption

10m+ 2

Km<p

<m +

——

if JV

and

universal bound of the form

u(x,t)<C(T,D,p,m), xeD,t>r>0

has been derived there for global nonnegative solutions provided

N + 2

1 <m<p< —Tz—m, if

> 1.

4 Cauchy problem for

semilinear equation

Consider the problem

ii\Vu\i +

ieR",(>0,

u(x,0)

(a;)>0

xeR

Herep, q > 1 and /u

0. If fi = 0 and (N-2)p < N + 2 then all global solutions decay

uniformly to zero (cf. [37]). (An earlier study of global solutions of this problem with

fi = 0 was performed in [21] where sign-changing solutions with fast decay (as

—>

oo)

were considered). If

> 0 then there exist global solutions such that (cf. [40])

lim u(x,t) = oo for all

x e

5 Stefan problem for

semilinear equation

Consider now the following reaction-diffusion problem with free boundary:

+ v?

(0, s(i)),

> 0,

u(x,

(x)

x e

(0,

), s(0)

> 0,

u(s(t),t)

{0,t) = 0

t>0,

s'(t)

Ua;

(s(t),t)

t>0,

where

p > 1.

This problem can be viewed as

simple model of

chemically reactive

and heat-diffusive liquid surrounded by ice. Here u > 0 represents the temperature of the

liquid phase, and the ice is assumed to be at temperature 0.

Global existence, stability and blow-up for this problem were studied in [18], for pre-

vious studies see references therein and in [15].

[15] we showed that

if u

global

solution then

u<C(s

, ||uo||

i([

]).

6 Cauchy problem

for a

semilinear equation

simi-

larity variables

Consider

solution

of the

problem

11,

A«

+ «

x e R

0<t<T,

u(x,0)

{x)>0 xeR

which blows

up at t = T. For a E R" one can

rescale

using backward similarity

variables

follows

-log(r-t),

= -^L,

w{y,s)

(T-t)^u{x,t).

Then

satisfies

= Am y

•

Vw + w

-w y € R", s >

—

logT,

p-1

(3)

w(y,-logT)

T^u

+ VTy)

yeR

Under some assumptions

on u

and p, it has

been shown

in [27]

that

for 6 > 0

there

exists

C =

C(p,

> 0

such that

for any

global solution

(3),

holds

w<C

for s> -\ogT

+ S.

the

original variables this

can be

equivalently formulated

as a

universal bound

(independent

«o)

for the

blow-up rate.

7 Nonlinear boundary conditions

7.1 Heat equation

The following problem

studied

in [4]:

= Au x

D, t > 0,

—

= f(u)

x£dD,t>0,

u(x,0)

UQ(X)

X&D.

Here

I? is a

bounded domain

in R and v is the

outer normal

to dD. The

result from

[4]

applied

to the

particular case with

f(u) =

\u\

u says that global solutions

are

bounded

if (N

—

2)p < N.

This means that

this case there

are no

global positive

solutions since there

is no

positive steady state

and the

trivial steady state

unstable

from above.

The

main result

[4]

and

several elements

of its

proof were generalized

[24].

7.2 Porous medium equation

In [25], Lieberman studied the problem

A$(u) x e D, t > 0,

/(«) xedD, t > 0,

«o(^) x £ D.

Here D is a bounded domain in R

and $ is a continuous increasing function. The

results from [25] say that global solutions are uniformly bounded if the growth of / is

super linear but subcritical (which may be faster than polynomial if N = 1,2). For the

precise meaning of "superlinear" and "subcritical" see [25].

7.3 Porous medium equation with absorption

The large time behavior of solutions of the problem

)

- au

u(x,0) =

UQ(X)

> 0

was discussed in [8]. Here a > 0 and p,q> m > 1. Among other things it was shown in

[8] that if p < 2q

—

m or if p =

2q —

m and ma < q then blow-up in infinite time does not

occur for any initial data while blow-up in finite time occurs for some initial data. On

the other hand, if p = 2q

—

m and ma = q then all solutions are global and unbounded.

Corresponding results for m = 1 can be found in [3].

