This section is devoted to the study the solution of (3.1.2) with asymptotically constant initial data. We first consider the following ODE system:
( Ut= −U/(U2+ V2), Vt= V /(U2+ V2), (3.3.1)
for t ≥ 0 with the initial condition (U (0), V (0)) = (M, 0) for some constant M > 0. Then it is easy to see that the solution is given by (U (t), V (t)) := (√
M2− 2t, 0). Note that the quenching time of this ODE system is T = T (M ) := M2/2.
Next, in order to estimate u(x, t) from below, we consider the following Cauchy problem:
ut= uxx− 1/u, x ∈ R, t ∈ [0, T ), u(x, 0) = u0(x), x ∈ R.
(3.3.2)
where [0, T ) is the maximal existence interval of u. Also, we consider the following ODE problem corresponding to the problem (3.3.2):
(3.3.3) Ut= −1/U , t ∈ [0, T ), U (0) = M.
Note that the solution of (3.3.3) is given by U (t) =√
M2− 2t with T = T (M ) := M2/2.
Motivated by an idea from [28], we have the following lemma. We also refer the reader to [41] for the Fujita equation, [49] for a quasilinear parabolic equation, and [51] for a cooperative parabolic system.
Lemma 3.3.1 Let U be the solution of (3.3.3) and let u be the solution of (3.3.2) defined on R × [0, T ). Suppose that there exist t0 ∈ [0, bT ), r0 ∈ (0, ∞) and a constant θ > 1 such that
u(x, t) ≥ θU (t), f or |x| ≤ r0, t0 ≤ t < bT .
where bT := min{T, T }. Then u has a positive lower bound in {|x| ≤ r0/2} × [t0, bT ).
Proof. We shall construct a suitable subsolution of (3.3.2) as follows w(x, t) := bθp
M2− 2t + h(x), where bθ ∈ (1, θ) and
h(x) := εcos2(πx 2r0) with small ε > 0 to be specified later.
By a simple computation, we obtain that
wt− wxx+ 1 if we choose ε > 0 sufficiently small such that
ε ≤ (M2− 2t0)
Then it follows from (3.3.4) and the comparison principle that w(x, t) ≤ u(x, t) for
|x| ≤ r0 and t0 ≤ t < bT . Therefore, we have
Hereafter, we assume
u0 ∈ C1(R), u0 ≥ M, u0 6≡ M, (3.3.5)
|x|→∞lim u0(x) = M.
(3.3.6)
Note that by (3.3.2), (3.3.3), and (3.3.5) we have U ≤ u. Therefore, we obtain T ≥ T and so bT = T .
The following lemma shows that quenching can occur only at space infinity.
Lemma 3.3.2 Let u be a solution of (3.3.2) satisfying (3.3.5) and (3.3.6) for some constant M > 0. Then u has a positive lower bound in Ω × [0, T ) for any compact set Ω ⊂ R.
Proof. In view of Lemma 3.3.1, since bT = T , it suffices to show that, for any given R > 0 there exist t0 ∈ [0, T ) and θ > 1 such that
u(x, t) ≥ θ√
M2− 2t, |x| ≤ 2R, t0 ≤ t < T.
(3.3.7)
For this purpose, we let γ(x, t) := u(x, t)/U (t). Then the function γ = γ(x, t) satisfies γt= γxx+ 1
U2
− 1 γ + γ
≥ γxx, since γ ≥ 1. Moreover, by (3.3.5) and (3.3.6) we obtain
γ(·, 0) = u0
M ≥ 1, γ(·, 0) 6≡ 1.
From the strong maximum principle, we have that γ(x, t) > 1 for all x ∈ R and t > 0.
Therefore, for any given R > 0, there exist θ > 1 and t0 ∈ (0, T ) such that γ(x, t) ≥ θ, |x| ≤ 2R, t0 ≤ t < T.
This gives (3.3.7). Therefore we complete the proof.
To investigate the behavior of the solution of (3.1.2) at space infinity, we recall the following useful property (cf. [28]). We also refer the reader to [51] for the blow-up problem for a cooperative parabolic system.
Theorem 3.3.1 Let u and bu be solutions of
( ut= Duxx+ f (u), x ∈ R, t > 0, u(x, 0) = u0(x), x ∈ R.
(3.3.8)
where u(x, t) = (u(x, t), v(x, t)) ∈ R2, f = (f1, f2) is a smooth mapping from R2 to R2, D = diag(1, 1), with initial data u0, ub0 ∈ (L∞(R) ∩ C(R))2, respectively. Suppose that there exist sequences {rn}∞n=1 ⊂ (0, ∞) and {an}∞n=1⊂ R with rn→ ∞ as n → ∞ such that
n→∞lim ||u0−ub0||L∞(B2rn(an))= 0.
