QUARTERLY OF APPLIED MATHEMATICS
VOLUME LXII, NUMBER 3
SEPTEMBER 2004, PAGES 553-568
QUENCHING FOR A DEGENERATE PARABOLIC PROBLEM DUE
TO A CONCENTRATED NONLINEAR SOURCE
By
C. Y. CHAN and X. O. JIANG
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010
Abstract. Let q, a, T, and b be any real numbers such that q > 0, a > 0, T > 0,
and 0 < b < 1. This article studies the following degenerate semilinear parabolic first
initial-boundary value problem with a concentrated nonlinear source situated at b:
xqut - uxx = a2S(x - b)f(u(x, t)) in (0,1) x (0, T],
u(:x, 0) = 0 on [0,1], u(0, t) = u(l, t) = 0 for 0 < t < T,
where <5 (x) is the Dirac delta function, / is a given function such that limu^c- f(u) = oo
for some positive constant c, and f{u) and f'(u) are positive for 0 < u < c. It is shown
that the problem has a unique continuous solution u before m&x{u(x,t) : 0 < x < 1}
reaches c~, u is a strictly increasing function of t for 0 < x < 1, and if max{u(x,i) :
0 < x < 1} reaches c~, then u attains the value c only at the point b. The problem is
shown to have a unique a* such that a unique global solution u exists for a < a*, and
max{u(x,t) : 0 < x < 1} reaches c~ in a finite time for a > a*; this a* is the same as
that for q = 0. A formula for computing a* is given, and no quenching in infinite time is
deduced.
1. Introduction. Let q, 0, a, and p be any real numbers with q > 0, 0 < (3 < a, and
p > 0. Let us consider the following degenerate semilinear parabolic first initial-boundary
value problem,
<^u7 - = 6(<;-0)F(u(c,7)) in (0,a) x (0,p], 1
u(<T, 0) = 0 on [0, a], u(0,7) = u(a, 7) = 0 for 0 < 7 < p, J
where S(x) is the Dirac delta function and F is a given function. These types of prob-
lems are motivated by applications in which the ignition of a combustible medium is
accomplished through the use of either a heated wire or a pair of small electrodes to
supply a large amount of energy to a very confined area. When q = 0, the problem
(1)
Received July 7, 2003.
2000 Mathematics Subject Classification. Primary 35K60, 35K65, 35K57.
Key words and phrases. Degenerate parabolic problem, concentrated nonlinear source, unique continu-
ous solution, single-point quenching, critical length, no quenching in infinite time.
E-mail address: chan01ouisicina.edu
E-mail address: xoj8985@louisiana.edu
©2004 Brown University
553
554 C. Y. CHAN and X. O. JIANG
(1) can be used to describe the temperature of a one-dimensional rod having a length a
and a concentrated nonlinear source at (3. When q = 1, it may also be used to describe
the temperature u of the channel flow of a fluid with temperature-dependent viscosity
in the boundary layer (cf. Chan and Kong [4]) with a concentrated nonlinear source at
/?; here, <r and 7 denote the coordinates perpendicular and parallel to the channel wall,
respectively.
Let <; = ax, 7 = aq+2t, /3 = <26, Lu = xqut - uxx, F(u(<.^,7)) = f(u(x,t)), D = (0,1),
D = [0,1], and tt = D x (0,T]. Then (1) is transformed into the following problem:
Lu = a25(x — b)f(u(x,t)) in !T2,
u(x, 0) = 0 on D, u(0, t) = u( 1, t) = 0 for 0 < t < T,
(2)
with 0 < b < 1 and T = p/aq+2. We assume that liinu^c- /(u) = 00 for some positive
constant c, and /(u) and /'(it) are positive for 0 < it < c.
The case q = 0 was studied by Deng and Roberts [7] by analyzing its corresponding
nonlinear Volterra equation at the site b of the concentrated source:
u(b,t) = a2 [ g(b,t;b,T)f(u(b,T))dT,
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where g(x,t;^,r) denotes Green's function corresponding to (2) with q = 0. By also
assuming that f" (u) > 0 for it > 0, they showed that there exists a length a* such
that for a < a*, the solution u(b,t) of the integral equation exists for all time and is
uniformly bounded away from c while for a > a*, there exists some finite tq such that
limt_>tfl u(b,t) = c and limt_tg ut (6, t) — 00.