8 Dynamical boundary conditions

8.1 Laplace equation

Consider next the problem

0 x e D, t> 0,

h(u) x&dD, t > 0,

(a;) > 0 xedD,

where A is the Laplacian in x, D is a bounded domain in R

, N > 2, with smooth

boundary, u is the outer normal to dD, h e C

and u

belongs to some appropriate

function space. We shall consider superlinear nonlinearities h for which blow-up in finite

time occurs if initial data

are "large". As an example we can take

h{u) = |w|

p_1

—

au, p > 1, a > 0.

It is shown in [13] that all global solutions are bounded provided the growth of h

is subcritical, i.e. (N

—

2)p < N if h is as above. But under this assumption only an

u(x,0)

\x\ < 1, t >0,

|x|

= 1, t> 0,

W<1,

du du

u(x,0)

a priori bound for limsup^^ ||w(-,t)||

(n)

obtained. To derive an a priori bound for

Poo ll

('i*)llc(ft)

stronger growth restriction on h is needed in [14]. As an application

of the latter result we proved existence of sign-changing solutions of the corresponding

stationary problem.

8.2 Parabolic equation

The paper

[14] is

devoted

to the

study

of the

problem

= Au + \\u\

u x£D,t>0,

du du , , i „„

-sr + w- = nM

u xedD, t > 0,

at dv

u(x,0) = u

{x)>0 xedD,

where Q is a bounded domain in R

, v is the outer normal on dQ, p, q > 1,

fj,

{—1,0,1}, max{A,

fj,}

= 1 and \p+/iq > 0. Under our assumptions on p, q,

and n, blow-

up in finite time occurs for some initial data. We proved in [14] that if (JV

—

2)p < N + 2

and (N

—

2)q < N then all global solutions are uniformly bounded.

To obtain an a priori bound for global solutions in terms of suitable norms of

stronger condition on p and q is needed. Namely, if

N--^)p<N + - and (N - -) q < N + -,

then

sup||u(-,£;u

)||z,~(n) < C (|KIU~(0), IKII/fi(n))

provided u is a global solution. As an application of this a priori bound we proved in [14]

the existence of sign changing solutions of the problem

Au - lu^u = 0 x £ D,

^ = Wo-^ xedD,

/ 5\ 1

Kp<q, (N--)q<N + -.

The existence of positive solutions of this stationary problem was studied before in

[3].

9 Weakly coupled system

Consider the system

«t

u(x,0)

v(x,0)

= Aw

= Av + u

= 0

= u

(x) > 0

= v

(x)>Q

x e D, t > 0,

xedD, t > 0.

xedD, t>0,

ID,

iefl,

where p, q > 1 and D is a bounded domain in R

It was shown in [16] that if max(p,g) < 1 + 2/(N + 1) and r > 0 then there is a

constant C(D,p,q,r) > 0 such that for all global solutions it holds

u(x,t) +v(x,t) < C(D,p,q,r), x e D, t > T.

An a priori bound (which depends on initial data) for global solutions has been estab-

lished in [35] under the assumption

max(p,g) + l

+ 1

pq - 1 " 2

and under the weaker assumption

max(p,

+ 1 TV

pq-1 ~^

if D = R

10 Methods

10.1 Energy methods

The basic idea of the proof of the boundedness of global solutions in [4]-[9], [11], [12],

[22]-[24] is similar. We illustrate it here for the simplest problem considered in Subsection

2.1.

We show by contradiction that the W

/1,2

(D)-norm of any global solution must be

bounded uniformly in t, and then we use known results to conclude that the C(Z))-norm

is also uniformly bounded. Suppose

limsup ||u(-,t)||

^i,2(£

)

) = oo.

t—*oo

A uniform L

(D)-estimate (or the concavity method) enables us to rule out the pos-

sibility that

hm||u(-,t)

11^1,2(0)

= oo.

In this part of the argument, the superlinear growth of the nonlinearity (as u

—>

oo)

plays a crucial role. On the other hand, if

limsup ||u(-,i)||vKi.2(D) < oo,

then the o;-limit set of u contains for every number B large enough a stationary solution

w with

||w||W^

(D)

= B. (The assumption that the growth of the nonlinearity (as u

—*

oo)

is subcritical is needed here.) This leads to a contradiction since there is a constant K

(depending on

UQ)

such that ||u||iyi,2(£)) < K for every stationary solution v in the oi-limit

set of u.