Then
n→∞lim ||u(·, t) −bu(·, t)||L∞(Brn(an))= 0.
for any t ∈ (0, eT ), where eT = min{T (u0), T (bu0)}.
Notice that the following corollary is applicable to our system (3.1.2). Since its proof is exactly the same as the one given in [28, Corollary 4.2], we omit is here.
Corollary 3.3.3 If some solutions of
Ut = f (U) (3.3.9)
quenches in a finite time, then there exists a spatially inhomogeneous solutions of (3.3.8) which quenches in a finite time.
In the following, we shall focus on the Cauchy problem for (3.1.2) with initial data satisfying (3.1.7) and (3.1.8).
Lemma 3.3.4 Let u be a solution of (3.3.2) satisfying (3.3.5) and (3.3.6) for some constant M > 0. Then u quenches at the finite time T = M2/2.
Proof. First, we set u(x, t) = u(x, t), bu(x, t) = U (t), |an| = 4n, and rn = n. By (3.3.6), we have
n→∞lim ||u0−ub0||L∞(B2rn(an))= 0.
(3.3.10)
Notice that u andu are solutions of (3.3.2) and (3.3.3) with initial data ub 0 andub0, respec-tively. Let f (u) = −1/u, f (u) = −1/U . Applying Theorem 3.3.1 to (3.3.2) and (3.3.3), web obtain
|x|→∞lim u(x, t) = U (t), ∀t ∈ [0, T ).
On the other hand, by (3.3.2), (3.3.3), (3.3.5), and the comparison principle, we have u(x, t) ≥ U (t) for all x ∈ R and t > 0. Combining the above two facts, we have the quenching time T = T = M2/2.
Now we prove the Theorem 3.1.2 by using Theorems 3.1.1 and 3.3.1.
Proof of Theorem 3.1.2. First, we have the local existence of (u, v) for t ∈ [0, σ] for some σ > 0. Let u(x, t) = (u(x, t), v(x, t)), u(x, t) = (U (t), V (t)) andb
where (u, v) and (U, V ) are solutions of (3.1.2) and (3.1.3), respectively. By applying Theorem 3.3.1 to (3.1.2) and (3.1.3) with |an| = 4n and rn = n, we have
|x|→∞lim u(x, t) = U (t), and lim
|x|→∞v(x, t) = V (t), ∀t ∈ [0, σ].
(3.3.11)
Also, it follows from (3.1.4) and (3.1.8) with N > 0 that lim
|x|→∞u(x, t)v(x, t) = U (t)V (t) = U (0)V (0) = lim
|x|→∞u0(x)v0(x) = M N > 0.
Hence the assumption (3.1.6) is satisfied for all x with |x| ≥ R at t = σ for some constants R sufficient large and K > 0.
Moreover, by the strong maximum principle, we obtain v > 0 in R × [0, σ]. It implies that the assumption (3.1.6) holds for all x with |x| ≤ R at t = σ with the positive constant K (taking a smaller one if necessary). Therefore, by applying Theorem 3.1.1 to the Cauchy problem (3.1.2) starting at t = σ, we obtain that the solution (u, v) exists globally in time and (u, v) converges to (0, ∞) as t → ∞. This completes the proof of Theorem 3.1.2.
Finally, we give a proof of Theorem 3.1.3.
Proof of Theorem 3.1.3. We choose u0 = u0. Then, by the comparison principle, we obtain
u(x, t) ≥ u(x, t), x ∈ R, for t > 0 such that u and u exist.
(3.3.12)
Suppose that the solution (u, v) quenches at time T∗. By (3.3.12), we have T∗ ≥ T . On the other hand, by Lemmas 3.3.2 and 3.3.4, the solution u quenches at finite time T = M2/2 only at space infinity. Thus the inequality (3.3.12) implies that
u ≥ u > 0 in R × [0, T ).
where (u, v) and (U, V ) are solutions of (3.1.2) and (3.3.1), respectively. Applying Theorem 3.3.1 to (3.1.2) and (3.3.1) with |an| = 4n and rn = n again, we have
|x|→∞lim u(x, t) = U (t), lim
|x|→∞v(x, t) = V (t) = 0, ∀t ∈ [0, T ).
(3.3.14)
Hence we obtain T∗ = T . From Lemma 3.3.2, u quenches only at space infinity. Combining this with (3.3.13), we conclude that the quenching of the solution (u, v) occurs only at space infinity. This proves the theorem.
Chapter 4
References
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