Simultaneous Quenching for Semilinear Parabolic System with Localized Sources in a Square Domain ()
1. Introduction
Let
, a and b be positive constants such that
,
,
,
. We also let L be the operator such that
. In this paper, we study the following semilinear parabolic system:
(1.1)
(1.2)
Problem (1.1)-(1.2) illustrates the instabilities in some dynamical systems in which certain reactions are localized to electrodes, catalytic membranes, or other surfaces and local sites are immersed in a bulk medium happening at the origin (0, 0), see [1] [2] . Additionally, (1.1) describes a thermal ignition driven by the temperature at a single point, see [3] . Chadam, Peirce, and Yin [4] examined the blow-up set of solutions when the initial data are nontrivial and nonnegative bounded functions.
The quenching problem was initiated by Kawarada [5] . The model describes polarization phenomena in ionic conductors. Quenching also illustrates the phase transition between liquids and solids [6] . Chang, Hsu, and Liu [7] discussed the quenching rate of the problem (1.1)-(1.2) in an n-dimensional ball.
The solutions u and v are said to quench if there exists a finite time
such that
To problem (1.1)-(1.2), there are critical values a* and b* (both are positive) such that the maximum of solutions u and v reaches 1 in a finite time if a > a* and b > b* while u and v exist globally and are bounded above by 1 if a < a* and b < b*, see [8] .
The purposes of this paper are to prove solutions u and v to quench simultaneously at (0,0), and use a numerical method to determine approximated values of a* and b* of the problem (1.1)-(1.2).
This paper is organized as follows. In Section 2, we prove that there are unique solutions u and v of the problem (1.1)-(1.2). In Section 3, we prove that either u or v quenches in a finite time. Then, we show that solutions u and v quench simultaneously at (0, 0). In Section 4, we calculate approximated values of a* and b*. This a* and b* associate with the existence of solutions of their steady state problem of the problem (1.1)-(1.2). Our numerical method is to evaluate an approximation of the steady solutions expressed in an integral representation form. For illustration, some examples are provided.
2. Properties of u and v
Let
and
be nontrivial, nonnegative, and bounded functions on
. Here is a comparison theorem
Lemma 2.1. Suppose that
and
are solutions of the following system:
Then,
and
on
.
Proof. Let
be a positive real number, and
where
is a positive real number to be determined. From the construction,
and
on
. By assumptions and a direction computation,
We choose
such that
. Thus,
Suppose
somewhere in
. Then, the set
is non-empty. Let
denote its infimum. Then,
because
on
. Thus, there exists some
such that
and
. On the other hand,
attains its local minimum at
. Therefore,
. Then, at
,
(2.1)
Follow a similar argument, we assume that
somewhere in
. Then, there exist some
and
such that
,
, and
attains its local minimum at
. Then at
(2.2)
Let us assume that
. Since
attains its local minimum at
,
. By inequality (2.2),
This gives a contradiction. Hence,
in
. Then, by (2.1), we show that
in
. Through a similar calculation, we obtain the same result when
. Let
, we have
and
in
. Follow
and
on
and
, u and v are non-negative on
. The proof is complete. ,
By Lemma 2.1, 0 is a lower solution of the problem (1.1)-(1.2). On the other hand, if u and v do not quench, then
and
, and solutions u and v are bounded above by 1. Further, u and v cease to exist when
and
. Therefore,
and
on
. The existence of classical solutions of the problem (1.1)-(1.2) is able to obtain by the Schauder fixed point theorem of [ [9] , pp. 502-504], and by Lemma 2.1 u and v are unique.
Theorem 2.2. Problem (1.1)-(1.2) has unique classical solutions u and
for some
such that
on
.
Lemma 2.3.
and
on
. Further,
and
in
.
Proof. From Theorem 2.2,
and
are nonnegative on
. Let h be a real number in
. Then,
and
on
, and
and
on
. By the mean value theorem, there exist u1 (between
and
) and v1 (between
and
) such that
By Lemma 2.1,
and
on
. This gives
and
on
.
Taking
, ut and vt are nonnegative on
, respectively. To show that ut and vt are positive, let us differentiate (1.1) with respect to t. Then, ut and vt satisfy
By the maximum principle [ [10] , p. 54],
and
in
. ,
3. Simultaneous Quenching and Global Existence
In this section, we show that either u or v quenches in a finite time first. Then, we prove that u and v quench in the same time at (0, 0). Afterward, we prove that the problem (1.1)-(1.2) has a global solution when a and b are sufficiently small.
Lemma 3.1. u and v both attain their maximum at (0, 0) for all
.
Proof. It suffices to prove that u attains its maximum along the x and y axes. Let us consider the first equation of (1.1) along the x-axis, we have
Differentiate the above equation with respect to x to yield
By the symmetry of D with respect to the x and y axes,
for all
. By the Hopf’s lemma [11] , p. 170],
for all
. At t = 0,
for all
. By the maximum principle,
for all
when
. Similarly, for all
, we obtain
when
. Therefore,
for all
when
. Likewise,
for all
when
. Thus,
on
. Similarly,
on
. Hence, u and v both attain their maximum at (0, 0) for all
. ,
Let
be the eigenfunction corresponding to the first eigenvalue
of the Sturm-Liuoville problem below,
This eigenfunction has the properties:
in D and
[ [12] , p. 10]. Let c be a positive real number such that
. We show that either u or v quenches in a finite time.
