Singular solutions of semilinear parabolic equations in several space dimensions

SINGULAR SOLUTIONS OF SEMILINEAR PARABOLIC EQUATIONS IN

SEVERAL SPACE DIMENSIONS

JEONGSEON BAEK AND MINKYU KWAK

ABSTRACT. We study the existence and uniqueness of nonnegative singular solutionsu(x,t) of the semilinear parabolic equation

defined in the whole spaceRN fort >0, with initial dataM8(x), a Dirac mass, with M >O. The exponentsp, qare larger than 1 and the direction vector a is assumed to be constant.

We here show that a unique singular solution exists for every M > 0 if and only if 1 < q < (N+l)j(N - 1) and 1 < p <

1+(2q*)j(N +1), where q* = max{q, (N+ l)jN}. This result agrees with the earlier one forN =1. In the proof of this result, we also show that a unique singular solution of a diffusion-convection equation without absorption,

exists if and only if 1<q<(N+l)j(N - 1).

1. Introduction

This paper is concerned with the Cauchy problem for the nonlinear diffusion-convection equation

(E) ut=L.\u-aoV(u^q), q>l

Received August 20, 1997. Revised September 10, 1997.

1991 Mathematics Subject Classification: 35K15.

Key words and phrases: existence and uniqueness, singular solution, semilinear heat equation.

This work was supported by BSRI-96-1426, KOSEF-GARC and KOSEF-95- 0701-01-01-3.

and for the nonlinear diffusion-convection equation with absorption (F)

in the domainQ = ^{(x,t) :x E RN,t >O}. We assume for convenience that the convection direction a is a constant vector in RN.

Following Escobedo and Zuazua (1991) (see [5]), it is easy to see that for given nonnegative initial datauo(x) E L1(R^N) there exists a unique nonnegative solutionu(x,t) of (E) and (F) such that

for every l E (1,00). Moreover, u(x, t) is positive unlessu=^{0 inQ by}

the Maximum Principle and becomesCoo smooth inQby the standard regularity theory.

A singular solutionisthe one which develops a singularity at (x,t) =

(0,0) as t -+ O. More precisely, a singular solution (it is also called a fundamental solution or a source-type solution) is the one correspond- ing to the initial data M6(x) (a Dirac mass) with M > 0, namely, a classical solution u(x, t) of (E) or (F) such that u(x,t) -+ M6(x) as t -+ 0 in the sense of measures, that is,

tim f u(x, t)<jJ(x)dx=M<jJ(O),

t-+o}RN

for every bounded continuous function <jJonRN.

By rotating the coordinate axes and rescaling, we may assume that the convection direction a satisfies a =^{qeN, eN} =(0,··· ,0,1). With the convenient notation x = ^{(x,y) E} R^N- 1 X R, (E) and (F) become

(1.1) and (1.2)

respectively.

Escobedo, Vazquez and Zuazua (see [4]) have studied the asymptotic behaviour of solutions of (1.1) with nonnegative initial data uo(x) E

L¹(R^N) in the range 1 < q < (N +1)jN. They have shown that the asymptotic behaviour of generic solutions of this problem is given in terms of the fundamental solution of the reduced equation

(R) ^{Ut -_ LJ.xU -A Uq-1U}y ,

where ~x denotes the Laplace operator acting only on the variable x which spans the hyperplane perpendicular to a in RN. Thus the diffusion in the direction a disappears in the limit as t ^{----T 00.} The question of existence of nonnegative singular solutions of (E) and (R) has been treated but not completed.

The main purpose of this paper is to complete the question of exis- tence of nonnegative singular solutions of(E), (R) and (F) for all the range ofp and q. In fact, we prove that

THEOREM A. There exists a unique singular solution of(E) (and R)) forevery totalmassM> 0 if andonlyif1 < q< (N+^{1)/(N -1).}

THEOREM B. There exists a unique singular solution of (F) for every M > 0ifand only if1 < q < (N+1)j(N - 1) and 1 < P <

1+(2q*)j(N+1). Here q* = max{q, (N +1)jN}.

We recall that the heat equation with absorption Ut = ~u - uP

admits a unique singular solution only for 1 < p < 1+2jN, see [2].

The case N = 1 has been considered in [1J and the existence range becomes 1<p<1+q* for any q > 1.

