Non-simultaneous quenching in a system of heat equations coupled at the boundary

Raúl Ferreira^{∗}, Arturo de Pablo^{∗},

Departamento de Matemáticas, U. Carlos III de Madrid, 28911 Leganés, Spain.

Fernando Quirós^{∗},

Departamento de Matemáticas, U. Autónoma de Madrid, 28049 Madrid, Spain.

Julio D. Rossi^{†}

Departamento de Matemática, F.C.E y N., UBA, (1428) Buenos Aires, Argentina.

Abstract

We study the solutions of a parabolic system of heat equations coupled at the boundary through a nonlinear flux. We characterize in terms of the parameters involved when nonsimultaneous quenching may appear. Moreover, if quenching is non-simultaneous we find the quenching rate, which surprisingly depends on the flux associated to the other component.

1 Introduction

We study the formation of singularities in finite time for solutions (u, v) of the parabolic system

\{ \begin{matrix}u_{t} = u_{xx} \\ v_{t} = v_{xx} \end{matrix} in (0, 1) × (0, T),  (1.1)

coupled at the boundary through a nonlinear flux at one border,

\{ \begin{matrix}u_{x}(0, t) = v^{−p}(0, t) \\ v_{x}(0, t) = u^{−q}(0, t) \end{matrix} in (0, T),  (1.2)

∗Partially supported by project BFM2002-04572 (Spain). †Partially supported by UBA grant EX046, CONICET and Fundación Antorchas (Argentina). 2000 AMS Subject Classification: 35B40, 35K45, 35K55 Keywords and phrases: quenching, parabolic system, non-simultaneous.

zero flux at the other border,

\{ \begin{matrix}u_{x}(1, t) = 0 \\ v_{x}(1, t) = 0 \end{matrix} in (0, T),  (1.3)

and initial data

\{ \begin{matrix}u(x, 0) = u_{0}(x) \\ v(x, 0) = v_{0}(x) \end{matrix} in (0, 1).  (1.4)

Throughout the paper we assume that the initial data u_{0}, v_{0} are positive, smooth and compatible with the boundary data. The constant T denotes the maximal existence time for the solution, which is to be understood in a classical sense. We also assume u′_{0}, v′_{0} ≥ 0 and u′′_{0}, v′′_{0} < 0.

We are interested in the quenching phenomenon, which has attracted a lot of attention in recent years, see [1], [2], [7] and references therein. There is quenching if T < ∞; in this case

lim inf_{t↗T} min{ min_{0≤x≤1} u(x, t), min_{0≤x≤1} v(x, t)} = 0,

and a singularity appears in the boundary condition. It is easy to see that quenching always happens for our system, see Proposition 2.1. However, a priori there is no reason why both components u and v should reach the level zero simultaneously at the quenching time T. If one of them remains bounded away from zero at t = T we say that quenching is non-simultaneous. An example of such a phenomenon was given in [8], where the authors considered a semilinear system. See also [9], [10], [11] for examples of non-simultaneous blow-up.

Our first aim is to characterize when non-simultaneous quenching is possible for problem (1.1)–(1.4).

Theorem 1.1 If v does not quench then q < 1. Moreover, if q < 1 then for every v_{0} there exist initial data u_{0} such that u quenches while v does not.

The analogous result holds for the component u. This means that if p, q ≥ 1, then quenching is always simultaneous, while if p < 1 or q < 1 non-simultaneous quenching indeed occurs. Nevertheless, if p, q < 1 simultaneous quenching is also possible.

Theorem 1.2 If 0 < p, q < 1 then there exist initial data which produce simultaneous quenching.

If p ≥ 1 > q we conjecture that only non-simultaneous quenching is possible. In this direction we prove that this is the case if q < 1 and p ≥ p_{0} = (1 + q)/(1 − q) > 1, see Lemma 3.1.

Next, we concentrate on the non-simultaneous case and we find the quenching rates, the quenching set and the quenching behaviour. The notation f ∼ g means that there exist finite positive constants c_{1}, c_{2} such that c_{1} g ≤ f ≤ c_{2} g.

Theorem 1.3 If quenching is non-simultaneous and, for instance u is the quenching variable, then u(0, t) ∼ (T − t)^{1/(q+1)}, u_{t}(0, t) ∼ −(T − t)^{−q/(q+1)} and u(x, T) ∼ x.

If u does not quench we have analogous estimates for v, replacing q by p. Observe that the quenching rate of the quenching component depends on the exponent appearing in the flux of the other component, something which is rather surprising. For instance, in the system of semilinear heat equations considered in [8], in the case of non-simultaneous quenching the rate exponent is 1.

2 Quenching

In this section we prove some a priori estimates, beginning with the fact that quenching always happens for our problem. To simplify the presentation of the proofs we define the functions

U(t) = u(0, t) = min_{0≤x≤1} u(x, t), V (t) = v(0, t) = min_{0≤x≤1} v(x, t).  (2.1)

Proposition 2.1 Quenching happens for system (1.1)–(1.4) for every initial data.

