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On the global existence of solutions of parabolic differential equations with a singular nonlinear term
Nonlinear Andvsis, Throry, Methods & Applicaftons. Vol. 2 No Q Pergamon Press Lid 1978. Printed tn Great Brrtain.
~362_546X/78!0?01-()499
4 pp. 499-504
%0200/o
ON THE GLOBAL EXISTENCE OF SOLUTIONS OF PARABOLIC DIFFERENTIAL EQUATIONS WITH A SINGULAR NONLINEAR TERM ANDREW ACKER and WOLFGANG WALTER Math~matisches Institut 1, Universit~t Karlsruhe (TH), Englerstrasse 2, Postfach 6380.7500 Karlsruhe 1, Federal Republic of Germany
Key word: Nonlinear parabolic differential equations, global existence, inequality methods.
THEFOLLOWING problem was recently posed by H. Karawada [ 1J. For any a > 0, let 0 < T, Z$ CXJ be the largest value such that a soIution z&x,t) < 1 exists for the boundary value problem:
21,= n,, + Ml - ~1 on (-a, a) x (0, ?,,I, u(x, 0) = 0
for 1x1 < a
uf-a,t)=u(a,t)=O
(1)
and
for O
(2)
If T, = c/3,then u(x, t) < 1 exists globahy on [-a, u] x [O, co). If ‘-r, < co, then max (u(x, &)I -a SZx ,( u> --+ 1 as t --f T, - 0. In the latter case, we say (using Karawada’s terminology) that u “quenches” at T,. Blow-up phenomena for parabolic differential equations, where solutions become infinite in finite time, have been extensively studied. Quenching is a related phenomenon in which the solution remains finite, but the derivatives blow up as t + T, - 0 due to the singularity in the nonlinear term. The problem is to determine the values of a > 0 for which T, = cs3.It is easily seen that a value a, > 0 exists such that T, = co for 0 < a < a, and T, < 00 for a > a,. Karawada has given the estimate a, $ ,/Z. The second author used differential inequality methods in [2] to obtain the bounds 0.765 < a, 4 n/4. In this paper, we show how to determine a, for the following more general problem: U, = UXX+ .#k uX) on (- 4 4 x (0,7J, u(x,O) = 0
for 1x1 < a
Z.&(--a,t) = u(a, t) = 0
(3)
and
for 0 < t < T@.
(4)
Heref(z, p) is a continuously-differentiable real function on CO,b) x (- co, xl) (where 0 < b < “o) such thatf(0, 0) > 0 andf(z, p) -+ + co uniformly over all p in any bounded interval as z --+b - 0. The determination of a, is based on the corresponding stationary problem: w,, + f(w, WJ = 0
on (-a, a),
wf - a) = w(a) = 0.
15) (6)
Let 0 < a, -=zcx3be the supremum of all values a > 0 for which a solution 0 d w < b of equations 499
500
A. ACKER AND W. WALTER
(5) and (6) exists. We will show that a, = a, whenf(z, p) satisfies certain regularity conditions. The proof that a, = a, in the simpler case wheref(z, p) = f(-)L is independent of p was obtained by the a_uthors in [3]. In the case of Karawada’s problem (equations (l), (2)), one can show that a, = J2 * M N 0.765, where M is the maximum for s > 0 of the function & exp(t* - s*) dt. Thus the relation u,* = a, gives a, explicitly. Our proof ts based on several qualitative properties of solution profiles u(x, to) (to fixed) which are of interest independent of the above problem (Theorem 1). They are obtained using differential inequality methods. 2. PARABOLIC DIFFERENTIAL INEQUALITIES Let G be a bounded region in R2 with boundary aG and closure G. The parabolic boundary a,G of G consists of those points (x, t) E aG such that for every E > 0 the lower neighbourhood N;(x, t): = ((x, 4 E RZI f -c t and (Z - x)” f (f - t)’ < c2} intersects the complement of G. The parabolic interior GP of G is defined by G, = G\(a,G). If G = (-a, a) x (0, T), then d,G = [-a, u] x (0) u { --a, u) x (0, T] and GP = (-a, a) x (0, 7’1. We always assume for a function u (or v, w, . . .) solving a parabolic equation or inequality on G that u is continuous on G and that u,, uX,and u,, are continuous in G. NAGUMO’SLEMMA.Assume for any values B E (0, b) and 0 < p0 < a that a constant L, exists such that f(Z, p) - jjz, p) d L&Z - z) whenever 0 < z < Z < B and IpI < po. If u, (or uX) is uniformly bounded in G, then n, - (urX + f(a, u,)) B 0, - (a,, + f(r, v.J
in G
and u 2 v ona,G implies u 2 u on G. STRONGMAXIMUMPRINCIPLE.Assume bounded functions c, d: G + R exist such that u, $ u,, + cu, + du
in G
and u2 0
on f3,G.
