This section is devoted to the proof of the main result, Theorem 2.1.3. We first construct a suitable Lyapunov function using a method of Zelenyak [59] (see also [25]). Define
(2.5.1) ER(s)[z](s) = Then, integrating by parts and using (2.5.2), we have
d where bg is defined in (2.4.5).
Let
where ψ is defined in (2.4.6)-(2.4.7). Then using (2.4.12) and by a simple computation we
Hence it follows from (2.5.4) and (2.5.5) that
Φ(y, z(R(s), s), zy(R(s), s)) ≤ |zy(R(s), s)|2exp−c∗z(R(s), s)(γ−1)/2R(s) ,
|Φw(y, z(R(s), s), zy(R(s), s))| ≤ |zy(R(s), s)| exp−c∗z(R(s), s)(γ−1)/2R(s) for large s for some constant c∗ > 0.
Next, we shall follow an idea from [24, p.54] to obtain an estimate of ux(1, t). Since ut< 0 in QT and ux > 0 in (0, 1] × (0, T ), we have
(um)xx < up in QT, (um)xx(um)x< up(um)x in (0, 1] × (0, T ).
An integrating of the last inequality from 0 to x ∈ (0, 1] gives 0 ≤ m(um−1ux)(x, t) = (um)x(x, t) ≤
r 2m
p + mu(p+m)/2(x, t)
for (x, t) ∈ (0, 1] × (0, T ). Therefore, by (2.1.1), we have 0 ≤ ux(1, t) ≤ c∗∗ for all t ∈ (0, T ) for some positive constant c∗∗. Then
(2.5.6) 0 ≤ zy(R(s), s) ≤ c∗∗mkm−1exp[(βm − α)s].
for some positive constant c. Since Φw(0, z(0, s), 0) = 0, together with (2.5.7) and (2.5.8), we obtain that J1 is bounded from above by a function that decays exponentially fast.
Next, we prove that J1 is bounded below. By (2.5.3) and noting thatbg(y, µ) = g(µ) for
Hence by (2.5.8) and (2.5.9) we obtain that J1 is bounded from below by a function that decays exponentially fast. Thus we have proved:
Lemma 2.5.1 We have where J1 satisfies the property R∞
s0 |J1(s)|ds < ∞. (2.3.4)-(2.3.7), applying parabolic Lp estimates to (2.1.8) and to the equation satisfied by zy, and then parabolic H¨older estimates, it follows that the sequence {zj}j∈N is compact in C2,1(ω) for any ω ⊂⊂ ˜Ω. Therefore, using a diagonal process, there exists a subsequence {jl} and a function z∞ ∈ C2,1( ˜Ω), such that zjl(y, s) → z∞(y, s) as l → ∞, locally uniformly in C2,1( ˜Ω). Moreover, z∞ satisfies (2.1.8) in ˜Ω and, due to (2.3.7),
(2.5.10) |∂yz∞(y, s)| ≤ c|y|, 0 < |y| ≤ 1, s ∈ R.
Consider the Lyapunov function ER(s)[z](s) defined in (2.5.1). Then we have ER(s)[z](s) ≤ ER(s)[z](s0) +
where c∗ is the upper bound for z(y, s) over y ∈ [0, 1]. From (2.5.11) we obtain
ER(s)[z](s) ≥ − Z R(s)
0
Z z(y,s)
κ bg(y, µ)P (y, µ, 0)dµdy ≥ − eC, and hence by using Lemma 2.5.1, we have
(2.5.12) γ
Z ∞ s0
Z R(s) 0
P (y, z(y, s), zy(y, s))zγ−1(y, s)|zs(y, s)|2dyds ≤ eC.
Note that P (y, v, w) is bounded below away from 0 for y, v, w in bounded sets. Also, z and zy are bounded for y bounded, z(y, s) ≥ D∗|y|δ in Ω0 and γ > 1. For each 0 < ε < 1 and M > eαs0, putting sM = α−1ln M , it follows from (2.5.12) that
Z ∞ sM
Z M ε
|zs(y, s)|2dyds < ∞.
