Singular solutions of semilinear parabolic equations

SINGULAR SOLUTIONS OF

SEMILINEAR PARABOLIC EQUATIONS

GEONGSEON BAEK AND MINKYU KWAK

1. Introduction

In this paper we discuss the existence and uniqueness of singular solutions for equations ofthe form

(F) _{Ut = U:n : - 1 1U} ^,-1_{U2: -} ^{I I}_U^{p - 1}_{U, p, q} > ,1

in the domain Q = {(x,t) :x E K,t >O}. This equation represents a model ofdiffusion-convection with absorption.

Following Escobedo and Zuazua (1991), it is easy to see that given initial dataUo E L1(R)there exists a unique solutionu(x,t) of(F)such that U E C((0,00);W²"(R» nCl«0,00);L'(K» for every I E (1,00).

Moreover, physical considerations lead us to assume that uo(x) 2:: 0, in which case u(x,t) is positive unlessU== 0 in Qfrom the Maximum Principle and becomes C(X) smooth inQfrom standard regularity the- ory.

The properties of solutions are usually explained in terms of the properties of special solutions such as singular ones. A singuJ.a.r solution (it is also called a fundamental solution or a source-type solution) is the one corresponding to initial data a Diracmass, M6(x)withM >0, namely a solution u(x,t) such that u(x,t) satisfies(F) in the classical sense andu(x,t) -+ M6(x) as t -+ 0 in the sense of measures, namely

lim

J

t_O

for all continuous, bounded functions 4>on K.

Received May 26, 1994.

1991 AMS Subject Classification: 35K15.

Key words: existence and uniqueness, singular solutions.

This work wassupported by KOSEF-GARC, BSRI-94-1426 and MRC-CNU.

Brezis and Friedman (1983) showed that the equation^Ut= ^Un-uP admits the unique singular solution only for 1 < ^P < 3. Moreover Escobedo, Vazquez and Zuazua (1991) considered equations of the type

(E) ^Ut =U xx - U^q-lU x

and proved its existence and uniqueness for every q>1. But it seems to us that no results are available for equations of mixed type.

As a first attempt to this direction, combining those two results we here show that the unique singular solution of (F) existsifand only if 1 < p < 3 or 3 ~ p < q+1. This reveals some interactions between absorption and convection for the existence of singular solution.

For the proof of existence and uniqueness we mainly use the com- parison with the equation (E) and for nonexistence following [1], we take a fun~tioIl of the f()rIJ1 1](k(lxJ(+t» ^as a test function to lead a contradiction, see section 5 for details. This method and the main estimates (see Lemma 2) turns out to be very useful and concise and applicable to wide class of diffusion equations. This will be discussed in other space.

2. Preliminary Estimates

We first recall some basic estimates which will be used. Any solution u(x, t) of (F) as .mentioned in introduction satisfies

(1) ~

J

Moreover for any two solutionstt,Vj by the standard technique of mul- tiplying by a sign function and integrating we get the contraction prin- ciple

(2) lu(·,t)-v(.,t)I£l(JIl.) ~ lu(·,r)-v(·,r)I£l(JB!.h for any t _~ r ~ O.

For the equation (E), it is shown in [31 (see estimate (2.32)in that paper) that

(3) 0~ u(x, t) ~ C(M)(C1/2+^{t(1-q)/2), 'it > 0,}

where C(M) is a constant depending only on M = luo(x)I£l(JB!.)' This estimate is not sharp and for 1<q ::; 2, Escobedo, Vazquez and Zuazua obtained a better bound for u:

(4)

LEMMA 1. ([2], Lemma 1.2) For 1< q ::; 2 we have

(

M )1/^q

0::; u(x,t)::; (/_ l)t '

where M is defined above.

The main estimate in this paper is a bound for q > 2. Accidently we obtained a similar estimate.

LEMMA 2. For q > 2 we llave

(5) _o::; u(x, t) ::; 4(q-1)/q² (2M)-t- ^l/q

Proof. We define the new varia.ble z =u^q- ¹• In terms of z,equation (E) reads

(6)

a""r~2

Zt +^{ZZx - I J -} =^Zrx

Z

where (3 =(2 - q)/(q - 1).

