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:
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   . Additionally, (1.1) describes a thermal ignition driven by the temperature at a single point, see  . Chadam, Peirce, and Yin  examined the blow-up set of solutions when the initial data are nontrivial and nonnegative bounded functions.
The quenching problem was initiated by Kawarada  . The model describes polarization phenomena in ionic conductors. Quenching also illustrates the phase transition between liquids and solids  . Chang, Hsu, and Liu  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  .
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 ,
Follow a similar argument, we assume that somewhere in . Then, there exist some and such that , , and attains its local minimum at . Then at
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 [  , 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 [  , 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  , 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 [  , 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:
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 [  , 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 [  , p. 11], we obtain
As , we have
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 [  , 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,
As and in D, and on , we choose a positive real number less than 1 such that
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  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
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
From Lemma 2.3, and on for all . Based on Theorem 10.4.2 of [  , 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
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  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 [  , pp. 288 and 304], the Green’s function is positive in Λ and is given by
where , and is a real harmonic function in Λ. With this , we map Λ onto S conformally. (4.5) is expressed as
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  , 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
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.
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.
The author thanks the anonymous referees for their suggestions.
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