Then u > 0 on G. Furthermore, if u(xO, to) = 0 at a point (x,, to) E Gp, then u(x, t) = 0 at every point (x, t) E G which can be connected to (x,, to) by a directed curve in G (beginning at (x, t)) along which t is (weakly) monotone increasing. Both the above theorems are stated here in a special form sufficient for our purposes. The proofs are found in [4; 24 VI] and [5, Chapter 3, Section 31. THEOREM1. Let u(.x,t) < b solve (3) and (4) on [--a, u] x [0, T,), where f~ C’([O, b) x R) and f(0,0) > 0. Then:
On the global existence of solutions of parabolic differential equations with a siniular nonlinear term
501
(a) u > 0 and u is strictly monotone increasing in t on (-a, a) x (0, TJ. (b) For each 0 < t < Ta, u(x, t) is monotone increasing in x on [-a, b(t)] and monotone decreasing in x on [6(t), a], where --a < (p(t) < a. (c) Iff(z, p) is monotone increasing in z on [0, b) x R, then for each 0 < t < T, there exist values -a < 4l(t) < 4(t) G (6*(t) < a such that uX(x,t) is monotone increasing in x on II-a, d&r)] u [&( t ), a I an d monotone decreasing in x on [4,(t), #z(t)]. ProoJ: (Part (a)). Let u&(x,t): = ct*cos@c/ta). Sincef(0, 0) > 0, there is for any T E (0, 7J an E > Ososmallthatu: < uzX+f(u”,Qon(--a,a) x (0, TJThusu & t? > Oon(-a,a) x (O,T] by Nagumo’s Lemma. Since f E C’([O, b) x R), u, is continuous and therefore bounded on [-a, u] x [O, T] for any T E (0, T,). Also u g B (. b on [-a, a] x [0, T] (I? a constant). Thus if v(x, t): = u(x, t + h) - ufx, t) then it follows from fo Cl([O, b) x R) that v, = yXXf c* uX+ d *v on (-a, a) x (0, T - h), 0 < T - h < T G T,, where c and d are uniformly bounded functions. Since u(x, 0) > 0 for Ix] c a and o( _t a, t) = 0 for t 2 0, the strong maximum principle implies that u > 0 on (-a, a) x (0, T - h], from which the strict monotonicity foilows. Part (b). For any 0 < h < a, let w = uh - U-~ on [-a C h, a - h] x [0, T,), where P(x, t} = u(xfh,t).Thenw(-a+h,t)~Oandw(u-h,t)~OforO~t~T,andw,=w,,+c~w,f d * w on (-a + h, a - h) x (0, T,), where the functions c and d are bounded on any finite subset. For any fixed 0 < T < T,, the component G, (respectively C_) of [-a -t h, a - h] x [O, T] containing f-u + h) x (0, T] ({a - h) x (0, T]) on which w > (0 0 is simply connected by the strong m~imum principle. By the continuity of w we have w = 0 on 8~. where y: = ([ - a + h, a - h] x [O, T])\(G+ u G_). Therefore w = 0 in y by the strong maximum principle. It is easily seen that f(T) n (G, u y) and i(T) n (G_ u y) are intervals, where I(T) = t--u + h, Q - h] x (Tj. Since 0 < T < T, is arbitrary, the assertion in (b) follows from this. Part (c). If for some 0 ( T < T, the value 4,(T) B [--a, #(T)] (such that u,,(x, T) 2 0 in (-a, #,(T)] and uXX(x,Tf B 0 in C+,(T), #(T)]) did not exist, then for 0 < h < a su~ciently small there would exist points --c1 < x1 < xz < 4(T) such that CI:= w(xI, T) = w(x2, T) > 0 and w(x, T) < a for x1 < x < x2. (We use the notation in Part (b).) Let G, be the component of (-a + h, a - h) x (0, T] containing (x1, x2) x {T) in which w < CI.It follows from the continuity ofw that w E E on (8G,) n (( - a + h,a - h) x (0, T)).Sincew = Oon~,wehaveG~ c G,. Since w( - a + h, t) > 0 is monotone increasing in t, one sees that w E CIon a,G,. Now w satisfies the equation w, = w,, + [f(u-” + w, u, h + WJ -~(u-~,u.;*]) in (-a + h,a - h) x (0, T], where f(z, p) is monotone increasing in z. Thus, Nagumo’s Lemma implies that w > a in G,, This contradiction implies that #,(T) exists.
Let z((x, t) < B -c b solve (3) and (4) on f-u, o] x [0, crs), where So C”([O, b) x R) and f(0, 0) > 0. Then /u,(x, t)f is uniformly bounded on [-a, u] x [0, ‘0) provided that a constant L(B) exists such that whenever 0 < , z < B and [pi 2 1 the following inequalities hold:
THEOREM 2.
f(z, P) G w9P2 f,(z, P) d w4 -pf,kpJ
(7) IPi
G W)p2.
(8)
(9)
proof. Sincefe Cl([O, b) x R), u, is continuous on [-a, a] x [0,00). We first prove that luxl is uniformly bounded on { -a> x [0, co). Choose L > max{L(B), (7c/4a)},and define
502
A.
