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Quasi traveling waves with quenching in a reaction-diffusion equation in the presence of negative powers nonlinearity.

1. Introduction. In this paper, we consider the quasi traveling waves with quenching (see Def. 3) of the following equation:

(1.1) [u.sub.t] = [u.sub.xx] + 1/[(1 - u).sup.[alpha]], t > 0, x [member of] R, [alpha] [member of] N.

First, we state the definition of "quenching" for the solution of (1.1).

Definition 1. We say that a solution u(t,x) of (1.1) quenches at point (T,[x.sub.0]) if

[mathematical expression not reproducible].

In order to consider the traveling waves of (1.1), we introduce the following change of variables:

[phi]([xi]) = 1 - u(t, x), [xi] = x - ct, c > 0.

We then seek the solution [phi]([xi]) of the following equation:

(1.2) c[phi]' = -[phi]" + [[phi].sup.-[alpha]], [xi] [member of] R, ' = d/d[xi],

or

[mathematical expression not reproducible].

Second, we state the definition of quasi traveling waves and quasi traveling waves with quenching as follows:

Definition 2. We say that a function u(t,x) [equivalent to] 1 - [phi]([xi]) is a quasi traveling wave of (1.1) if the function [phi]([xi]) is a solution of (1.2) on a finite interval or semi-infinite interval.

Definition 3. We say that a function u(t, x) [equivalent to] 1 - [phi]([xi]) is a quasi traveling wave with quenching of (1.1) if the function u(t, x) is a quasi traveling wave of (1.1) on a finite interval (resp. semi-infinite interval) such that [absolute value of [phi]'] becomes infinite (namely, [phi] reaches 0) at both ends of the interval (resp. finite end point of the semi-infinite interval). More precisely, we have the following three cases:

(I) The function [phi]([xi]) is a solution of (1.2) on a semi-infinite interval (-[infinity], [[xi].sub.*]) ([phi]([xi]) [member of] [C.sup.2](-[infinity],[[xi].sub.*]) [intersection] [C.sup.0](-[infinity],[[xi].sub.*], [absolute value of [[xi].sub.*]] < [infinity]), and satisfies

[mathematical expression not reproducible].

(II) The function [phi]([xi]) is a solution of (1.2) on a semi-infinite interval ([[xi].sub.*], +[infinity]) ([phi]([xi]) [member of] [C.sup.2]([[xi].sub.*], +[infinity]) [intersection] [C.sup.0][[xi].sub.*], +[infinity]), [absolute value of [[xi].sub.*]] < [infinity]), and satisfies

[mathematical expression not reproducible].

(III) The function [phi]([xi]) is a solution of (1.2) on a finite interval ([[xi].sub.-],[[xi].sub.+]) ([phi]([xi]) [member of] [C.sup.2]([[xi].sub.-], [[xi].sub.+]) [intersection] [C.sup.0][[[xi].sub.-],[[xi].sub.+]], -[infinity] < [[xi].sub.-] < [[xi].sub.+] < +[infinity]), and satisfies the followings

[mathematical expression not reproducible].

Remark 1. The definition of quasi traveling wave (with quenching) implies that it satisfies (1.1) only on semi-infinite interval or finite interval. In this paper, we do not discuss the behavior of the solutions of (1.3) after [psi] becomes infinity. It is necessary that more detailed (and hard) analysis in order to study the solutions after quenching (outside of the interval on that [phi]([xi]) satisfies (1.2)), and so we leave it open here.

In this setting, Matsue [4] proved the following theorem.

Theorem 1 (Theorem 4.21 of [4]). Assume that [alpha] > 1 with [alpha] [member of] N. Then, the quasi traveling waves with quenching for (1.1) are, if exist, characterized by trajectories whose initial data are on the stable manifold of a equilibrium at infinity ([phi], [psi]) = (0, +[infinity]) of (1.3). The quenching rates, namely, the extinction rate of [phi] and blow-up rate of [psi], are

[mathematical expression not reproducible]

with [absolute value of [[xi].sub.*]] < [infinity] and C [not equal to] 0.