Instead of studying a solution it (b, t) of the nonlinear Volterra equation, we would like
to investigate a solution u(x,t) of the degenerate problem (2). Since u(x,t) need not
be differentiate at b, we say that a solution of the problem (2) is a continuous function
satisfying (2). In Sec. 2, we show that the problem (2) has a unique solution u, and uxx >
0 for x G (0, b) and x € (b, 1). It follows from xqut (x, t) = uxx (x,t) + a25(x-b)f {u (x, t))
that ut (6, t) = 00 for each t > 0. Hence, we say that a solution it of the problem (2) is
said to quench if there exists some tq such that max{u(x,t) : x € D) —> c~ as t —> tq
(cf. Chan and Liu [5]). If tq is finite, then u is said to quench in a finite time. On the
other hand, if tq = 00, then it is said to quench in infinite time. We also show that u is
a strictly increasing function of t in D, and if u quenches, then b is the single quenching
point.
The length a* is called the critical length (cf. Chan and Kong [3]) if a unique global
solution u exists for a < a*, and if the solution it quenches in a finite time for a > a*.
In Sec. 3, we show existence of a unique critical length, and that it is the same as that
for q = 0. By making use of limu^c- f(u) = 00, we show that for a = a*, u exists for
0 < t < 00, and is uniformly bounded away from c. This shows that quenching does not
occur in infinite time. We also derive a formula for calculating a*.
2. Existence, uniqueness, and single-point quenching. Green's function
G(x,t-,^,r) corresponding to the problem (2) is determined by the following system:
QUENCHING DUE TO A CONCENTRATED NONLINEAR SOURCE 555
for x and £ in D, and t and r in (—00,00),
LG = 6{x - £)5{t - t),
G(x, t; £, t) = 0 for t < r, G(0, t\ £, r) = G(l, t; £, r) = 0.
By Chan and Chan [1], we have
OO
G{x,t;Z,r) = x)MZ)e~Xi(t~T),
i=i
where Aj (« = 1, 2, 3,...) are the eigenvalues of the problem
<t>" +\xq<j) = 0, 0(0) = 0(1) = 0,
and their corresponding eigenfunctions are given by
(t£*"+2,/2)
c^-(x) = (<? + 2)1/V/2-
2A
1/2
"^1+<I+2 \ 1+2
with Ji/(q+2) denoting the Bessel function of the first kind of order 1 /(q + 2). From
Chan and Chan [1], 0 < Ai < A2 < A3 < • • ■ < A, < AI+i < • • •. The set {(j>i(x)} is a
maximal (that is, complete) orthonormal set with the weight function xq (cf. Gustafson
[9, p. 176]).
To derive the integral equation from the problem (2), let us consider the adjoint
operator L*, which is given by L*u = —xqUt — uxx. Using Green's second identity, we
obtain
u(x,t) = a2 I G(x,t;b,T)f(u(b,r))dT. (3)
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For ease of reference, let us state below Lemmas 1(a), 1(b), 1(d), and 4 of Chan and
Chan [1] as Lemma 1(a), 1(b), 1(c), and 1(d), respectively; we also state below Lemma
2.2(a), 2.2(b), 2.2(c), and 2.2(d) of Chan and Tian [6] as Lemma 1(e), 1(f), 1(g), and
1(h), respectively.
Lemma 1. (a). For some positive constant ki, \<j>i (x)| < k\x~q/4 for x G (0,1].
(b). For some positive constant /c2, \4>i (^)l < k2XX^\\ for x G D.
(c). For any Xo > 0, and x G [xo, 1], there exists some positive constant k<$ depending
on xo such that |<^ (x)| < £3 A J .
(d). In {(x,t;^,r) : x and £ are in D, T > t > t > 0}, G(x,t; £,r) is positive.
(e). For (x,t;€,r) G (D x (r,T]) x (£> x [0, X1)), G{x,t\^,r) is continuous.
(f). For each fixed (£,r) G D x [0,T), Gt(x,i;^,r) G C (D x (r,T]).
(g). For each fixed (£, r) G D x [0, T), Gx (x, t\r) and Gxx(x, t; £, r) are in C ((0,1]
x(r,T]).
(h). If r G C ([0, T]), then J0' G(x, t: b, T)r{r)dT is continuous for x G D and t G [0, T],
We modify the techniques in proving Lemma 2.3 and Theorems 2.4 and 2.6 of Chan
and Tian [6] to show that the integral equation (3) has a unique nonnegative continuous
solution. Unlike theirs, we achieve this without using the contraction mapping and
556 C. Y. CHAN and X. O. JIANG
without considering the integral equation (3) at x — 6; also by making use of (3), we
prove further that u is a strictly increasing function of t in D.
