SELF-SIMILAR RADIAL SOLUTIONS TO A SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS
MODELLING CHEMOTAXIS
By
Yutaka MizuTANi and Toshitaka NAGAi
(Received November 25, 1994)
(P)
1. Introduction
We are concerned with the system of partial differential equations Ou - = 7• (7u - xu7v),
ot
Ov 6-=]V-7V+ctU
ot
for xGRN and tÅr O, where x, e,7 and ct are positive numbers, and u(x, t) and v(x, t) are positive functions.
This system is a mathematical model of chemotaxis (aggregation of organisms sensitive to a gradient of a chemical substance) proposed by Keller-Segel [7]. u(x, t) and v(x, t) denote the cell density of cellular slime molds and the concentration of the chemical at place x and time t, respectively. It is assumed that u and v satisfy
(1.1) u(•, t), v(•, t)EL'(RN) for tÅr O.
Blow-up phenomenon is conjectured to the system (P) by Nanjundiah [9], Childress and Percus [2] and Childress [3], but the blow-up problem is still open. Yagi [10] has studied some norm behavior of maximal solutions to more general parabolic systems including (P). He has also shown in [11] that there exists a soiution of the Cauchy problem to (P) globally in time if IIu(•, O)llLi is sufficiently small.
A limiting system as Åí.O in (P) has been considered by Jager and Luckhaus [6], and the global existence and biow-up of solutions have been studied under -
homogeneous Neumann boundary conditions on a bounded domain in R2. Nagai [8] and Diaz and Nagai [5] have considered another limiting system as 6.0 in (P).
[8] has studied the global existence and blow-up of radial solutions to the limiting
system under homogeneous Neumann boundary conditions on a bounded domain in RN. [5] has shown the global existence of solutions to the limiting system under
homogeneous Dirichlet boundary conditions on a bounded domain in RN without
20 Yutaka MizuTANi and Toshitaka NAGAi
radially symmetric assumptions.
For the study about self-similar solutions we impose the assumption (A) on 7.
(A) 7- O.
Under (A) we look for special positive solutions (u, v) of (P) called self-similar solutions having the form
(1.2) u(x, t) == -l- (p(111Zli), v(x, t) =: ifr(IllZll)
for xERN and tÅrO. By (1.1) the positive function (op, W) must satisfy (i.3) jl,OO (p (r) r"-' dr Åq oo, jl,co u(r) r"-' dr Åq oo.
For the existence and non-existence of (op, ut) we confine ourselVes to the case N == 2.
THEoREM 1. Under (A) and N == 2, there exists a positive self-similar radial solution (u, v) of (P), having the form (1.2), rvhich satisfies (1.1) and
,1.im. u(x, t) = O, Jlm. v(x, t) -= c for xER2,
where c is a positive constant.
We next consider another type of positive self-similar radial solutions (u, v) of (P) having the form
(1.4) u(x,t)-T!,q(Xl ), v(x•t)-w(Sl)
for xER" and tÅq T. This solution (u, v) blows up at t=Tif it exists.
THEoREM 2. Under (A) and N =2, there are no positive solutions of (P) having the form (1.4) which blow up in finite time.
2. Self--similar solutions
Under (A), (P) is invariant under the similarity transformation uA(x, t) = 12u(Ax, Z2t), v,(x, t) = v(Zx, Z2t)
forZÅr O, that is, if (u, v) is a solution of (P) then so is (uA, v2). We look for solutions invariant under this transformation such that •
(2.1) u(x, t)=u2(x, t), v(x, t) == vA(x, t) for all ZÅrO.
Letting A = 1/N/i, we see that (u, v) has the special form
u(x, t) =l op(i), v(x, t) = !,L((i)
for xERN and tÅr O. By a direct computation, it is shown that (u, v) satisfies (P) if and only if (op(x), W(x)) satisfies
(2.2)
1
7• (7Åë - xop7W) +ix•7op + op - O,
e
AW +-x• VW + ctq =O 2
for all xER". When we confine ourselves to radial solutions of (2.2) depending only on lxl, the self-similar radial solution (u, v) of (P) has the form (1.2). The function (op(r),W(r)) on [O, oo) in (1.2) must now satisfy the system of ordinary differential equatlons
(2.3)
N-1 r
((p' - x(p iP(')' + , (q)' - xq)IPf') + - ll q)' + q) = O,
ut" +("i' +gr) w• + ctop -o
for rÅr O. At r=O we need
(2.4) q'(o)-v'(o) =o.
When we have Z= 1/N/fii =i in (2.1), solutions of (P) have the form
u(x, t) = t op (t), v(x, t) = W( ti ),
and (q(x), W(x)) satisfies
1
7• (7q - xop7W) --x• 7op - op = O,
2 8
Aifr - -x • VW + ctq) == O.