$(x) = (l/L) *Ln
ACKERAND W. WAL.TER
sin(c: f L *(x + a)) sin(s) C 1
forO
L-(x+
a)
where 0 < E < (7c/2)solves sin(s) = exp( - L *B). Letx, = (l/L)*@/2 -8) - a&(--,a).Thenrl/(-a) = 0,$(x,) = B 3 u(x,,t)forallt 3 0,and JI,, = - L* (1 + ($J”) on (- a, x,,). It follows using (7) and Nagumo’s Lemma that 0 < u(.x,t) ,< tl/(x) on [-a, x0] x [O, co). Therefore 0 ,< uX(-a, t) f $,(--a) for all t 3 0. The proof that ~ux~is uniformly bounded on (u) x [0, co) is similar, Now, define v = u, + y(u) on C-u, u] x [0, c;o),where y(s) = exp(3Ls)and L = max{ 1, L(B)). Clearly, there is a constant M such that IuX]d M and 1u) G M f y(B) on the parabolic boundary. Assume for 0 < T -c co faed that V: = sup(~(x, t)l - a d x d a, 0 4 t ,< 7’) 2 A4 -+-y(B) -t- 2. We will show this leads to a contradiction. The set S c f-a, a] x [0, T] on which 0(x, t) = Y is a compact subset of (-a, a) x (0, T]. Thus, for CI> 0 sufficiently small there is a compact c-neighbourhood S, of S (relative to [--a, u] x [O, T]) such that ,S, c (-a, a) x (0, T] and u d Y - a in ([--a, u] x [O, T))\S,. For 0 < h < 2u, define w = 6u t y(u) on [-a, a - h] x CO,co), where 6u = (l/h)(u(x + h, t) - u(x, t)). Due to the uniform continuity ofn, on [--a, a] x [0, T] we have sup{{&{n, t) - u,{x, t)l: -a < x < a - h, 0 G t ,< T> + 0 zssh-+O.Therefore, if h > 0 is su~ciently small, then W: = sup(w(x,t):-a~~xu--_,0dt,
w, = w,, + jlpw,- 9~~(~~~~~)~+ 3L~~~)~~-~~~*) + .Du, where at each (x, t), fp = fp(u{.x*, t), u,(x*, t)) and fz = j&(x*, t), u&x*, t)) with 0 < x* < x + h. At the point (x, t) E S’ c S, with minimum t-coordinate we have w, 2 0, w’, = 0, and w,, Q 0, from which it follows that 9~~(~)~2 < 3L~(~~~(~- &p) + jx,Su (where p = u,(x, r)). Since p > 1 in SE,the estimates (8) and (9) lead to 9~~(~)pz f ~LY~~)(~L~~)+ LP’ + 3L~~~)~(& -
j*,)+
jy(S, - P) + (j”, - f,)~.
Since S, is compact, we arrive as h -+ 0 at the inequality 9cp2 < (M? + L)p2 * 3L 6 1, which is a contradiction. Thus, u, < u < M + y(B) -I- 2 in E-4 aJ x [0,7’3 for all T > 0. The same argument using u = -u, + y(u) bounds u, from below on [-a, u] x [0, co). THEOREM
on [-u,uJ
3. Under the assumptions ofTheorem 2, Qx, t) and uX,(x, t) are both uniformly bounded x (0,‘~).
Proof. Under the assumptions onfl u, and uXXare continuous on [--a, u] x [O, “3) except at the points (-a, 0) and (a, 0). The proof of Theorem 3 from Theorem 2 is based on the functions v = u, + y(u) and w = 6,~ + y(u), where 6,u(x, t) = (l/h)(u(x, t + h) - u(x, t)). The proof is omitted, since the steps are essentially analogous to those in the proof of Theorem 2.
On the global existence of solutions of parabolic differential equations with a singular nonlinear term
503
3. THE MAIN RESULTS assume throughout this section that ,#‘EC’([O, b) x R), f(0,0) > 0,and f(z, p) -+ + CQ uniformly on any finite p-interval as z -+ b - 0. Furthermore, for any B F (0, b) there is a constant We
L(B) such thatf(z, p) satisfies (71,(8), and (9) whenever 0 6 z $ B and \p[ >, 1. If u solves (3) and (4) on [-a, u] x [O, T] and w(x, T) B B K b for all 1x1 6 a, then u can be extended to a solution of (3) and (4) on [-a, a] x [O, 7’ C s] for some E > 0. Therefore either T, = a (giobai existence) or else the solution of (3) and (4) on [-a, u] x [0, ‘I’J quenches at T,. Nagumds lemma implies that the solutions of (3) and (4) are monotone increasing in a. This means if u and u’ solve (3) and (4) on respectively [-a, a] x [O, T] and [-u’ a’] x [O, T], 0 -c a < a’, then u‘ B u 2 0 on [-a, u] x [0, ?“‘I).Therefore T, is a monotone decreasing function of a. It follows that a value a, 3 0 exists such that T, = 00 for 0 < a < a, and T, < GOfor a > a,. LEMMA 4. Assume 0 < a < a,. Then a constant 0 < B < b exists such that the solution u of (3)
and (4) on [-a,
u] x LO,co) satisfies u(x, t) 6 B.