Lemma 3.2. If
, where
, then either u or v quenches on
in a finite time
.
Proof. By Lemma 3.1,
and
on
. Let
and
be the solutions to the following parabolic system:
(3.1)
(3.2)
By the maximum principle,
and
on
. Further,
and
satisfy the system below:
By
and
on
and
, and the maximum principle, we have
and
on
. It suffices to prove either
or
to quench over
in a finite time.
Multiplying
on both sides of (3.1) and integrating expressions over the domain D, we obtain
Use the Green’s second identity [ [10] , p. 96] and (3.2), it yields
By the Maclaurin’s series, we have
Let
and
. Adding above inequalities together and using the Jensen’s inequality [ [12] , p. 11], we obtain
(3.3)
As
, we have
Hence,
Then, differential inequality (3.3) becomes
Let
. Then,
in
and
Using separation of variables and integrating both sides over (0, t), we obtain
Suppose that
exists for all
. By the assumption
, we have
But,
is bounded above by
. This is a contradiction. It implies that
ceases to exist in a finite time
. This shows that either
or
does not exist when t approaches
. Thus, either
or
quenches on
at
. Since
and
, either u or v quenches on
in a finite time
where
. ,
From the result of Lemma 3.1, we know that (0, 0) is a quenching point of u and v if they quench. Let
be the supremum of the time
for which the problem (1.1)-(1.2) has unique solutions u and v.
Theorem 3.3. If
, then either
or
quenches at
.
Proof. Suppose that both u and v do not quench at (0,0) when
. Then, there exist positive constants
and
such that
and
for all
. This shows that
and
for some positive constants
and
when
. Then, by Theorem 4.2.1 of [ [13] , p. 139], u and
. This implies that there exist positive constants
and
such that
and
for all
. In order to arrive a contradiction, we need to show that u and v can continue to exist in a larger time interval
for some positive
. This can be achieved by extending the upper bound. Let us construct upper solutions
and
, where f(t) and g(t) are solutions of the following differential system:
By
, and the Picard iteration, f(t) and g(t) are positive functions, and
and
. This implies that f(t) and g(t) are increasing functions of t. Let
be a positive real number determined by
and
for some positive constants
and
greater than
and
respectively. By our construction,
and
satisfy,
By Lemma 2.1,
and
on
. Therefore, we find solutions u and v to the problems (1.1)-(1.2) on
. This contradicts the definition of
. Hence, either
or
quenches at
. ,
Let
such that
,
in D, and
on
and
. Let
be the solution to the following first initial-boundary value problem:
By the maximum principle,
in
and is bounded above by
, and it satisfies
Let
such that
for some positive constant
. Then,
(3.4)
As
and
in D, and
on
, we choose a positive real number
less than 1 such that
(3.5)
Also,
for all
. Let us define
. By inequalities (3.4) and (3.5),
on
. Let
where
is a positive real number less than 1. Similar to the previous argument, we choose
such that
on
. We modify the proof of Lemma 3.4 of [7] to obtain the result below.
Lemma 3.4.
and
on
.
Proof. By a direct computation,
From the above expression, we have
By
on
,
, and
for all
, it gives
in
. In addition,
on
, and
on
. By the maximum principle,
on
. Similarly,
on
. ,
Here is the result of simultaneous quenching.
Theorem 3.5. If either u or v quenches at (0, 0) when
, then u and v quench simultaneously at (0, 0) when
.
Proof. If not, let us assume that v quenches at (0, 0) when
but u continues to exist beyond
. That is, there exists a positive constant
such that
for all
for some
. By Lemma 3.4, we have
By Lemma 3.1, u and v both attain their maximum at (0,0) for all
. Then,
and
on
. Combine (1.1) with above inequalities to give
From them, we get a compound inequality
(3.5)
From the left-side inequality, we have
Since
for all
, there exists a positive constant
such that
for all
. Integrating both sides over the interval
where
, we obtain
By assumption, v quenches at (0, 0) when
, we have
as
. Since
and
are both bounded, the above inequality implies
as
. Therefore, u quenches at (0, 0) when
. It contradicts that u exists on
. Follow the second half of inequality (3.5), we can prove that v quenches at (0, 0) when
if u quenches at (0, 0) when
. The proof is complete. ,
Now, we prove that u and v exist globally when a and b are sufficiently small. Our method is to construct global-exist upper solutions of the problem (1.1)-(1.2).
Lemma 3.6. If a and b are sufficiently small, then there is a global solution to the problem (1.1)-(1.2).