For the proof, we first obtain the so-called L1 - LOO regularizing effect for (E):

1 N±l

0:::; u(x,t) :::;C(q,N)M-qC ^{2q ,}

for any q > 1, which generalizes the main estimate of [4]. Here and in the sequel, C(q,N) willdenote a constant depending only onq and N. In fact, the estimate is essential in proving both existence and nonexistence of singular solutions. Together with this estimate we use a comparison argument for the proof of existence and uniqueness, see section 3 and 4.

For nonexistence, we treat three cases separately. For 1 < q <

(N+1)j(N-1) and 1<P< 1+(2q*)j(N+1),following [lJ, we take a function of the form7](k(lxI²+jylQ+t)) with a= N+1-~i-l)q. ^{as a}

test function to lead to a contradiction. For q_~ (N+¹⁾j (N - 1) and p ~ 1+(2q*)j(N+1), using a similarity transformation

N+l-(N-l)q

Y ---+ A : ! q , t ---+ At, wewill show that a family {u>..} defined by

.!y.±.!. 1 (N+l)-(N-l)q

u>..(x,y,t) = A 2q U(A²X,A 2q y,At)

has a limit as A^---t 0and its limit is a singular solution of the reduced equation (R),which contradicts the nonexistence result of singular so- lution for (R).For the completion of proof of limiting process, we bring a compensated compactness argument from Evans [6] and Tartar [9].

In case q~ (N+ l)j(N -1) and 1 < p < 1+(2q*)j(N +1), we use the same similarity transformation and consider the limit of

1

1

as A---+ 0 to lead to another contradiction. See section 5 for details.

This nonexistence argument is also applicable for (E) in the range q~ (N+l)j(N -1).

The asymptotic behaviour of solutions of (F) as t ^---t 00 (also as t ---+ 0) will be discussed in other space.

2. Preliminary Estimates

For the diffusion-convection equation (E), it is shown in [5] (see estimate (2.32) in that paper) that

(2.1) o~u(x,t) ~G(M)(r^N/2+tC1 -^Nq)/2), \It>0,

where G(M) is a constant depending only on M = JRN^{uo(x)dx. This}

estimate is not sharp and for 1<^q ~ 2, Escobedo, Vazquez and Zuazua obtained a better (for larget) bound for u:

LEMMA 2.1 ([4], LEMMA 2.2). For 1 <q ~ 2 we have (2.2) 0 ~u(x,t) ~ G(q,N)M¹/qrCN+l)/C2q\ t> O.

We here extend the above estimate (2.2) to the whole range of q.

LEMMA 2.2. Forq>2 we have

(2.3) 0 :S u(x,t) :S C(q,N)M¹/qc CN +l)/C2q), t> O.

Proof. We define the new variableZ = u^q- 1 as explained in [4]. In terms of z, equation (E) reads

. lV'zl2

(2.4) ^{Zt - 6.z}+^{ZZy -} f3-- = 0,

Z

wheref3 ⁼ (2 - q)j(q - 1).

Let vex,t) be a solution of the Burgers type equation

(2.5) ^{Vt - 6.v}+vVy = 0

with initial data v(x,O) = uq- 1(x, s), s > O. For q > 2, f3 < 0 and vex, t) is a supersolution of (2.5) and Nagumo's Lemma implies that

uq

- 1(x, t+s) :Svex, t), t> O.

Moreover from Lemma 2.1 we obtain

o~ ^v(x,t) ~ ^C(2,N)

(1

and from the conservation of mass of solutions of (2.5) we get (2.6)

O:Suq-1(x,t+s):S C(2,N)

(1

We also have

1

1

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

wet) = sup ^7CN+¹)/C2q)ju(x,⁷ +^E)IVX>CRN),

O<T~t

then, for t >0, 0 :S wet) < ⁰⁰ and

O:SW(t)q-1 :Sw(2t)q-1 :SC(q,N)M1/2W(t)Cq-2)/2.

Therefore we have

(2.7) wet) :SC(q,N)M1/q, "'It> O.

This implies that 0 :S t CN+l)/C2q)u(x, t +E) :S C(q, N)M1/q and (2.3)

in the limit as E-+O. 0

3. Existence

We discuss in this section the existence of fundamental solutions for equations (E) and (F).

THEOREM 3.1. For1<^q <(N+l)/(N -1), there exists a funda- mental solution of (E) for everyM >O.