Pmctp. We assume that no upper bound B < b exists and obtain a contradiction. if this were the case, we would have lim ol(&t), t) = b. (This is because u(#(t), t) is monotone increasing in t on [O, xi), where # was bezned in Theorem I (b).) Thus, one could choose 6 > 0 such that tl + 6 < a*, then 0 < E < 1 such that a2 f 2~/t5* G S(z, p) for all (z, p) E [b - I, b) x f - 2/6,2/J], and then t, Y 0 such that u(#(t& co) 2 b - 8. Let tid be the solution of (3) and (4) on E-u - 5, a + S] x [O, co). For each 6’ E [-S, S], Nagumo”s Lemma shows that n&x + 8, 1) 2 U(X,t) on [--a, a] x [O, r;)). Thus (using Theorem I (a)) u&x, t) B b - E on the strip S: = f&t& - b, dt(Gf + a] x P,> “c). On the other hand, the function ti(x, t): =b - E + (r - t,)fb’ - (x ~(~~~~2~ is equal to ti - E on the parabolic boundary of S. Furthermore, vr G o, i- _QY,u_~)in S, : =[#t,) - 6, rfift,) + 6] x [to, t, -t c/PJ. Hence, t16> D in S,. In particular, ~~~~(~~),fQ + s/S”) 2 ~~~(~~),t, + s/s”) = b. This shows that u&x, E)quenches in contradiction to the choice of6 > 0.
Proof. For the proof that a, 3 a,, we show for every a E (0, a,) that a G a,. TfO < a < a,, then (51 and (6) are solved by a function 0 d w i(x) < h on [ - LI~* GJ for some a, E (a, aJ Pu’agumo’s Lemma implies that u < wX< b on C-n, af x [O, T,). Thus, T = m and Q G a*. To prove that G, d a,, we show for every a E (0, a+) that a < izo’If0 < a < a,, then by Lemma 4 a constant B E (0, b) exists such that the solution u of (3) and (4) on [-a, G] x [OF60) satisfies u d B. By Theorem 3, faXXl f SZC (a constant) on [-a, u] x (0, mf. Thus, given an unbounded sequence ft,), there is by Ascoti-Arzel& a subsequent (t,J and a Lipschitz continuous function c(x) (with Lipschitz constant C) such that u&, t,,) -+ U(X)uniformIy on [ - ~1,a] as la -+ ocj. Let I W(X) = f”-, v(x’) dx” on [--a, a]. Then U(X~ tJ -+ W(X)uniformly on [--a, a] as n + e. Since U(X,t) is monotone increasing in t for each x E [- u, a], we have s(t): = max~w(.~)- u(x, t): -a < x d of + 0 as t -+ CO(uniform convergence). Clearly w( -n) = w(a) = 0 and 0 d W(X)< I3 on [--a, a]. We will now show that u,(x, t) -+ wX(x) uniformly on C-u, a] as t -+ m. For 0 < h < 2a, define &w(x) = (l/~)(w{~ f h) - W(X))on [-a, a - h) and Gwjx,t) = (l/h)(u(x C h, t) - U(X,t)) on E-a, a - h] x [O, cry)).Then (~w(x) - W,(X)/d Clz and l&(x, t) - zc,(%t)\ d Ch for -a < x < a - h and t 3 0. Also, j&(x, t) - Gw(x)j & (2/h) E (c). By combining inequalities, we obtain (u&, t) - w,(x)/ $ 2Ch + (2/h) E (t). The choice h = &@j yields
504
A. ACKER AND W.WALTER
1u,(x, 0 - w,cdI G 2(C + 1) x/m
(10)
for all x E [-a, a - ,/$J, for each t 2 0. The same argument based on left-hand differences shows that (10) holds for all x E [-a + ,/$), a] and t 2 0, so that the assertion is proved. Now define U(x, t) on [-a, LX]x [0, cc) by U(x, t) = Si” u(x, r) dr. Then ffl u(x, t + 1) - u(x, t) = UXX(X, t) -If(u(x, r), ux(x, r)) dr sf on [-a, u] x [0, CD).Sincef(z, p) is uniformly continuous on any compact subset of [0, b) x R, it follows from the above convergence results that U,,(x, t) + -f(w(x), w,(x)) uniformly on [ - a, a) as t + CD.Therefore, one concludes from the Taylor formula U(x, t) = U(o, t) + x . U,(O, t) + (for/x\ < a
‘(x - x’). UJx’, t) dx’ s0
and t 2 0)
that w(x) satisfies the equation w(x) = w(0) + x *w,(O) -
x (x - x’) .f(w(x’), w,(x’)) dx’ (1x1< a).
s0 Therefore, w(x) satisfies (5) and (6) on [-a,
a], implying that a d a,.
Remark. The result a, = a0 does not determine
whether or not the solution of (3) and (4) is global when a = a,. Of course if a solution 0 < w(x) < b of (5) and (6) on [-a,, a,] exists, then q, = co.
Remark. Theorem 5 shows that a solution of (5) and (6) on [-a,
u] exists for every a E (0, a,).
REFERENCES 1. KARAWADA H., On solutions of initial-boundary problem for u, = u_ + l/(1 -g), Publ. RIMS, Kyoto Univ. 10, 729-736 (1975). 2. WALTER W., Parabolic differential equations with a singular nonlinear term, Funkcialaj Ekuacioj 19,271-277 (1976). 3. ACKER A. & WALTER W., The quenching problem for nonlinear partial differential equations, Ordinary and Partial Dij,‘,rential Equations, Dundee 1976; Lecture Notes in Mathematics, No. 564. pp. l-12. Springer-Verlag (1976). 4. WALTER W., Differential and Integral Inequalities, Ergebnisse der Mathematik und Ihrer Grenzgebiete, Vol. 55. Springer-Verlag-(1970). 5. PROTTEXM. H. & WEINBERGERH. F.. Maximum Princioles in Differential Eauations. Prentice Hall. Enelewood Cliffs. __ _ N.J. (1967).