Proof. By the maximum principle we have u ≤ M = ∥u_{0}∥_{∞}, v ≤ N = ∥v_{0}∥_{∞}. Hence, integration of (1.1) in space gives the following mass estimates,

∫_{0}^{1} u(x, t) dx ≤ M − N^{−p}t,  ∫_{0}^{1} v(x, t) dx ≤ N − M^{−q}t,  (2.2)

which yield a contradiction if u and v are positive for all times. □

Some authors understand quenching as the blow-up of the time derivative while the solution itself remains bounded, see for example [5], [6]. This indeed happens for our problem if u and v are strictly decreasing in time, see Corollary 2.1. This will follow for instance if the initial values are strictly concave, something that we assume. We first prove an auxiliary result.

Lemma 2.1 There exists ϵ > 0 such that

U′(t) ≤ −ϵU^{−q}(t),  V′(t) ≤ −ϵV^{−p}(t).  (2.3)

Proof. We define the functions F = u_{t} + ϵv_{x}, G = v_{t} + ϵu_{x}. They are solutions to the heat equation. Choosing ϵ > 0 small enough, we have F (x, 0) < 0 and G(x, 0) < 0 for x ∈ [0, 1]. Also, since u and v are decreasing in time, we have F (1, t) < 0 and G(1, t) < 0 for t ∈ (0, T). As to the flux at x = 0 we obtain

\begin{aligned}F_{x}(0, t) & = −(pv^{−p−1} − ϵ)v_{t} ≥ −CG(0, t), \\ G_{x}(0, t) & = −(qu^{−q−1} − ϵ)u_{t} ≥ −CF (0, t), \end{aligned}

if ϵ is small. The maximum principle implies F (x, t), G(x, t) ≤ 0 for every x ∈ [0, 1] and t ∈ [0, T). In particular for x = 0 we obtain (2.3). □

Direct integration of inequalities (2.3) yields the following estimates.

Corollary 2.1

\begin{matrix}U(t) ≥ C(T − t)^{1/(q+1)}, & V (t) ≥ C(T − t)^{1/(p+1)}, \\ U′(t) ≤ −C(T − t)^{−q/(q+1)}, & V′(t) ≤ −C(T − t)^{−p/(p+1)}. \end{matrix}

Remark. (i) In the case of non-simultaneous quenching, the flux at the boundary of the quenching variable remains bounded. Nevertheless, its time derivative blows up.

(ii) In fact, both time derivatives blow up. Hence quenching is always simultaneous in the sense of [5].

With these estimates we are able to prove the continuity of the quenching time.

Theorem 2.1 The quenching time is continuous with respect to the initial data.

Proof. Let (u, v) and (ũ, ṽ) be the solutions corresponding to the initial values (u_{0}, v_{0}) and (ũ_{0}, ṽ_{0}), and let T and T̃ be their quenching times. Assume for instance that T < T̃, and also that u quenches. Given ε > 0, there is a δ > 0 as small as desired such that if T − δ < t_{0} < T then U(t_{0}) ≤ ε/2. Since up to time t_{0} the solutions are classical, by the continuous dependence with respect to the initial data we have that, if |u_{0}−ũ_{0}|+|v_{0}−ṽ_{0}| ≤ μ then Ũ(t_{0}) ≤ ε. Therefore, using the lower estimate given in Corollary 2.1, we get |T − T̃| ≤ |T − t_{0}| + |T̃ − t_{0}| ≤ δ + ε^{q+1}. □

Theorem 2.2 The set of initial data such that one of the components quenches while the other one remains bounded is open.

Proof. Let (u, v), (ũ, ṽ) and T, T̃ be as before. Assume for instance that u quenches and v does not. Let t_{0} near T and (ũ_{0}, ṽ_{0}) near (u_{0}, v_{0}) be such that V (t_{0}) = K and Ṽ(t_{0}) ≥ K/2. Using the representation formula for ṽ we have

\frac{1}{2}Ṽ(t) ≥ C(t_{0}) − ∫_{t_{0}}^{t}(T̃ − s)^{−q/(1+q)}(t − s)^{−1/2} ds.

As t → T̃ we obtain

\frac{1}{2}Ṽ(T̃) ≥ K − (T̃ − t_{0})^{/1−q)/(2(1+q))}.

By the continuity of the quenching time shown above we can choose T̃ − t_{0} small enough to make this quantity strictly positive. Therefore Ṽ does not quench. □

We now establish a lower bound for the time derivatives. These estimates will be used in the next section to establish an upper bound for the quenching rates in the case of non-simultaneous quenching.