The proof is given in [4].

Remark 2. We can obtain the equilibria at infinity (of (1.3)) not only ([phi], [psi]) = (0, +[infinity]) but also other equilibria by applying the Poincare compactification (see [4] and Sec. 3 for the details).

We note that the existence of the quasi traveling waves has not been proved yet. In this paper, we give the proof of the existence of them by considering the restricted case of [alpha] [member of] 2N. The proof is based on Poincare compactification (that is also used to prove Theorem 1 in [4]) and basic theory of dynamical systems. We then state the main theorem of this paper (see also Figure 1).

Theorem 2. Assume that [alpha] [member of] 2N. Then, the Eq. (1.1) possesses a family of "quasi traveling waves with quenching on a finite interval". Moreover, each quasi traveling wave with quenching u(t,x) = 1 - [phi]([xi]) (which satisfies (1.2) on a finite interval ([[xi].sub.-],[[xi].sub.+])) satisfies the followings

[mathematical expression not reproducible].

* [phi]([xi]) < 0 holds for [xi] [member of] ([[xi].sub.-],[[xi].sub.+]).

* There exists a constant [[xi].sub.*] [member of] ([[xi].sub.-],[[xi].sub.+]) such that the following holds: [psi]([xi]) < 0 for [xi] [member of] ([[xi].sub.-], [[xi].sub.*]), [psi]([[xi].sub.*]) = 0 and [psi]([xi]) > 0 for [xi] [member of] ([[xi].sub.*],[[xi].sub.+]).

In addition, quenching rates are

[mathematical expression not reproducible]

and

(1.4) [mathematical expression not reproducible]

with C > 0.

In order to prove Theorem 2, it is necessary to seek a family of orbits that connect ([phi],[psi]) = (0, -[infinity]) and (0, +[infinity]) of (1.3) (see Sec. 4 for the details). As shown in [2,4], the Poincare compactification is useful, and applicable for this problem. In the next section, we briefly introduce the Poincare compactification for the convenience of readers.

2. Preparation. In this section, we briefly introduce the Poincare compactification.

Let

X = P([phi],[psi])[partial derivative]/[partial derivative][phi] + Q([phi],[psi])[partial derivative]/[partial derivative][psi]

be a polynomial vector field on [R.sup.2], or in other words

[mathematical expression not reproducible],

where denotes d/dt, and P, Q are polynomials of arbitrary degree in the variables [phi] and [psi].

First, we consider [R.sup.2] as the plane in [R.sup.3] defined

by

([y.sub.1],[y.sub.2],[y.sub.3]) = ([phi],[psi],1).

We consider the sphere

[S.sup.2] = {y [member of] [R.sup.3] | [y.sup.2.sub.1] + [y.sup.2.sub.2] + [y.sup.2.sub.3] = 1}

which we call Poincare sphere. We divide the sphere into

No. 1]

[H.sub.+] = {y [member of] [R.sup.3] | [y.sub.3] > 0}, [H.sub.-] = {y [member of] [R.sup.3] | [y.sub.3] < 0}

and

[S.sup.1] = {y [member of] [R.sup.3] | [y.sub.3] = 0}.

Let us consider the projection of vector field X from [R.sup.2] to [S.sup.2] given by

[f.sup.+] : [R.sup.2] [right arrow] [S.sup.2] and [f.sup.-] : [R.sup.2] [right arrow] [S.sup.2],

where

[f.sup.[+ or -]]([phi],[psi]) := [+ or -]([phi]/[DELTA]([phi],[psi]), [psi]/[DELTA]([phi],[psi]), 1/[DELTA]([phi],[psi]))

with [DELTA]([phi],[psi]) = [square root of ([[phi].sup.2] - [[psi].sup.2] + 1)].