For each S > 0, we thus deduce Z S
−S
Z M ε
|∂sz∞(y, s)|2dyds ≤ lim inf
j→∞
Z ∞ sj−S
Z M ε
|zs(y, s)|2dyds = 0,
hence ∂sz∞(y, s) = 0 and z∞= z∞(y) satisfies
(2.5.13) z00− αγyzγ−1z0 + βzγ− zq = 0, y > 0.
Using (2.5.13) and (2.5.10), it follows that z∞can be extended to a C2solution of (2.5.13) on [0, ∞) with ∂yz∞(0) = 0, hence to a symmetric C2 solution on R, in view of the symmetry of z in y. Note that z∞ is monotone in y > 0. Therefore, the conclusion follows from (2.3.4)-(2.3.5) and Proposition 2.4.1. This completes the proof of the theorem.
Chapter 3
Dynamics for a complex-valued heat equation
3.1 Introduction
In this chapter, we study the following equation zt= zxx− 1
z, (3.1.1)
where z = z(x, t) is a complex-valued function of the spatial variable x ∈ R and the time variable t ≥ 0. If we set z(x, t) = u(x, t) + iv(x, t), where i =√
−1 and u(x, t), v(x, t) ∈ R, then (3.1.1) can be written as a system of parabolic equations
( ut= uxx− u/(u2+ v2), vt = vxx+ v/(u2+ v2).
(3.1.2)
If z(x, t) is real-valued (i.e., v ≡ 0), then the system is reduced to the equation ut = uxx− 1
u.
An initial boundary value problem for the above equation was first studied by Kawarada [37] in 1975. For more general negative power nonlinearity, we refer the reader to, e.g., [22, 31, 38] and the references cited therein. The goal of this chapter is to study the dynamics of solutions of the system (3.1.2) with v 6≡ 0.
First of all, we consider a spatially homogeneous solution of (3.1.2), namely, (u, v) = (U (t), V (t)). We obtain that (U (t), V (t)) satisfies the following ODE system:
( Ut= −U/(U2+ V2), Vt= V /(U2+ V2).
(3.1.3)
Given (U (0), V (0)) ∈ R2\ {(0, 0)}. By a simple computation, we obtain that U (t)V (t) = U (0)V (0) := C, ∀t ≥ 0.
(3.1.4)
for some constant C ∈ R.
If U (0) = 0, then the trajectory stays on the the V -axis, exists globally and tends to
±∞ as t → ∞. On the other hand, if V (0) = 0, then V (t) ≡ 0 and U tends to zero in finite time. When C 6= 0, by (3.1.3) and (3.1.4) we have (U (t), V (t)) → (0, ±∞) as t → ∞.
In this chapter, we consider the initial value problem (P) for (3.1.2) with the initial condition
(u(·, 0), v(·, 0)) = (u0, v0).
(3.1.5)
In the sequel, we shall always assume that
u0 > 0, v0 ≥ 0, u0, v0 ∈ L∞(R) ∩ C(R), inf
R
u0+ inf
R
v0 > 0.
Then the problem (P) has a unique solution (u, v) ∈ (C([0, T ); L∞(R)))2, where T = T (u0, v0) ∈ (0, ∞] is the maximal existence time of the solution. Furthermore, we have either T = ∞, or
T < ∞ and lim inf
t→T {inf
x∈Ru(x, t) + inf
x∈Rv(x, t)} = 0.
In the first case, we have the global existence. For the second case, we say that the solution of (P) quenches in a finite time T in which T is called the quenching time. Moreover, we say that xQ ∈ R is a (finite) quenching point for (u, v) if there exists a sequence {(xj, tj)}
such that xj → xQ, tj ↑ T and u(xj, tj) + v(xj, tj) → 0 as j → ∞. We shall investigate the global and non-global existence of solutions of (P).
The first result is about the global existence and (time) asymptotic behavior of solution of the problem (P).
Theorem 3.1.1 Suppose that the initial data satisfy
u0(x) > 0, v0(x) > 0, ∀x ∈ R, u0 and v0 are bounded in R, u0(x)v0(x) ≥ K, ∀x ∈ R, f or some constant K > 0.