2
Assuming that (ep, W) is radial, we see that (u, v) has the form (1.4) and (op, W) satisfies
(2.5)
N-1 r
(q)' - x(p ifr')' + , (q)' - xop (fr') - i go' - (p - O,
wtt .(-N.rm; ' . g,) w• + ctop - o
22 Yutaka MizuTANi and Toshitaka NAGAi
for rÅrO. We aiso need the condition (2.4) at r=:O.
3. Existence
In what follow, we assume N == 2. Let us look for a positive solution (op, W) of (2.3) satisfying (1,3). By N= 2, we see that the first equation in (2.3) implies
{2r(q)' - xq)ifr') + r2 q)}' =O for rÅr O, from which it follows that
2r(op' - xoput') + r2q = O.
Dividing this relation by 2rop, we have
r
(log op - zut)' +- - O.
2 Hence,
(3.1) ep=J,e-"214exW
for some ZÅr O. Substituting (3.1) into the second equation in (2.3), we get lfr" + (;. + IIii- r) lfr' + ctAe-"2f`eXut .. O
forrÅrO. By putting
w(r) =: xut(r) and " = ctzZ,
our problem is reduced to finding a positive solution w on [O, oo) of
(3.2)
w" +(-ili + g r) w' + "e-"2!4ew =O for rÅr O,
w' (o) =: o satisfying the condition
(3.3) j],co w(r)rdrÅqoo,
To find a positive solution of (3.2) satisfying (3.3), we rely on the shooting method (see [1, 4]). Let us begin by the following lemma for a solution of (3.2) with w(O) År O.
LEMMA 1. w' (r) ÅqO for rÅr O.
PRooF. It follows from (3.2) that w satisfies
(eÅír214rw')' .. - Ltre(E-1)"2!4eW Åq O for rÅr O. Hence,
eÅír214rw'ÅqO for rÅrO,
which implies that w'(r) ÅqO for rÅr O.
By (3.2), w' is represented as
(3•4) w'(r) =- -pa
r-e-Åí'2f` jj pe(Åí-')p2!4ew(p)dp,
from which it follows that
(3.s) w(r) =w(o) - pt jlg -l e-ec2!4[jlj: pe(e-op2!4ew(p)dp]d4
for rÅrO. Let us define I(6) by
I(e) == f,co te-eC2i` If,C pe(e-i)p2f4dp]dc
LEMMA 2. I(s) is finite and is represented as
log6 if 6# 1, 6-1
I(e) =-
1 ifÅí== 1.
PRooF. WhenÅí= 1, it is easy to see I(e) = 1. Hence, we shall prove in the case s# 1. We note that I(Åí) is rewritten as
i(e) = , 2 i f,oo (e-4i` - e-Ee2i`) tdc = g2-i i,.
Let us claculate Io in the following way.
Io == ,-1,i,rp-. (Åí" e-e2f4 idc - Åí" e-Eg2!4 }d4)
ff
- ,-i,i',m.. (jll/,2 e'p2 i dp - j'I)ll",I,2 e-p2 ; dp)
Here we used the coordinate transformations e = 2p in the first integral and VE4 = 2p
in the second one. Further calculations yield that
24 Yutaka MizuTANi and Toshitaka NAGAi
Io == ,.1,ilep.. (f,f',Oi2 e-"2 ; dp - f.f',"!2 e-p2 ; dp)
= }i-m,, f,f',ji2 e-P2 ; dp - .llm. f.f',"!2 e-p2 ; dp
- }i.m, ÅírllYf2 ,-p2 ;; d,
= }i.., (ÅíIE,b'i2 l; dp + ÅíIIIib'f2 (e-p2 - i) l; dp)
- gioge+ }i.m, Åíf',j/2 (e-p2 - i) l; dp
i =-loge.
2
Thus, we have obtained the desired relation.
By (3.5) and w(r) Åq w(O) (r År O), we see that w(r) ;}i w(O) - ptI(s)eW(O) for rÅr O,
which together with lemma 1 gives us the following lemma.
LEMMA 3. Iim,-.w(r) always exists and is finite.
Let us put
V = pt(8).
With v, the nonexistence of positive solutions is stated as follows.
PRoposiTioN 1. if v }ir 1, then w(oo)ÅqO. Hence, there are no positive solutions of (3.2).
PRooF. Since w(r) is decreasing, by (3.4) we have w'(r) Åq - ll e-er2!4 jll pe(E-op2i4dpew(r),
from which it follows that
(- e-w(r))' Åq - IIt e-er2i4 jlj pe(e-op2!4dp.
Integrating this inequality on (O, oo) yields that
e-W(O) - e-W(OO) Åq - Izl(6) = - V•
Hence,
w(oo) Åq - log (v + e-w(O)),
which together with v21 implies w(co)ÅqO. Thus the proposition is completely proved.
LEMMA 4. lf w(O) s{ v, then w(co)ÅqO.
PRooF. Since w(r) is decreasing, it follows from (3.5) that w(co) Åq w(O) - ve"(co).