Proof. The proof of existence for the reduced equation (R) given in [4] can be directly applied to thiscase. The only different point is when checking the initial data where they use Lemma 2.1 and the restriction (N+l)/(N-1) ~2 appears. Due to Lemma 2.2, no such a restriction is needed. (See [4], Theorem 8.1 for details.) D

We now turn to the equation (F). We denote by EM(x, t) the sin- gular solution for (E) cOITewonding to the initial data, a Dirac mass, M8(x). We note that J^{EM(X,t)dx = M for allt} >O. We also denote q* =^max{q,(N+l)IN} and we prove

THEOREM 3.2. For 1 < q < (N + l)/(N - 1) and 1 < p < 1+

(2q*)/(N +1), there exists a fundamental solution of(F) for every M>O.

Proof. Let us define^Un(x,t) asthe solution of (F) for t > 11n with initial data un(x,l/n) = EM(x,1/n) at t =1/n. By the Maximum Principle (see [8] for example) we know that

Vt ~ 11n and x E RN.

Since {un(x,t)} is monotone decreasing asn ---t 00, its limit u(x, t) = !im un(x, t)

n-+oo

exists andu(x, t) is a weak solution of (F) inQ. By standard regularity results, we may conclude that u(x,t) is a classical solution inQ.

It is clear that u(x, t) ~ EM(x,t) in Q and u(x,O) = 0 for every

x ⁼¹⁼O. Moreover, for l/n< t ::; 1

In(t) = liN (un(x, t) - un(x, l/n)) dxl

= t { _u~(x,^s)dxds

Jl/nJRN

::; t ( _E~(x,^s)dxds

Jl/n JRN

::; t MsupE~\.,s)ds

Jl/n RN

::;C(q,M,N) (t s-(N+l)(p-l)/(2q*) ds (from (2.1) and (2.3)) Jl/n

2q* 2q*-(N+l)(p-l)

::;C(q, M, N) 2q* _ (N+l)(p _ 1)t ^{2q* •} Note that 2q* - (p - l)(N+1) >0 ifp< 1+(2q*)/(N+1).

Since Jun(x, l/n)dx = M by the definition ofUn, first takingn--+

00 and then taking t --+ 0 we may conclude that limt-+oJ^{u(x, t)dx}=

M. For every continuous bounded function <jJ,

0::; j u(x, t)[<jJ(x) - <jJ(O)]+dx :::; j EM(x,t)[<jJ(x) - <jJ(O)]+dx and since the second integral tends to0 ast --+ 0, the samewillbe true for the first integral. Similarly

limju(x, t)[<jJ(x) - <jJ(O)]_dx= 0

t-+O

and writing

<jJ(x) =<jJ(O) +[<jJ(x) - <jJ(O)]+ - [<jJ(x) - <jJ(O)]_,

we finally obtain

limj u(x, t)<jJ(x)dx= M <jJ(O).

t-+O

This completes the proof. D

4. Uniqueness

The uniqueness of singular solutions of (E) and (R) is proved in [4]' section.3. For equation (F), we prepare the following lemmas.

LEMMA 4.1. Letu(x,t) be a fundamental solution of (F), then (4.1)

J

-it ^J

In particular Jo^oo JuPdxds :::; M and Ju(x, t)dx:::; M for every t >O.

Proof An integration gives

J

J

-it ^J

for t >T > O. By definition limr-+oJ^{u(x, T)dx} = M and (4.1) holds.

The last inequality also holds obviously. 0

LEMMA 4.2. Let u(x,t) be a fundamental solution of (F), then u(x, t) :::; EM(x, t) in Q.

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

Wt = b..w - (w^qIq)y, w(x,l/n) =u(x, 1/n)

fort > 1/n. Similarly to the proof of Theorem 3.2, one can see thatWn

is increasing andu(x, t) :::; wn(x, t) for t > 1/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

f

J

J

Since Ju(x, t)dx goes to M as t tends to 0, the same will be true for

J^{w(x, t)dx.} We now apply the same argumentas the proof of Theorem 3.2 to conclude that

(4.2) limf w(x, t)</>(x)dx=M </>(0)

t-+O

for every continuous bounded function</>. 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 'l/J(x) = I</>Iu'" - [</>(x) - </>(0)]+ as a test function. Then

o~ f u(x, t)'ljJ(x)dx ~ f w(x, t)'l/J(x)dx ~ MI</>Iu"'·

Since limt-+oJu(x, t)'l/J(x)dx=MI</>Iu"',we obtain (4.3) lim f w(x, t)[</>(x) - </>(O)]+dx= o.

t-+O

Similarly we have

and (4.2).

lim fw(x, t)[</>(x) - </>(O)]_dx= 0 t-+O

o

We now prove the uniqueness.