~362_546X/78!0?01-()499
4 pp. 499-504
%0200/o
ON THE GLOBAL EXISTENCE OF SOLUTIONS OF PARABOLIC DIFFERENTIAL EQUATIONS WITH A SINGULAR NONLINEAR TERM ANDREW ACKER and WOLFGANG WALTER Math~matisches Institut 1, Universit~t Karlsruhe (TH), Englerstrasse 2, Postfach 6380.7500 Karlsruhe 1, Federal Republic of Germany
Key word: Nonlinear parabolic differential equations, global existence, inequality methods.
THEFOLLOWING problem was recently posed by H. Karawada [ 1J. For any a > 0, let 0 < T, Z$ CXJ be the largest value such that a soIution z&x,t) < 1 exists for the boundary value problem:
21,= n,, + Ml - ~1 on (-a, a) x (0, ?,,I, u(x, 0) = 0
for 1x1 < a
uf-a,t)=u(a,t)=O
(1)
and
for O
(2)
If T, = c/3,then u(x, t) < 1 exists globahy on [-a, u] x [O, co). If ‘-r, < co, then max (u(x, &)I -a SZx ,( u> --+ 1 as t --f T, - 0. In the latter case, we say (using Karawada’s terminology) that u “quenches” at T,. Blow-up phenomena for parabolic differential equations, where solutions become infinite in finite time, have been extensively studied. Quenching is a related phenomenon in which the solution remains finite, but the derivatives blow up as t + T, - 0 due to the singularity in the nonlinear term. The problem is to determine the values of a > 0 for which T, = cs3.It is easily seen that a value a, > 0 exists such that T, = co for 0 < a < a, and T, < 00 for a > a,. Karawada has given the estimate a, $ ,/Z. The second author used differential inequality methods in [2] to obtain the bounds 0.765 < a, 4 n/4. In this paper, we show how to determine a, for the following more general problem: U, = UXX+ .#k uX) on (- 4 4 x (0,7J, u(x,O) = 0
for 1x1 < a
Z.&(--a,t) = u(a, t) = 0
(3)
and
for 0 < t < T@.
(4)
Heref(z, p) is a continuously-differentiable real function on CO,b) x (- co, xl) (where 0 < b < “o) such thatf(0, 0) > 0 andf(z, p) -+ + co uniformly over all p in any bounded interval as z --+b - 0. The determination of a, is based on the corresponding stationary problem: w,, + f(w, WJ = 0
on (-a, a),
wf - a) = w(a) = 0.
15) (6)
Let 0 < a, -=zcx3be the supremum of all values a > 0 for which a solution 0 d w < b of equations 499
500
A. ACKER AND W. WALTER
(5) and (6) exists. We will show that a, = a, whenf(z, p) satisfies certain regularity conditions. The proof that a, = a, in the simpler case wheref(z, p) = f(-)L is independent of p was obtained by the a_uthors in [3]. In the case of Karawada’s problem (equations (l), (2)), one can show that a, = J2 * M N 0.765, where M is the maximum for s > 0 of the function & exp(t* - s*) dt. Thus the relation u,* = a, gives a, explicitly. Our proof ts based on several qualitative properties of solution profiles u(x, to) (to fixed) which are of interest independent of the above problem (Theorem 1). They are obtained using differential inequality methods. 2. PARABOLIC DIFFERENTIAL INEQUALITIES Let G be a bounded region in R2 with boundary aG and closure G. The parabolic boundary a,G of G consists of those points (x, t) E aG such that for every E > 0 the lower neighbourhood N;(x, t): = ((x, 4 E RZI f -c t and (Z - x)” f (f - t)’ < c2} intersects the complement of G. The parabolic interior GP of G is defined by G, = G\(a,G). If G = (-a, a) x (0, T), then d,G = [-a, u] x (0) u { --a, u) x (0, T] and GP = (-a, a) x (0, 7’1. We always assume for a function u (or v, w, . . .) solving a parabolic equation or inequality on G that u is continuous on G and that u,, uX,and u,, are continuous in G. NAGUMO’SLEMMA.Assume for any values B E (0, b) and 0 < p0 < a that a constant L, exists such that f(Z, p) - jjz, p) d L&Z - z) whenever 0 < z < Z < B and IpI < po. If u, (or uX) is uniformly bounded in G, then n, - (urX + f(a, u,)) B 0, - (a,, + f(r, v.J
in G
and u 2 v ona,G implies u 2 u on G. STRONGMAXIMUMPRINCIPLE.Assume bounded functions c, d: G + R exist such that u, $ u,, + cu, + du
in G
and u2 0
on f3,G.