By w(O) s; v,
w(oo) + veW(co) Åq v,
which implies that w(oo)Åq O. This completes the proof.
Our next consideration is to give some condition on w(O) under which the positivity of w(cx)) holds. For this purpose, we use (3.5) to get
w(cxD) År w(O) - vew(O).
Let us define the function f(s) on R by f(S) = s - veS.
When O Åq v Åq 1/e, there exist positive numbers co, ci, satisfying v Åq co Åq ci, such that f(c,) = f(c ,) = O, f(s) År O (s e (c,, c,)), f(s) Åq O (s Åë [c,, ci] ).
When v == 1/e, f(1) = O andf(s) Åq O for s l 1. Hence, we have the following lemma.
LEMMA 5. (i) Let OÅqvÅq1/e. if co :{g w(O) f{{ ci, then w(oo)ÅrO.
(ii) Let v = 1/e. If w(O) = 1, then w( co) År O.
For a given aER, we denote the solution w of (3.2) with w(O) =a by w(r; a).
LEMMA 6. w(co;a) is continuous with respectto a.
PRooF. By (3.5), w(r;a) is rewritten as
(3•6) w(r; a) -= a-"Sg g(r, 4)ew(e;a)d4,
where
g(r, C) ,. 4e(E-i)C214 Åí' e-Åíp2!4 ; dp.
26 Yutaka MizuTANi and Toshitaka NAGAi
Let ai Åqa2. Using (3.6) yields that
lw(r; ai) - w(r; a2)l -Åq lai - a2l + itt jlj g(r, olew(e;ai) - ew(e;a2)td4
g la, - a,1 + uea2 jjg(r, olw(4; a,) - w(g; a,)1de.
Here we used w(r;ai)sa2 for i= 1,2. Note that
g(r, C) -Åq h(4) i Ce(E-')42!` feco e-EP21` ; dp
and h is integrable on (O, oo). We then have
(3•7) lw(r; ai) - w(r; a2)1 g lai - a21 + pte"2 jlj h(4)Iw(g"; ai) - w(e; a2)ld4.
By using Gronwall's inequality, (3.7) implies
lw(r; ai) - w(r; a2)I s{ exp IC(a2) jlj h(C)d4] lai - a2l,
where C(a2) ="exp {a2}. Letting r. cx), we obtain
lw(co ; ai) - w(oo ; a2)i g exp IC(a2) f,co h(e)d4] lai - a21,
which implies that w(oo;a) is continuous with respect to a.
PRoposiTioN 2, Let OÅqv sg 1/e. Then there exists a positive solution w qf (3.2) satis.fying w(oo) == O. Moreover, the solution w satisLfies
w(r) r' co R(r) ,fbr some cÅr O, where
R(r) = f,co ee-E421` If,S: pe(E-')p2!`dp] d4
PRooF. By Intermediate-value theorem there exists an initial value aÅrv such that w(oo;a) == O. Integrating (3.4) on (O, oo), we have
w(r, a) == iz jl,CD ee-sg2f4 Ijlli pe(e-i)p2i`ew(p•")dpl d4
for r2 O. By L'Hospital's rule,
(3.g) ,i!m. -W X"i)") = ,i-im. I" jlj pe(s-i)p2i4ew(p;a)dp/jlj pe(c-i)p2!4dp]
When OÅq6Åq 1, both the integrals in (3,9) converge as r. oo. Hence, lim w(r; a) =c
r-co R(r)
for somecÅrO. Whens2 1, by aplying L'Hospital's rule to (3.9) again, it is shown that
w(r; a)
lim =pa
r'co R(r) Thus the proof is complete.
We are now in a position to give the proof of Theorem 1.
PRooF oF THEoREM 1. Let w be a positive solution with w(oo)=O stated in Proposition 2. Define the function W(r) by W(r)=w(r)/x and op(r) by (3.1),
respectively. The positive function (op, W) is a solution of (2.3) satisfying op'(O) = W'(O) = O. By (3.1), rop(r) is integrable on (O, oo). Note that
(3.10) R(r) f{g Ce'K'2 (rÅrO),
where C and K are positive constants depending only on 6. By (3.8) and (3.10), rifr(r) is integrable on (O, oo). As concerns the asymptotic behavior of (u, v) defined by (1.2), it is easy to see that
lim u(x, t) =O, lim v(x, t) =W(O)ÅrO
t--, oo t- co
for xER2. This completes the proof of Theorem 1.
4. Proof of Theorem 2
We shall prove the nonexistence of positive solutions (u,v) having the form (1.4), Let (op, V) be a function in (1.4). As in Sect. 3, by the first equation in (2.5)
we have
(4.1) op=Ae'2!4eXut
for some Z År O. Putting w(r) = zW(r) and substituting (4.1) into the second equation in (2.5), we get
wn + (.l. - g r) w' + pter2!4ew = o, rÅr O,
(4.2)