(3.1.6)
Then the solution of (3.1.2) with (3.1.5) exists globally in time and (u, v) converges to (0, ∞) as t → ∞ uniformly in R.
For t ≥ 0, we set
R(t) :=(u(x, t), v(x, t)) ∈ R2; x ∈ R
to be the image of the solution on (u, v)-plane. We remark that, under the hypothesis of Theorem 3.1.1, the closure of the convex hull of R(0) lies in the first quadrant of (u, v)-plane. Indeed, under the condition (3.1.6), we shall see that R(t) stays in the first quadrant for all t > 0. This implies the global existence of solutions.
On the other hand, if the initial data do not satisfy (3.1.6), in view of the dynamics of (3.1.3), it is interesting to see what happen. One question is to see under what conditions the quenching occurs. From (3.1.2) it is easy to see that both u and v quench simultaneously whenever quenching occurs. On the contrary, there might be non-simultaneous quenching in which just one component quenches and the other remains bounded away from zero. For this, we refer the reader to, e.g., [9, 43, 61, 45, 60].
To find solutions quenching in finite time, we consider the case when the initial data are asymptotically constants. Namely, we impose the following conditions on initial data:
u0, v0 ∈ C1(R), u0 ≥ M, u0 6≡ M, v0 ≥ 0, v0 6≡ 0, (3.1.7)
|x|→∞lim u0(x) = M, lim
|x|→∞v0(x) = N (3.1.8)
for some constants M > 0 and N ≥ 0.
The following theorem shows that the solution of (3.1.2) with initial data satisfying (3.1.7) and (3.1.8) with N > 0 behaves like the solution the ODE system (3.1.3) with (U (0), V (0)) = (M, N ).
Theorem 3.1.2 Let (u, v) be a solution of (3.1.2) with initial data (u0, v0) satisfying (3.1.7) and (3.1.8). If N > 0, then the solution of (3.1.2) with (3.1.5) exists globally for all t ≥ 0 and (u, v) converges to (0, ∞) as t → ∞ uniformly in R.
On the other hand, if the initial data of (3.1.2) satisfy (3.1.7) and (3.1.8) with N = 0, then the solution of (3.1.2) and (3.1.5) quenches only at space infinity. Namely, there are no (finite) quenching points, while there exists a sequence {(xj, tj)} such that |xj| → ∞, tj ↑ T and u(xj, tj) + v(xj, tj) → 0 as j → ∞.
Theorem 3.1.3 Let (u, v) be a solution of (3.1.2) and (3.1.5) with the initial data (u0, v0) satisfying (3.1.7) and (3.1.8) with M > 0 and N = 0. Then the solution of (3.1.2) with (3.1.5) quenches at the finite time t = T = M2/2. Moreover, the solution quenches only at space infinity.
Note that the problem of quenching at space infinity for scalar equation was studied by Giga-Seki-Umeda [20, 21]. In [20], they characterized that, with suitable initial data, solutions of the following Cauchy problem ut = uxx/(1 + u2x) − (n − 1)/u quenching only
at space infinity. In [21], they estimated its profile at the quenching time from above and below.
The motivation of this study is from a work of Guo-Ninomiya-Shimojo-Yanagida [28].
In [28], they considered, instead of (3.1.1), the following complex-valued equation:
zt= ∆z + z2 (3.1.9)
where z = z(x, t) = u(x, t) + iv(x, t) is a complex-valued function of x ∈ Rm (m ∈ N) and t ≥ 0. To obtain the asymptotical behavior of the solution, our method is close to that in [28] by using an invariant set argument. But, instead of considering the invariant subset in (u, v)-plane, we transform our problem in (u, w)-plane where w := 1/v. Also, the solu-tion blows up non-simultaneously at space infinity for the case (3.1.9) with asymptotically constant initial data. But, in our case (3.1.1), quenching can only occurs simultaneously.
This chapter is organized as follows. In section 3.2, we provide a sufficient condition for the existence of global solutions and study the asymptotic behavior of solutions as t → ∞.
In section 3.3, we study the solution of (3.1.2) with asymptotically constant initial data.