Then u > 0 on G. Furthermore, if u(xO, to) = 0 at a point (x,, to) E Gp, then u(x, t) = 0 at every point (x, t) E G which can be connected to (x,, to) by a directed curve in G (beginning at (x, t)) along which t is (weakly) monotone increasing. Both the above theorems are stated here in a special form sufficient for our purposes. The proofs are found in [4; 24 VI] and [5, Chapter 3, Section 31. THEOREM1. Let u(.x,t) < b solve (3) and (4) on [--a, u] x [0, T,), where f~ C’([O, b) x R) and f(0,0) > 0. Then:
On the global existence of solutions of parabolic differential equations with a siniular nonlinear term
501
(a) u > 0 and u is strictly monotone increasing in t on (-a, a) x (0, TJ. (b) For each 0 < t < Ta, u(x, t) is monotone increasing in x on [-a, b(t)] and monotone decreasing in x on [6(t), a], where --a < (p(t) < a. (c) Iff(z, p) is monotone increasing in z on [0, b) x R, then for each 0 < t < T, there exist values -a < 4l(t) < 4(t) G (6*(t) < a such that uX(x,t) is monotone increasing in x on II-a, d&r)] u [&( t ), a I an d monotone decreasing in x on [4,(t), #z(t)]. ProoJ: (Part (a)). Let u&(x,t): = ct*cos@c/ta). Sincef(0, 0) > 0, there is for any T E (0, 7J an E > Ososmallthatu: < uzX+f(u”,Qon(--a,a) x (0, TJThusu & t? > Oon(-a,a) x (O,T] by Nagumo’s Lemma. Since f E C’([O, b) x R), u, is continuous and therefore bounded on [-a, u] x [O, T] for any T E (0, T,). Also u g B (. b on [-a, a] x [0, T] (I? a constant). Thus if v(x, t): = u(x, t + h) - ufx, t) then it follows from fo Cl([O, b) x R) that v, = yXXf c* uX+ d *v on (-a, a) x (0, T - h), 0 < T - h < T G T,, where c and d are uniformly bounded functions. Since u(x, 0) > 0 for Ix] c a and o( _t a, t) = 0 for t 2 0, the strong maximum principle implies that u > 0 on (-a, a) x (0, T - h], from which the strict monotonicity foilows. Part (b). For any 0 < h < a, let w = uh - U-~ on [-a C h, a - h] x [0, T,), where P(x, t} = u(xfh,t).Thenw(-a+h,t)~Oandw(u-h,t)~OforO~t~T,andw,=w,,+c~w,f d * w on (-a + h, a - h) x (0, T,), where the functions c and d are bounded on any finite subset. For any fixed 0 < T < T,, the component G, (respectively C_) of [-a -t h, a - h] x [O, T] containing f-u + h) x (0, T] ({a - h) x (0, T]) on which w > (0 0 is simply connected by the strong m~imum principle. By the continuity of w we have w = 0 on 8~. where y: = ([ - a + h, a - h] x [O, T])\(G+ u G_). Therefore w = 0 in y by the strong maximum principle. It is easily seen that f(T) n (G, u y) and i(T) n (G_ u y) are intervals, where I(T) = t--u + h, Q - h] x (Tj. Since 0 < T < T, is arbitrary, the assertion in (b) follows from this. Part (c). If for some 0 ( T < T, the value 4,(T) B [--a, #(T)] (such that u,,(x, T) 2 0 in (-a, #,(T)] and uXX(x,Tf B 0 in C+,(T), #(T)]) did not exist, then for 0 < h < a su~ciently small there would exist points --c1 < x1 < xz < 4(T) such that CI:= w(xI, T) = w(x2, T) > 0 and w(x, T) < a for x1 < x < x2. (We use the notation in Part (b).) Let G, be the component of (-a + h, a - h) x (0, T] containing (x1, x2) x {T) in which w < CI.It follows from the continuity ofw that w E E on (8G,) n (( - a + h,a - h) x (0, T)).Sincew = Oon~,wehaveG~ c G,. Since w( - a + h, t) > 0 is monotone increasing in t, one sees that w E CIon a,G,. Now w satisfies the equation w, = w,, + [f(u-” + w, u, h + WJ -~(u-~,u.;*]) in (-a + h,a - h) x (0, T], where f(z, p) is monotone increasing in z. Thus, Nagumo’s Lemma implies that w > a in G,, This contradiction implies that #,(T) exists.
Let z((x, t) < B -c b solve (3) and (4) on f-u, o] x [0, crs), where So C”([O, b) x R) and f(0, 0) > 0. Then /u,(x, t)f is uniformly bounded on [-a, u] x [0, ‘0) provided that a constant L(B) exists such that whenever 0 < , z < B and [pi 2 1 the following inequalities hold:
THEOREM 2.
f(z, P) G w9P2 f,(z, P) d w4 -pf,kpJ
(7) IPi
G W)p2.
(8)
(9)
proof. Sincefe Cl([O, b) x R), u, is continuous on [-a, a] x [0,00). We first prove that luxl is uniformly bounded on { -a> x [0, co). Choose L > max{L(B), (7c/4a)},and define
502
A.