Second, we consider six local charts on [S.sup.2] given by

[U.sub.k] = {y [member of] [S.sup.2] [absolute value of [y.sub.k] > 0} and [V.sub.k] = {y [member of] [S.sup.2]] [y.sub.k] < 0} for k = 1, 2, 3. Consider the local projection

[g.sup.+.sub.k] : [U.sub.k] [right arrow] [R.sup.2] and [g.sup.-.sub.k] : [V.sub.k] [right arrow] [R.sup.2]

defined as

[g.sup.+.sub.k]([y.sub.1],[y.sub.2],[y.sub.3]) = -[g.sup.-.sub.k] ([y.sub.1],[y.sub.2],[y.sub.3]) = ([y.sub.m]/[y.sub.k],[y.sub.n]/[y.sub.k])

for m < n and m, n [not equal to] k. The projected vector fields are obtained as the vector fields on the planes

[[bar.U].sub.k] = {y [member of] [R.sup.3] | [y.sub.k] = 1}

and

[[bar.V].sub.k] = {y [member of] [R.sup.3] | [y.sub.k] = -1}

for each local chart [U.sub.k] and [V.sub.k]. We denote by (x, [lambda]) the value of [g.sup.[+ or -].sub.k] (y) for any k.

For instance, it follows that

([g.sup.+.sub.2] o [f.sup.+])([phi],[psi]) = ([phi]/[psi], 1/[psi]) = (x,[lambda]),

therefore, we can obtain the dynamics on the local chart [[bar.U].sub.2] by the change of variables [phi] = x/[lambda] and [psi] = 1/[lambda]. The locations of the Poincare sphere, ([phi], [psi])-plane and [[bar.U].sub.2] are expressed as Figure 2. We refer to [2] and references therein for more details. We also refer to [5] and [4] for the Poincare type compactification and its applications, respectively.

Throughout this paper, we follow the notations used here for the Poincare compactification. It is sufficient to consider the dynamics on [H.sub.+] [union] [S.sup.1], which is called Poincare disk, to obtain our main result.

3. Dynamics on the Poincare disk of

(1.3). In order to study the dynamics of (1.3) on the Poincare disk, we desingularize it by the timescale desingularization

(3.1) ds/d[xi] = [{[phi]([xi])}.sup.-[alpha]] for [alpha] [member of] 2N.

Since we assume that a is even, the direction of the time does not change via this desingularization. Then we have

(3.2) [mathematical expression not reproducible].

It should be noted that the time scale desingularization (3.1) is simply multiplying the vector field by [[phi].sup.[alpha]]. Then, with excepting the singularity {[phi] = 0}, the solution curves of the system (vector field) remain the same but are parameterized differently. Still, we refer to Section 7.7 of [3] and references therein for the analytical treatments of desingularization with the time rescaling. In what follows, we use the similar time rescaling (reparameterization of the solution curves) repeatedly to desingularize the vector fields.

Now we can consider the dynamics of (3.2) on the charts [[bar.U].sub.j] and [[bar.V].sub.j].

3.1. Dynamics on the chart [[bar.U].sub.2]. To obtain the dynamics on the chart [[bar.U].sub.2], we introduce coordinates ([lambda], x) by the formulas

[phi](s) = x(s)/[lambda](s), [psi](s) = 1/[lambda](s).

Then we have

[mathematical expression not reproducible].

Time-scale desingularization d[tau]/ds = [lambda][(s).sup.-[alpha]] yields

(3 3) [mathematical expression not reproducible],

where [[lambda].sub.[tau]] = d[lambda]/d[tau] and [x.sub.[tau]] = dx/d[tau]. The system (3.3) has the equilibria

[p.sup.+.sub.0] : ([lambda], x) = (0, 0) and [p.sub.c] : ([lambda],x) = (0, -1/c).

The Jacobian matrices at these equilibria are

[mathematical expression not reproducible].