Let v(x,t) be a. solution of the Burgers' equation

with initial data v(x,O) = uq- 1(x,s), ^S > 0. For q > 2, (3 < °^and

v(x, t) is a supersolution of (6) and a comparison principle implies that uq- 1(J:,t+s)::;v(x,t), t>O.

Moreover from Lemma 1 we obtain

° ( ) (2JV(X,O)dx)1/2

::;vx,t::; t

and

(7) (

?J q-1( )d )1/2

°^{::;uq-1 .(x,t+s::;) - u t x, s x ' t >0.}

We also have

We now let s= t +e, with e> 0 and define

then, fort >0, 0 ~w(t) < ⁰⁰ and

Therefore we have.

(8) Vt>0.

This implies that0~ t1/qu(x,t+e) ~4(q-l)/q2(2M)1/q and (5)in the limit as e-4 O. 0

3. Existence

We denote by EM(X,t) the singular solution for (E) corresponding to initial datit, M5(x). We note that f ^EM(X,t)dx = M for all t > O.

We also denoteq* = max{q,2} and we prove

THEOREM3. For1 <P< ^q*+^1,there existsafundamentalsolution of(F) for evezy M > O.

Proof. Let us define un(x,t) as the solution of(F)fort > l/n with initial data un(x,l/n) =^{EM(X,l/n) at t} = ^l/n. By the Maximum Principle (see [5] for example) we know that

"It ~ 1/n and x EJR.

Sinceun(x,t) is monotone decreasing asn -400, its limit u(x,t)= lim un (x,t)

n-oo

exists and u(x,t) is a weak solution of (F) in Q. By a standard reg- ularity result, we may conclude that u(x,t) is a classical solution in

Q. .

It is clear that u(x,t) ~ EM(X,t) in Q and u(x,O) = 0 for every x'#0. Moreover, for 1/n<t ~ 1

(9) fn(t) =

1/

= t ^I u~(x,s)dxds

Jl/nJ.

~_A/nt J.^I E~(x,s)dxd8

=:; It M supEti¹(.,s)ds

J1/n :I

~C t ^8-(P-1)/Q-ds (from (3) and (5»

J1/n

< ^Cq* t(l-P+Q- }/Q- •

-1-p+q*

Note that 1 - p+^q* >o.

Since Jun(x, 1/n)dx = M by the definition of Un, we conclude that un(x, t)dx will be close to M for sufficiently small t uniformly in n and in the limit the same will be true for u. Consequently limt_o J^{u(x, t)dx = M.} For every continuous bounded function 4>,

o=:; / u(x,t)[4>(x) - 4>(O)]+dx =:; / EM(X,t)[4>(x) - 4>(O)]+dx

and since the second integral tends to 0 ast -+0, the same will be true for the first integral. Similarly

lim/tt(x,t)[4>(x) - 4>(O)]_dx = 0

t_O

and writing

4>(x) =4>(0)+[4>(x) - 4>(0)]+ - [4>(x) - 4>(0)]-,

we finally obtain (10)

o

lim

J

t~O

4. Uniqueness

For the proof of uniqueness, we prepare the following lemmas.

LEMMA 4. Let u(x, t) be a fundamental solution of(F), then (11)

J

-it ^J

Inparticular limt.-oJu(x, t)dx = M and Ju(x,t)dx =:; M for every t > O.

Proof. An integration gives

J

J

-it

for t > r > O. By definition limr:-oJu( x, r )dx = M and (11) holds.

The last inequality also holds obviously. 0

LEMMA5. Letu(x,t) beafundamentalsolutionof(F), thenu(x,t) =:;

EM(X,t) in Q.