THEOREM 4.3. There exists at most one fundamental solution of (F) for each M > O.

Proof Let u, v be fundamental solutions of (F) with initial data

M8(x). Then from the contraction principle and Lemma 4.1 we have

f lu(x, t) - v(x,t)ldx ~ f lu(x, 1/n) - v(x,1/n)ldx

~ ^{f (EM(X,}1/n) - u(x, 1/n))dx

+f (EM(x,1/n) - v(x, 1/n))dx

= 2M - f u(x, 1/n)dx - f v(x, 1/n)dx, which tends to 0 as n ^---t 00 from lemma 4.1. Hence u ==v in Q. 0

5. Nonexistence

We first consider the case 1 < q < (N +l)j(N - 1) and p ;::: 1+

(2q*)j(N+1). Recall that q* = max{q, (N+1)/N}. Let u(x, t) be a fundamental solution of (F) with initial data Md, then from Lemma 4.1 we see that u(x,t) is LP-integrable over Q.

Following [IJ, we prove that

THEOREM5.1. For1< ^q< ^{(N +1)/(N-1)} andp;::: 1+(2q*)j(N+

1) thereisno fundamentalsolution of (F).

Proof. Let1J(s) be any smooth non-decreasing function onR such that

1J(S)= {~ ^forfor· s::;^s;:::0,¹ and set 1Jk(S) =^1J(ks).

Ifwe definet/>k(X,y,t) =^1Jk(lxI2+!Yla+t), wherea;::: 2 is a constant to be determined, then we know that

for 0<^E<T. We first have

and in the limit as E--t0 we get

We claim that all the integral in the right tends to 0 as k ^--t 00. It

iT I

ilk

iT I

ilk

(5.2)

then follows that J:[J^{uPdxdt= 0 and u} ==0 in Q. We have

iT I

Ilk

iT I

I kk U,

where Dk = {(x,y,t);t > 0, 0 < Ixl²+Iyla +^t ~ Ilk}. ByHolder inequality we get:

(5.3)

iLk

(iLk

iLk

(ILk

Here we note that

2q* N-l

P -q> 1+ - --q> ^{1-- - q}> ^O.

- N+l - N+l

We also have (5.4)

and our claim will follow from the fact that JJ^Dk uP ---+ 0 as k ---+ 00 if the next inequalities hold:

(5.5) 1 - ( ~1 +- 2 -N+l) (1 - P

1)

As an example we take a = N+l-(N-l)q*'2q* then Nq* - (N+1) a - 2= 2 N +1- (N-1)q* 2::0 and the left sides of (5.5) and (5.6) become

_N+1_2pq* (P-1-~)_{N +1}

and

_ N +1 (p( q* _ q) +q(p _ 1 _ ~»)

2pq* . N +1

respectively. Both of them are nonpositive from assumption. Hence

the proof is completed. 0

We now turn to the case q 2:: (N + l)j(N - 1). We introduce a similarity transformation

and define

U>.(x,y,t) = A!3U(A~X,A-Yy,At).

Suppose that u(x,y,t) be a fundamental solution of (F) with initial data M8(x) and if we assume that

(5.7) then since

(3 - - 2- - ,N-1 = 0,

f

f

(5.8)

If

and

(5.9) limju)..(x,y,t)dXdY = M

t-+O

uniformly in 0 < A~1. It is easy to see that u).. satisfies (5.10)

We also assume that 1 - i - (q-1).B = 0, then together with (5.7), we have

.B=N+t, 2q Now observe that

(N+^{1) -} (N - l)q

i= .

2q

(5.11) (5.12)

N ( N+1)

1- 2i= q ^q-- r ^{> 0,}

N +^{1 (} 2q )

.B+1 - .Bp= - - 1+ - - - ^{P .}

2q N+1

We split into two cases:

Case I: 1 <P<1+(2q)/(N+1).

In this case (5.12) becomes positive. Hence we may consider equa- tion (5.10) as a small perturbation of the reduced equation (R) with convection in the y-direction and diffusion in the orthogonal directions:

(R) Ut -^_ .u.xU -^A U^q-lu y.

The nonexistence result will consist precisely in showing that in the limit A-+ 0, we will obtain a fundamental entropy solution of (R) with mass M > O. This will lead to a contradiction.

It is easy to check that the uniform LOO-estimates of^u).. is still valid.