Proof. It suffices to construct upper solutions which exist all time. Let
and
where A and B are positive real numbers such that
and
. Clearly,
for all
. In addition,
If a is sufficiently small, then we have the inequality:
. This leads to
Similarly, if b is sufficiently small, we have
and
By Lemma 2.1,
and
on
. Hence, u and v both exist globally. The proof is complete. ,
Lemma 3.7. u and v are non-decreasing functions in a and b respectively.
Proof. Let
and
be solutions to the problem (1.1)-(1.2) corresponding to
and
, and
and
be solutions when
and
, where
and
. Then,
and
satisfy the parabolic system:
As
and
on
and
, we have
and
by Lemma 2.1. ,
4. Approximated Values of a* and b*
Let U(x, y) and V(x, y) be steady-state solutions of the problem (1.1)-(1.2). They satisfy
(4.1)
(4.2)
From Lemma 2.3,
and
on
for all
. Based on Theorem 10.4.2 of [ [10] , pp. 532-533], we have the following result.
Lemma 4.1. If
and
on
, then u and v converge monotonically to U(x,y) and V(x,y) on
respectively as
.
Let
be the Green’s function of the operator:
over the domain D. The integral representation of the solution of the problem (4.1)-(4.2) is given by
(4.3)
(4.4)
By Lemma 3.7, u and v are respectively non-decreasing functions in a and b. Then, by Lemma 3.6, there exist a* and b* for which u and v exist globally and less than 1 if
and
. By Lemma 4.1, u and v converge to U and V when
and
. Thus, U and V exist and they are bounded above by 1 when
and
, and
and
if
and
.
Let us construct sequences of integral solutions:
and
such that
, and they satisfy
for
. We follow Theorem 4 of [14] to obtain the following result.
Theorem 4.2. Suppose that
and
, the sequences
and
converge monotonically to solutions U and V of the Equations (4.3) and (4.4) where
and
in D for
.
To determine
, we map the domain D onto the unit disk S:
through a conformal mapping. Let J denote this mapping. By the Riemann Mapping Theorem, J exists and is unique. This theorem is stated below.
Theorem 4.3 (Riemann Mapping Theorem). Suppose that z is a point locating in Λ which is a simply-connected two-dimensional domain with more than one boundary point, and
is a point of Λ, then there exists a unique analytic function
which is regular in Λ and maps Λ conformally onto the unit disk S:
in such a way that
and
.
Let
and
be some points in a simply-connected two dimensional domain Λ. From the result of [ [15] , pp. 288 and 304], the Green’s function is positive in Λ and is given by
(4.5)
where
, and
is a real harmonic function in Λ. With this
, we map Λ onto S conformally. (4.5) is expressed as
(4.6)
The Taylor series representation of
with respect to
is given by
where
is a complex number given by
. Let
where
and
is the angle between the line segment
and the positive x-axis. Then, the above series is represented by
.
To determine an approximation of
, we let
. By
is a real function, we have
.
From the symmetry of D with respect to the x-axis, y-axis, and y = x, we have
for
. The truncated Taylor polynomial of p(x, y) (that is, p(z)) at some finite 4n terms, where n is a positive integer, is given by
.
Let n = 8. By the result of [16] ,
is given by
By (4.6) and the above expression, an approximation of G at
is given by
Thus, approximated solutions of U and V of (4.3) and (4.4) are able to evaluate through an iterative scheme. A numerical method of finding an approximation of a* and b* is stated below.
Step 1: Assign a positive value for a. Choose a positive value for b (say b1). Set
and
for all
. Let
and
be approximated solutions of
and
given by
(4.7)
(4.8)
Compute
and
for
and
. At this
,
and
are bounded by 1 and converge. That is,
and
for
and satisfy
for
for some positive integer N.
Step 2: With the same value of a in Step 1, choose another value for b (say b2). Set
and
for all
. To each
. evaluate iterative integral (4.7) and (4.8) for
. At this
,
and
do not exist. That is,
and
for some positive integer j. Calculate
. Then, at
, evaluate (4.7) and (4.8) and compute
and
for
.
Step 3: Set
if
and
for
, and satisfy
for
for some positive integer N. Otherwise, set
if
and
for some positive integer j. This procedure stops when
and
(or
if
). Then, set
and
. Otherwise, calculate
. Then, at
, evaluate (4.7) and (4.8), and compute
and
for
. Then, repeat Step 3.
When we evaluate (4.7) and (4.8), the domain D is divided into 225 (
) grid points uniformly. The B-Spline interpolation is used to interpolate the function value at these grid points. We use Mathematica to evaluate (4.7) and (4.8). As examples, we compute two groups of approximated values of
and
. In the first group, we set
and vary the value of b. In the second one, we let a = b, then they change together. The results are listed in Table 1.
5. Conclusion
In this paper, we prove that u and v reach their maximum at (0, 0) for all
. Lower and upper bounds of
and
at (0, 0) are obtained. From these results, we then show that u and v quench simultaneously at (0, 0). A numerical
Table 1. Approximated critical values.
method is introduced to compute approximated critical values of the semilinear parabolic system, and two sets of result are reported.
Acknowledgements
The author thanks the anonymous referees for their suggestions.