Proof. Let wn(x,t) be a solution of

Wt = ^Wxx - ^(wq/q)x, w(x, l/n)= u(x, l/n)

for t > l/n. Similarly to the proof of Theorem 3, one can see that Wn

is increasing and u( x, t) =:; wn (x, t) for t > l/n. Let W = limn . -oo^Wn ,

then u(x, t) =:; w(x,t) for t > 0 and w(x, t) is a weak solution of (E) and also becomes a classical solution in Q.

We now note that

J

J

J

for t > 1/n. From the Monotone Convergence Theorem, we obtain

j w(x, t)dx ;:; M.

Since J^{u(x, t )dx goes to M as t} tends to 0, the same will be true for

Jw( x, t )dx. We now apply the same argument as the proof of Theorem 3 to conclude that

(12) limj w(x, t)4>(x )dx= M 4>(0)

t-O

for every continuous bounded function4>. Thusw(x,t)is a fundamental solution of(E) and the completion of the proof of Lemma follows from the uniqueness of the fundamental solution of (E). But here we take .,p(x) = 14>ILoo - [4>(x) - 4>(0)]+ as a test function. Then

0;:; j u(x, t).,p(x)dx

;:; j w(x,t).,p(x)dx ;:; MI4>ILoo.

Since limt_o Ju(x, t).,p(x)dx = MI</>ILoo, we obtain

(13) limjw(x,t)[</>(x) - </>(O)]+dx= O.

t_O

Similarly we have

and (12). 0

limjw(x,t)[</>(x) - 4>(O)l-dx = 0

t-O

We now prove the uniqueness.

THEOREM 6. There existsa.tmost one fundamental solution of (F) for each M >O.

Proof. Let u, v be fWldamental solutions of (F) with initial data MS(x). Then from the contraCtion principle(2) andLemma4we have

f

f

~ j(EM(x,Ijn)-u(x,I/n»dx

+^{!(EM(x,I/n) -} vex, I/n))dx

=^{2M -}

J

J

5. Nonexistence

We here consider the casep ~ q*+^1. Recall that q* = max{q,2}.

Letu(x, t) be a fundamental solution of(F) with initial dataMD, then from Lemma 4 we see that u(x, t) is V -integrable overQ.

Following [I}, we prove that

THEOREM 7. There is nofundamentalsolutionof(F).

Proof. Let '1](s) be any smooth non-decreasing function on R such that

{

I for s> 1

"l(s)= .0 for ⁸ ~ ^0,

and set '1]1:(8) =^'I](ks).

H we define1>k(X,t)= 'l]k(lxl^q• +^t), then we know that

for 0 <^E< T. We first have

and in the limit a.s ^€ -+0 we get

We claim that all the integral in the right tends to 0 a.s k -+ 00. It then follows that JoT JuP^dxdt =^{0 and} u^{== 0 in} Q. We have

iT

!

iT

!

iT

!

where Dk = {(x,t)jt > 0, 0 < Ixl^q• +t 5 IJk}. By Holder inequality we finally get:

!l~ u⁵

(!!Dk

ilk

(!lk

Now note that IDkl 5 2k-(l+1/q·), 2Jq* 5 1 and

( 1 ) ( 1) p - q* - 1

1 - 1+ - 1 - - = - 5 0,

q* p pq*

2. _(1 ⁺2.) (1 _2) ^{= _p( q* - q)+P - q* -} 15^o.

q* q* p pq*

Our claim follows from the fact that JJDk uP ^{-+ 0 a.s k -+ 00. 0}

References

1. H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. pures et appl. 62 (1983),73-97.

2. M. Escobedo, J.L. Vazquez and E. Zuazua, Source-type solutions and asymp- totic behaviour for a diffusion-convection equation, Arch. Rational Mech. Anal.

124(1993),43-65.

3. M. Escobedo and E. Zuazua,Large time behavior for convection-diffusion equa- tions inRN,Journal of functional analysis 100 (1991), 119-161.

4. S. Kamin and J .L. Vazquez, Singular solutions of some nonlinear parabolic equations, IMA Preprint series 834 (1991).

5. M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equa- tions, Springer-Verlag, New York, 1984.

Department of Mathematics Chonnam National University Kwangju, 500-757, Korea