Both parts of the argument rely on the existence of a suitable Lyapunov functional.

In general, this method does not yield any a priori estimate of ||u(-, i)||iv

(D) in terms of

some norms of

UQ.

Other energy and interpolation arguments were used in [2]. The method from [2] was

further developped in [31, 33] by bootstrapping and employing maximal regularity results.

10.2 Scaling arguments

The variational structure together with scaling invariance are the main tools in [19]. If

we rescale a solution u of the equation

= AH + u

v(y, s) = \r-

u{x + Xy,t + As),

then v satisfies the same equation. This fact, combined with the existence of a Lyapunov

functional, yield a contradiction argument to prove the a priori bound in [19].

A different contradiction argument, relying on scaling, energy and Hardy's inequality,

gives the universal bound in [32]. In order to handle problem (3), this approach was

extended in [27].

For problems without variational structure, a scaling argument was used in [35] in

order to prove a priori estimates for global solutions.

10.3 Zero number techniques

The method of our proof of boundedness of global classical solutions in [12] is based on

the intersection-comparison with a particular self-similar solution. It was inspired by the

proof of Theorem 15.1 in [17] where a similar result was shown for a different problem.

However, unlike the case in [17], the nonincrease of the number of zeros of the difference

of two solutions, by

itself,

is not sufficient. We also employ the more subtle "zero number

diminishing property" which says that the number of zeros drops when a multiple zero

occurs.

The nonincrease of the zero number also plays a crucial role in [41].

10.4 Smoothing effect in weighted Lebesgue spaces

To prove (2.2) in [16], we start from the classical argument of Kaplan which yields that

for any r > 0 there is T\ £ (0, r/2) such that

Ju"(T

,x)<pi{x)dx<C{n,p,T), (4)

where ipi > 0 is the first eigenfunction of —A in

HQ(SV),

normalized by /

ipi

— 1. Since

Ci<5 < ipi < C2<5, <5(a;) = dist(x,dD), the bound (4) suggests that it is rather natural to

consider the problem in

= L

(D, S(x) dx) spaces. With this motivation, we developped

in [16] both linear and local nonlinear theories in these spaces.

Our main result in the linear theory is an optimal Lj - L

estimate for the heat

semigroup. In view of this estimate for the heat semigroup, the local existence and

uniqueness theory in L\ is similar to the usual L

-theory in dimension

JV+1.

In particular,

we proved local existence and uniqueness for solutions in L\ for q > q

= |(JV + l)(p

—

and local nonexistence for q < q

. We also established a nonlinear version of the L\

—

estimate whose particular case q = p, r = oo, plays a crucial role in deriving the universal

bound (2) from (4).

A similar idea is used in [27] but weighted Lebesgue spaces are replaced by convolution

Lebesgue spaces there.

11 Applications

11.1 Blow-up

The first application concerns blow-up in finite time. It requires some knowledge of the

corresponding stationary problem. For example, if one can show that there are no steady

states then all solutions of the parabolic problem blow up in finite time (cf. [1, 5]). A

slightly more sophisticated application of this type says that a solution blows up in finite

time if it starts above a maximal steady state provided this steady state is unstable from

above, cf. [3, 4, 6, 8], for instance.

11.2 Existence of stationary solutions

The second application goes in the opposite direction. Some knowledge of the parabolic

problem is used to draw conclusions on the corresponding stationary problem. We describe

roughly a simple illustration of this idea. Assume a stable steady state is known to exist.

belongs to the boundary of its domain of attraction and the solution u starting from

is global then there must exist another (unstable) steady state which is in the w-limit

set of u. For dynamical proofs of existence of solutions of elliptic problems we refer to

[13,

14, 29, 30, 34].

11.3 Existence of periodic solutions

The a priori bound discussed in Subsection 2.6 was used in [34] to prove the existence of

solutions which are periodic in t.

11.4 .^-connections

The result from [12] mentioned in Subsection 2.2 plays a fundamental role in the study

of connections between equilibria by solutions which blow up but continue to exist in a

weak sense (see [10, 11]).