$(x) = (l/L) *Ln
ACKERAND W. WAL.TER
sin(c: f L *(x + a)) sin(s) C 1
forO
L-(x+
a)
where 0 < E < (7c/2)solves sin(s) = exp( - L *B). Letx, = (l/L)*@/2 -8) - a&(--,a).Thenrl/(-a) = 0,$(x,) = B 3 u(x,,t)forallt 3 0,and JI,, = - L* (1 + ($J”) on (- a, x,,). It follows using (7) and Nagumo’s Lemma that 0 < u(.x,t) ,< tl/(x) on [-a, x0] x [O, co). Therefore 0 ,< uX(-a, t) f $,(--a) for all t 3 0. The proof that ~ux~is uniformly bounded on (u) x [0, co) is similar, Now, define v = u, + y(u) on C-u, u] x [0, c;o),where y(s) = exp(3Ls)and L = max{ 1, L(B)). Clearly, there is a constant M such that IuX]d M and 1u) G M f y(B) on the parabolic boundary. Assume for 0 < T -c co faed that V: = sup(~(x, t)l - a d x d a, 0 4 t ,< 7’) 2 A4 -+-y(B) -t- 2. We will show this leads to a contradiction. The set S c f-a, a] x [0, T] on which 0(x, t) = Y is a compact subset of (-a, a) x (0, T]. Thus, for CI> 0 sufficiently small there is a compact c-neighbourhood S, of S (relative to [--a, u] x [O, T]) such that ,S, c (-a, a) x (0, T] and u d Y - a in ([--a, u] x [O, T))\S,. For 0 < h < 2u, define w = 6u t y(u) on [-a, a - h] x CO,co), where 6u = (l/h)(u(x + h, t) - u(x, t)). Due to the uniform continuity ofn, on [--a, a] x [0, T] we have sup{{&{n, t) - u,{x, t)l: -a < x < a - h, 0 G t ,< T> + 0 zssh-+O.Therefore, if h > 0 is su~ciently small, then W: = sup(w(x,t):-a~~xu--_,0dt,
w, = w,, + jlpw,- 9~~(~~~~~)~+ 3L~~~)~~-~~~*) + .Du, where at each (x, t), fp = fp(u{.x*, t), u,(x*, t)) and fz = j&(x*, t), u&x*, t)) with 0 < x* < x + h. At the point (x, t) E S’ c S, with minimum t-coordinate we have w, 2 0, w’, = 0, and w,, Q 0, from which it follows that 9~~(~)~2 < 3L~(~~~(~- &p) + jx,Su (where p = u,(x, r)). Since p > 1 in SE,the estimates (8) and (9) lead to 9~~(~)pz f ~LY~~)(~L~~)+ LP’ + 3L~~~)~(& -
j*,)+
jy(S, - P) + (j”, - f,)~.
Since S, is compact, we arrive as h -+ 0 at the inequality 9cp2 < (M? + L)p2 * 3L 6 1, which is a contradiction. Thus, u, < u < M + y(B) -I- 2 in E-4 aJ x [0,7’3 for all T > 0. The same argument using u = -u, + y(u) bounds u, from below on [-a, u] x [0, co). THEOREM
on [-u,uJ
3. Under the assumptions ofTheorem 2, Qx, t) and uX,(x, t) are both uniformly bounded x (0,‘~).
Proof. Under the assumptions onfl u, and uXXare continuous on [--a, u] x [O, “3) except at the points (-a, 0) and (a, 0). The proof of Theorem 3 from Theorem 2 is based on the functions v = u, + y(u) and w = 6,~ + y(u), where 6,u(x, t) = (l/h)(u(x, t + h) - u(x, t)). The proof is omitted, since the steps are essentially analogous to those in the proof of Theorem 2.
On the global existence of solutions of parabolic differential equations with a singular nonlinear term
503
3. THE MAIN RESULTS assume throughout this section that ,#‘EC’([O, b) x R), f(0,0) > 0,and f(z, p) -+ + CQ uniformly on any finite p-interval as z -+ b - 0. Furthermore, for any B F (0, b) there is a constant We
L(B) such thatf(z, p) satisfies (71,(8), and (9) whenever 0 6 z $ B and \p[ >, 1. If u solves (3) and (4) on [-a, u] x [O, T] and w(x, T) B B K b for all 1x1 6 a, then u can be extended to a solution of (3) and (4) on [-a, a] x [O, 7’ C s] for some E > 0. Therefore either T, = a (giobai existence) or else the solution of (3) and (4) on [-a, u] x [0, ‘I’J quenches at T,. Nagumds lemma implies that the solutions of (3) and (4) are monotone increasing in a. This means if u and u’ solve (3) and (4) on respectively [-a, a] x [O, T] and [-u’ a’] x [O, T], 0 -c a < a’, then u‘ B u 2 0 on [-a, u] x [0, ?“‘I).Therefore T, is a monotone decreasing function of a. It follows that a value a, 3 0 exists such that T, = 00 for 0 < a < a, and T, < GOfor a > a,. LEMMA 4. Assume 0 < a < a,. Then a constant 0 < B < b exists such that the solution u of (3)
and (4) on [-a,
u] x LO,co) satisfies u(x, t) 6 B.
Pmctp. We assume that no upper bound B < b exists and obtain a contradiction. if this were the case, we would have lim ol(&t), t) = b. (This is because u(#(t), t) is monotone increasing in t on [O, xi), where # was bezned in Theorem I (b).) Thus, one could choose 6 > 0 such that tl + 6 < a*, then 0 < E < 1 such that a2 f 2~/t5* G S(z, p) for all (z, p) E [b - I, b) x f - 2/6,2/J], and then t, Y 0 such that u(#(t& co) 2 b - 8. Let tid be the solution of (3) and (4) on E-u - 5, a + S] x [O, co). For each 6’ E [-S, S], Nagumo”s Lemma shows that n&x + 8, 1) 2 U(X,t) on [--a, a] x [O, r;)). Thus (using Theorem I (a)) u&x, t) B b - E on the strip S: = f&t& - b, dt(Gf + a] x P,> “c). On the other hand, the function ti(x, t): =b - E + (r - t,)fb’ - (x ~(~~~~2~ is equal to ti - E on the parabolic boundary of S. Furthermore, vr G o, i- _QY,u_~)in S, : =[#t,) - 6, rfift,) + 6] x [to, t, -t c/PJ. Hence, t16> D in S,. In particular, ~~~~(~~),fQ + s/S”) 2 ~~~(~~),t, + s/s”) = b. This shows that u&x, E)quenches in contradiction to the choice of6 > 0.