Therefore, [p.sub.c] is a source, and [p.sup.+.sub.0] is not hyperbolic. In order to determine the dynamics near [p.sup.+.sub.0], we desingularize [p.sup.+.sub.0] by introducing the following blow-up coordinates:

[lambda] = [r.sup.[alpha] - 1][bar.[lambda]], x = [r.sup.[alpha]+1] [bar.x]

(see Sec. 3 of [2] for the desingularizations of vector fields by the blow-up). Since we are interested in the dynamics on the Poincare disk, we consider the dynamics of blow-up vector fields on the charts {[bar.[lambda]] = 1} and {[bar.x] = [+ or -] 1}.

Dynamics on the chart {[lambda] = 1}. By the change of coordinates [lambda] = [r.sup.[alpha]-1], x = [r.sup.[alpha]+1][bar.x], we have

[mathematical expression not reproducible].

The time-rescaling [mathematical expression not reproducible] yields

(3.4) [mathematical expression not reproducible].

The equilibria of (3.4) on {r = 0} are

[mathematical expression not reproducible].

The Jacobian matrices at these equilibria are

[mathematical expression not reproducible].

Moreover, since [absolute value of 1/([alpha] - 1)] < 2 holds, trajectories near [[bar.p].sup.+.sub.[alpha]] are tangent to [mathematical expression not reproducible] as [eta] [right arrow] +[infinity]. The solutions are approximated as

[mathematical expression not reproducible].

Dynamics on the chart {[bar.x] = -1}. By the change of coordinates [lambda] = [r.sup.[alpha]-1][bar.[lambda]], x = - [r.sup.[alpha]+1], and time-rescaling [mathematical expression not reproducible], we have

[mathematical expression not reproducible].

The equilibria on {r = 0} are

[mathematical expression not reproducible].

By the further computations, we can see that (0, 0) is a saddle, and [mathematical expression not reproducible] is a sink.

Dynamics on the chart {[bar.x] = 1}. The change of coordinates [lambda] = [r.sup.[alpha]-1][bar.[lambda]], x = [r.sup.[alpha]+1], and time-rescaling [mathematical expression not reproducible] yield

[mathematical expression not reproducible].

The equilibrium on {r = 0, [bar.[lambda]] [greater than or equal to] 0} is (0, 0). The linearized eigenvalues are [([alpha] + 1).sup.-1] and -([alpha] - 1)/ ([alpha] + 1) with corresponding eigenvectors (1, 0) and (0,1), respectively. Therefore, (r, [bar.[lambda]]) = (0, 0) on the chart {[bar.x] = 1} is a saddle.

Combining the dynamics on the charts {[bar.[lambda]] = 1} and {[bar.x] = [+ or -] 1}, we obtain the dynamics on [[bar.U].sub.2] (see Figure 3).

Still, we continue to study the dynamics on other charts in order to obtain the whole dynamics on the Poincare disk.

3.2. Dynamics on the chart [[bar.V].sub.2]. The change of coordinates

[phi](s) = -x(s)/[lambda](s), [psi](s) = -1/[lambda](s)

give the projected dynamics of (1.3) on the chart [[bar.V].sub.2]:

(3.5) [mathematical expression not reproducible],

where [tau] is the new time introduced by d[tau]/ds = [lambda][(s).sup.[alpha]]. The system (3.5) can be transformed into (3.3) by the change of coordinates ([lambda],x) [right arrow] (-[lambda], x). Therefore, it is sufficient to consider the blow-up of singularity [p.sup.-.sub.0] : ([lambda], x) = (0, 0) by the formulas

[lambda] = [r.sup.[alpha]-1][bar.[lambda]], x = [r.sup.[alpha]+1][bar.x] with [bar.[lambda]] = 1.

Then we have

(3.6) [mathematical expression not reproducible],

where [eta] satisfies [mathematical expression not reproducible]. The equilibria of (3.6) on {r = 0} are

[mathematical expression not reproducible].