Theorem 1. There exists some tq (< oo) such that for 0 < t < tq, the integral equation
(3) has a unique nonnegative continuous solution u, and u is a strictly increasing function
of t in D. If tq is finite, then u reaches c~ at tq.
Proof. Let us construct a sequence {u{\ in CI by uq = 0, and for i = 0,1,2,...,
L'Ui+i — a2d(x — b)f(ui(x,t)) in 17,
ui+i(x,0) = 0 on D, ui+i(0,t) = ui+i(l,t) = 0 for 0 < t < T.
Let dCl denote the parabolic boundary (D x {0}) U ({0,1} x (0, T]) of fi. We have
L(u\ — u0) = a25(x — b)f(0) in fl, u\ — u0 = 0 on dfl.
By Lemma 1(d) and 1(e), G(x,t;£,r) is positive and continuous. From (3), u\ > u0 in
fl. Let us assume that for some positive integer j.
0 < Ui < U2 < ■ ■ ■ < Uj-1 < Uj in CI.
We have
L(uj+1 — Uj) = a2(5(x — b)(f(uj) — f(uj-1)) in fi, Uj+\ — Uj = 0 on dCl.
Since / is a strictly increasing function and Uj > Uj-1, it follows from (3) that Uj+1 > uj.
By the principle of mathematical induction,
0 < u\ < U2 < ■ ■ ■ < iin_i < un in Cl
for any positive integer n.
To show that each un is an increasing function of t, let us construct a sequence {wn}
such that for n = 0,1,2,..., wn (x,t) = u„(x,t + h) - un(x,t), where h is any positive
number less than T. Then, wo(x,t) = 0. By (3), we have
w\(x, t) = u\(x, t + h) — Ui (x, t)
= a2/(0) G(x,t + h;b,T)d,T - J G(x, t; b, r)dr^ .
We note that G(x,t + h;b,r) = G(x, t + h — r; 6,0). Let a — r — h. Then,
We have
ri-tn pn pi
/ G(x,t + h-b,T)dr = / G(x,t + h;b,T)dr + / G(x,t;b,a)
Jo Jo Jo
rh
Wi(x,t) = a2f(0) / G(x,t + h;b,T)dT,
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QUENCHING DUE TO A CONCENTRATED NONLINEAR SOURCE 557
which is positive for 0 < t < T — h. Let us assume that for some positive integer j,
Wj > 0 for 0 < t < T — h. Let a — r — h. We have
a2
ri-\-n
/ G(x,t + h;b,T)f(uj(b,T))d,T
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a2 ( f G(x,t + h',b,T)f(uj(b,T))dT+ f G(x, t;b,a)f(uj(b, a + h))da
\Jo Jo y
J G(x,t + h;b,T)f(uJ(b,T))dT + s: G(x, t; b, a)f{uj (b,cr))daj .
> a2
In D x (0, T — h],
Wj+i(x,t) = Uj+i (x, t + h) - Uj+\(x, t)
: f G(x,t + h;b,T)f(uj(b,T))dT>0.
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> a2
By the principle of mathematical induction, wn > 0 for 0 < t < T — h and all positive
integers n. Thus, each un is an increasing function of t.
By Lemma 1(h), G(x,t;b,r) is integrable. Thus for any given positive constant M
(< c), it follows from
un(x,t) = a2 G(x,t;b,T)f(un-i(b,T))dT (4)
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and un being an increasing function of t that there exists some t\ such that un < M for
0 < t < t\ and n = 0,1, 2,.... In fact, t\ satisfies
rti
a2f(M) / G(x,ti;b,r)dT < M.
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Let u denote limn^oo un. From (4) and the Monotone Convergence Theorem (cf. Royden
[10, p. 87]), we have (3) for 0 < t < t\. Thus, u is a nonnegative solution of the integral
equation (3) for 0 < t < t\.
To prove that u is unique, let us assume that the integral equation (3) has two distinct
solutions u and u on the interval [0,£i]. Let 0 = maxg^o^jlw — u\. From (3),
u(x, t) - u(x, t) =a2 [ G(x, t; b, r) r)) - f(u(b, r))) dr. (5)
J 0
Using the Mean Value Theorem, we have
I f(u{b,r)) - f(u(b,r))\ < f'{M)G.