For every A>0 and everyt >0, we have

(5.13) o~ u)..(x, y, t) ~ CC^E±.l2q for every (x, y) ER.^N

Hence the family {u>..} is uniformly bounded inLoo(R^N X (r,T» and we may extract a subsequence {U>..J~l which converges in the weak star topology ofLOO. We now apply Theorem 26, page 202, of Tartar [9]

(see also Theorem 6, page 57, of Evans [6]) with a minor modification (when we apply the Div-Curl Lemma, we mainly consider the vector fields

in the variables (x,y,t) E R^N- l X R x (0,00), see page 58, [6].) to conclude that along such a sequence

(5.14) u'._{A3 ~} U ill LOO_lac(Q)_, U^q_>"j ~ U^q In LOO_lac(Q)_•

Thus Uis an entropy solution of(R), see [4] for the precise definition.

We now want to check the initial condition. We see from (5.9) that the total mass ofU is M. Moreover

v>..(x,t) =

J

J

(5.15) v>..,t - ~xv>.. ⁼ _>..{3+l-{3P

J

with initial data a Dirac mass, M8(x) in R^{N- l•} Therefore v>.. is bounded from above by the fundamental solution G(x,t) of the lin- ear heat equationin R^N- 1 with massM. Now for any r >0,

{ {u>..(x,y,t)dydx= { v>..(x,t)dx::; f ^G(x,t)dx,

J,xl>rJR J1xl>r J1xl>r

which tends to 0 as t ^--t 0 uniformly in 0< ^{>.. ::;}1. One also sees that

{ 1

1

JRN-l Iyl>r JRN-l 117 12:>"'Yr

which also tends to 0 as t -+ 0 uniformly in 0 < ,X :s; 1. Notice that 'Y :s; 0 and X'Yr ~r. Hence we have from (5.9) that

(5.16) lim { ( u>.(x, y, t)dydx = M

t-+Ollxl<r llyl<r

uniformly in 0 < ,X :s; 1 and limt-+oU(x,y,t) is a Dirac mass with mass M. But this contradicts obviously to the nonexistence result of fundamental entropy solution of (R), see Corollary 5.5, [4].

The above argument does not apply to the second case.

Case 11. p~ 1+(2q)/(N +1).

Nevertheless (5.9), (5.15) and the arguments leading to (5.16) still holds and in particular (5.16) implies

(5.17) t { { u>.(x, y, t)dydxdt ~ ^E

lo l'xl<l llyl<l

for all 0< ,X:s; 1 and for some^E> O. On the other hand

t { { u>.(x,y,t)dydxdt lo l/xl<l lly/<l

= t { { ^{,X/3u(,Xl/2x ,,X"Y}y , 'xt)dydxdt lo llxl<l l'YI<l

= {>. { { ,X-lu(~,'fJ,s)~d'fJds,

lo llf.I<>.1/2 ll'TJI<>.-r .

which is bounded from the Holder inequality by

(

>. ) ^lip

(5.18) ,X~tql(P-l-;%l) ^{{ { {} uP(~,'fJ,s)~d'fJds

lo llf.I<>.1/2 ll'TJI<>.-r

Lemma 4.1 implies theLP(Q)-integrability ofu and this in turn implies that (5.18) tends to 0 as ,X tends to 0, which contradicts to (5.17).

Therefore we have proved

THEOREM5.2. Forq ~ (N+1)!(N-1), thereisnosingular solution of(F).

The above compensated compactness argument used in Case I also works for the equation (E) and we may conclude that

THEOREM5.3. For q~ (N+1)!(N -1), thereisnosingular solution of(E).

References

(1) J. S. Baek and M. Kwak, Singular solutions of semilinear parabolic equations, J. Korean Math. Soc. 32 (1995),483-492.

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

(3) M. Escobedo, J. L. Vazquez and E. Zuazua, Source-type solutions and as- ymptotic behaviour for a diffusion-convection equation, Arch. Rational Mech.

Anal. 124(1993), 43-65.

[4) M. Escobedo, J. L. Vazquez and E. Zuazua, A diffusion-convection equation in several space dimensions, Indiana Univ. Math. Jour. 42 (1993), 1413-1440.

(5) M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in)RN, Journal of functional analysis100 (1991), 119-161.

[6) L. C.Evans, Weak convergence methods for nonlinear partial differential equa- tions, vol. 74, Conference board of the mathematical sciences, Regional con- ference series in mathematics, A. M. S., Providence,1988.

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

[8) M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.

[9) L. Tartar, Compensated compactness and applications to partial differential equations, Research notes in Mathematics39 (1979), 136-212.

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

E-mail: [email protected] E-mail: [email protected]