The equilibrium [[bar.p].sup.-.sub.0] is a saddle with the eigenvalues [([alpha] - 1).sup.-1] and -2[([alpha] - 1).sup.-1] whose corresponding eigenvectors are (1,0) and (0,1), respectively. Further, [[bar.p].sup.-.sub.[alpha]] is a source with the eigenvalues [([alpha] - 1).sup.-] and 2 whose corresponding eigenvectors are (1, 0) and (0,1), respectively.

3.3. Dynamics on the chart [[bar.U].sub.1]. Let us study the dynamics on the chart [[bar.U].sub.1]. The transformations

[phi](s) = 1/[lambda](s), [phi](s) = x(s)/[lambda](s)

yield

(3.7) [mathematical expression not reproducible]

via time-rescaling d[tau]/ds = [{[lambda](s)}.sup.-[alpha]]. The equilibria of (3.7) are (0, 0) and (0, -c) whose Jacobian matrices are

[mathematical expression not reproducible],

respectively. Then the center manifold theory is applicable to study the dynamics near (0, 0) (for instance, see [1]). It implies that there exists a function h([lambda]) satisfying

h(0) = dh/d[lambda](0) = 0

such that the center manifold of (3.7) is represented as {([lambda],x) | x = h([lambda])} near (0, 0). Differentiating it with respect to [tau], we have

-[lambda]h([lambda]) dh/d[lambda]([lambda]) = -ch([lambda]) + [[lambda].sup.1+[alpha]] - [{h([lambda])}.sup.2].

Then we can obtain the approximation of the (graph of) center manifold as follows:

{([lambda], x) | x = [[lambda].sup.[alpha]+1]/c + O([[lambda].sup.2[alpha]+2])} .

Therefore, the dynamics of (3.7) near (0,0) is topologically equivalent to the dynamics of the following equation:

[[lambda].sub.[tau]] = - [[lambda].sup.[alpha]+2]/c + O([[lambda].sup.2[alpha]+3]).

These results give us the dynamics on the chart [[bar.U].sub.1].

3.4. Dynamics on the chart [[bar.V].sub.1]. The transformations

[phi](s) = -1/[lambda](s), [psi](s) = -x(s)/[lambda](s)

yield

(3.8) [mathematical expression not reproducible]

via time-rescaling d[tau]/ds = [{[lambda](s)}.sup.-[alpha]]. We can see that the system (3.8) can be transformed into the system (3.7) by the change of variables: ([lambda],x) [??] (-[lambda], x). Therefore, the dynamics of (3.8) is equivalent to the reflected one of (3.7) with respect to {[lambda] = 0}.

4. Proof of Theorem 2. Since the point ([y.sub.1],[y.sub.2],[y.sub.3]) = (0,1,0) on the Poincare disk corresponds to [p.sup.+.sub.0], we denote it by [p.sup.+.sub.0] as well. Similarly, we denote by [p.sup.-.sub.0] the point ([y.sub.1], [y.sub.2], [y.sub.3]) = (0, -1, 0). In order to prove Theorem 2, it is necessary to find the orbits that connect [p.sup.-.sub.0] and [p.sup.+.sub.0] on the Poincare disk. The phase portrait on the Poincare disk of (1.3) is shown in Figure 4 for the convenience of readers. Proof. (I): For a given compact subset W [subset] [H.sub.+], there are no equilibria or closed orbits in W. Therefore, by the Poincare-Bendixson theorem, any trajectories starting from the points in W can not stay in W with increasing s. This implies that the trajectories in [H.sub.+] go to [S.sup.1], which corresponds to {[parallel]([phi],[psi])[parallel] = [infinity]}.

(II): The line {[phi] = 0} is invariant under the flow of (3.2). Therefore, any trajectories start from the points in {y [member of] [H.sub.+] | [y.sub.1] < 0} can not go to {y [member of] [H.sub.+] | [y.sub.1] > 0}.