Proof. For the proof that a, 3 a,, we show for every a E (0, a,) that a G a,. TfO < a < a,, then (51 and (6) are solved by a function 0 d w i(x) < h on [ - LI~* GJ for some a, E (a, aJ Pu’agumo’s Lemma implies that u < wX< b on C-n, af x [O, T,). Thus, T = m and Q G a*. To prove that G, d a,, we show for every a E (0, a+) that a < izo’If0 < a < a,, then by Lemma 4 a constant B E (0, b) exists such that the solution u of (3) and (4) on [-a, G] x [OF60) satisfies u d B. By Theorem 3, faXXl f SZC (a constant) on [-a, u] x (0, mf. Thus, given an unbounded sequence ft,), there is by Ascoti-Arzel& a subsequent (t,J and a Lipschitz continuous function c(x) (with Lipschitz constant C) such that u&, t,,) -+ U(X)uniformIy on [ - ~1,a] as la -+ ocj. Let I W(X) = f”-, v(x’) dx” on [--a, a]. Then U(X~ tJ -+ W(X)uniformly on [--a, a] as n + e. Since U(X,t) is monotone increasing in t for each x E [- u, a], we have s(t): = max~w(.~)- u(x, t): -a < x d of + 0 as t -+ CO(uniform convergence). Clearly w( -n) = w(a) = 0 and 0 d W(X)< I3 on [--a, a]. We will now show that u,(x, t) -+ wX(x) uniformly on C-u, a] as t -+ m. For 0 < h < 2a, define &w(x) = (l/~)(w{~ f h) - W(X))on [-a, a - h) and Gwjx,t) = (l/h)(u(x C h, t) - U(X,t)) on E-a, a - h] x [O, cry)).Then (~w(x) - W,(X)/d Clz and l&(x, t) - zc,(%t)\ d Ch for -a < x < a - h and t 3 0. Also, j&(x, t) - Gw(x)j & (2/h) E (c). By combining inequalities, we obtain (u&, t) - w,(x)/ $ 2Ch + (2/h) E (t). The choice h = &@j yields
504
A. ACKER AND W.WALTER
1u,(x, 0 - w,cdI G 2(C + 1) x/m
(10)
for all x E [-a, a - ,/$J, for each t 2 0. The same argument based on left-hand differences shows that (10) holds for all x E [-a + ,/$), a] and t 2 0, so that the assertion is proved. Now define U(x, t) on [-a, LX]x [0, cc) by U(x, t) = Si” u(x, r) dr. Then ffl u(x, t + 1) - u(x, t) = UXX(X, t) -If(u(x, r), ux(x, r)) dr sf on [-a, u] x [0, CD).Sincef(z, p) is uniformly continuous on any compact subset of [0, b) x R, it follows from the above convergence results that U,,(x, t) + -f(w(x), w,(x)) uniformly on [ - a, a) as t + CD.Therefore, one concludes from the Taylor formula U(x, t) = U(o, t) + x . U,(O, t) + (for/x\ < a
‘(x - x’). UJx’, t) dx’ s0
and t 2 0)
that w(x) satisfies the equation w(x) = w(0) + x *w,(O) -
x (x - x’) .f(w(x’), w,(x’)) dx’ (1x1< a).
s0 Therefore, w(x) satisfies (5) and (6) on [-a,
a], implying that a d a,.
Remark. The result a, = a0 does not determine
whether or not the solution of (3) and (4) is global when a = a,. Of course if a solution 0 < w(x) < b of (5) and (6) on [-a,, a,] exists, then q, = co.
Remark. Theorem 5 shows that a solution of (5) and (6) on [-a,
u] exists for every a E (0, a,).
REFERENCES 1. KARAWADA H., On solutions of initial-boundary problem for u, = u_ + l/(1 -g), Publ. RIMS, Kyoto Univ. 10, 729-736 (1975). 2. WALTER W., Parabolic differential equations with a singular nonlinear term, Funkcialaj Ekuacioj 19,271-277 (1976). 3. ACKER A. & WALTER W., The quenching problem for nonlinear partial differential equations, Ordinary and Partial Dij,‘,rential Equations, Dundee 1976; Lecture Notes in Mathematics, No. 564. pp. l-12. Springer-Verlag (1976). 4. WALTER W., Differential and Integral Inequalities, Ergebnisse der Mathematik und Ihrer Grenzgebiete, Vol. 55. Springer-Verlag-(1970). 5. PROTTEXM. H. & WEINBERGERH. F.. Maximum Princioles in Differential Eauations. Prentice Hall. Enelewood Cliffs. __ _ N.J. (1967).