(III): Let [mathematical expression not reproducible] be a stable manifold of [[bar.p].sup.+.sub.[alpha]] (which is the equilibrium of the system (3.4)). We denote by [W.sup.s]([[bar.p].sup.+.sub.[alpha]]) the stable set, which corresponds to [mathematical expression not reproducible] on the blow-up vector filed (3.4), of the equilibrium [p.sup.+.sub.0] of (3.3). Similarly, we denote by [W.sup.u]([[bar.p].sup.-.sub.[alpha]]) the unstable set of [p.sup.-.sub.0], corresponding to the unstable manifold of [[bar.p].sup.-.sub.[alpha]] on the blow-up vector field (3.6). Consider the trajectories start from the points on [W.sup.u]([[bar.p].sup.-.sub.[alpha]]) [subset] {y [member of] [H.sub.+] | [y.sub.1] < 0}. The trajectories can not stay in any compact subset on [H.sub.+], and can not go to {y [member of] [H.sub.+] | [y.sub.1] > 0}, therefore, they go to [p.sup.+.sub.0] with lying on [W.sup.s]([[bar.p].sup.+.sub.[alpha]]). This implies that the system (3.2) possesses the orbits that connect [p.sup.-.sub.0] and [p.sup.+.sub.0] on the Poincare disk. It is easy to see that d[phi]/d[psi] takes the same values on the vector fields defined by (3.2) and (1.3) by excepting the singularity {[phi] = 0}. Thus, there are orbits connecting ([phi], [psi]) = (0, -[infinity]) and (0, +[infinity]) on the original vector field (1.3). (IV): As shown in [4], we can obtain the quenching rates of [phi]([xi]) and [psi]([xi]). Indeed,

[mathematical expression not reproducible]

holds with a constant C. This yields

[mathematical expression not reproducible].

Set [mathematical expression not reproducible], then we have

[mathematical expression not reproducible].

Therefore,

[mathematical expression not reproducible]

holds. Finally, we obtain

[mathematical expression not reproducible]

with C > 0. Similarly, we can obtain the quenching rates for [psi]([xi]) as [xi] [right arrow] [[xi].sub.+] and (1.4).

This completes the proof.

doi: 10.3792/pjaa.96.001

References

[1] J. Carr, Applications of centre manifold theory, Applied Mathematical Sciences, 35, Springer-Verlag, New York, 1981.

[2] F. Dumortier, J. Llibre and J. C. Artes, Qualitative theory of planar differential systems, Universitext, Springer-Verlag, Berlin, 2006.

[3] C. Kuehn, Multiple time scale dynamics, Applied Mathematical Sciences, 191, Springer, Cham, 2015.

[4] K. Matsue, Geometric treatments and a common mechanism in finite-time singularities for autonomous ODEs, J. Differential Equations 267 (2019), no. 12, 7313 7368.

[5] K. Matsue, On blow-up solutions of differential equations with Poincaree-type compactifications, SIAM J. Appl. Dyn. Syst. 17 (2018), no. 3, 2249 2288.

By Yu ICHIDA *) and Takashi OKUDA SAKAMOTO **)

(Communicated by Masaki KASHIWARA, M.J.A., Dec. 12, 2019)

2010 Mathematics Subject Classification. Primary 35C07, 34C05, 34C08.

*) Graduate School of Science and Technology, Meiji University, 1-1-1 Higashimita, Tama-ku, Kawasaki, Kanagawa 214-8571, Japan.

**) School of Science and Technology, Meiji University, 1-1-1 Higashimita, Tama-ku, Kawasaki, Kanagawa 214-8571, Japan.

Caption: Fig. 1. Schematic picture of the quasi traveling wave with quenching on [xi] [member of] [[[xi].sub.-],[[xi].sub.+]] obtained in Theorem 2.

Caption: Fig. 2. Locations of the Poincare sphere and chart [[bar.U].sub.2].

Caption: Fig. 3. Schematic pictures of the dynamics of the blow-up vector fields and [[bar.U].sub.2].

Caption: Fig. 4. Compactification of the system (1.3).
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Author:Ichida, Yu; Sakamoto, Takashi Okuda
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Date:Jan 1, 2020
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