Lemma 2.2 There exists a constant C > 0 such that,

U′(t) ≥ −CV^{−p−1}(t)U^{−q}(t),  V′(t) ≥ −CU^{−q−1}(t)V^{−p}(t).  (2.4)

Proof. We adapt an idea from [3]. Let J = u_{x} − ϕ(x)v^{−p}, L = v_{x} − ϕ(x)u^{−q}, where ϕ : [0, 1] → [0, 1] is a nonnegative, nonincreasing, convex, C^{2} function such that ϕ(0) = 1, ϕ(1) = 0, and ϕ(x) ≤ u′_{0}(x)v_{0}^{p}(x), ϕ(x) ≤ v′_{0}(x)u_{0}^{q}(x) for x ∈ [0, 1]. This implies that J and L are nonnegative at t = 0. Differentiating J we get

J_{t} − J_{xx} = ϕ′′v^{−p} − 2pϕ′v^{−p−1}v_{x} + p(p + 1)ϕv^{−p−2}(v_{x})^{2} ≥ 0,

and the same for L, so that both J and L are supersolutions of the heat equation. In addition, they vanish at the border, x = 0 and x = 1. Therefore J(x, t), L(x, t) ≥ 0 for every x ∈ [0, 1], t ∈ [0, T). In particular this implies J_{x}(0, t) ≥ 0 and L_{x}(0, t) ≥ 0, i.e.

u_{t}(0, t) = u_{xx}(0, t) ≥ ϕ′(0)v^{−p}(0, t) − pv^{−p−1}v_{x}(0, t) ≥ −Cv^{−p−1}u^{−q}(0, t),

and the analogous estimate for v. □

As a byproduct we obtain the quenching set.

Lemma 2.3 The only quenching point is the origin x = 0.

Proof. Since J(x, t) ≥ 0, we have u_{x}(x, t) ≥ ϕ(x)v^{−p}(x, t) ≥ N^{−p}/2 for every 0 ≤ x ≤ x_{0} such that ϕ(x_{0}) = 1/2. Thus u(x, t) ≥ u(0, t) + Cx. The same happens for v. □

3 Non-simultaneous quenching

This section is devoted to the characterization of non-simultaneous quenching. We first finish the proof of Theorem 1.3. The upper estimate of the non-simultaneous rate is then used to prove Theorem 1.1.

Proof of Theorem 1.3. Lemma 2.1 gives the lower bound of the non-simultaneous rate. The upper bound follows easily by integrating the first estimate in (2.4) using that V ≥ c > 0:

U′(t) ≥ −CU^{−q}(t) ⇒ U^{q+1}(t) ≤ C(T − t).

As to the behaviour of the final profile (t = T) as x ∼ 0, we have the lower estimate given by Lemma 2.3. The upper estimate follows directly from the fact that u is concave; therefore u_{x}(x, t) ≤ u_{x}(0, t) = v^{−p}(0, t) ≤ C. □

Proof of Theorem 1.1. We use the representation formula obtained from the heat kernel. Let Γ(x, t) = (4πt)^{−1/2}e^{−x^{2}/4t} be the fundamental solution of the heat equation in ℝ. For x ∈ (0, 1) and t ∈ (0, T), we have

\begin{aligned}v(x, t) & = ∫_{0}^{1} Γ(x − y, t)v_{0}(y) dy − ∫_{0}^{t} v(0, s)\frac{∂Γ}{∂x}(x, t − s) ds \\ & + ∫_{0}^{t} v(1, s)\frac{∂Γ}{∂x}(x − 1, t − s) ds − ∫_{0}^{t} u^{−q}(0, s)Γ(x, t − s) ds. \end{aligned}

Taking limits as x → 0, and using the jump relation (see for instance [4]), we get

\begin{aligned}\frac{1}{2}V (t) & = ∫_{0}^{1} Γ(y, t)v_{0}(y) dy + ∫_{0}^{t} v(1, s)\frac{∂Γ}{∂x}(−1, t − s) ds − ∫_{0}^{t} U^{−q}(s)Γ(0, t − s) ds \\ & = I_{1} + I_{2} + I_{3}. \end{aligned}

The first two integrals are easily bounded by a constant times N. Introducing the upper estimate of the quenching rate in I_{3} we get

V (t) ≤ C_{1} − C_{2} ∫_{0}^{T} s^{− \frac{q}{q+1} − \frac{1}{2}} ds.

We conclude by observing that this integral diverges if q ≥ 1, leading to a contradiction with the fact that v is bounded from below.

Now we use again the above representation formula to show that there exist constants C_{1}, C_{2} > 0 such that

V (t) ≥ C_{1} − C_{2} ∫_{0}^{T} (T − s)^{− \frac{q}{q+1} − \frac{1}{2}} ds = C_{1} − C_{2}T^{\frac{1−q}{2(1+q)}} .

The integral converges since q < 1. On the other hand,

T ≤ min{M_{v}(0)u_{0}^{q}(0), M_{u}(0)v_{0}^{p}(0)},  (3.5)

where M_{u}(0) and M_{v}(0) are the initial masses of the variables u and v. Thus T can be made as small as we please, just taking u_{0}(0) small enough. We conclude that V (t) ≥ C_{2}/2 for every 0 < t < T. □

If 0 < p, q < 1, we have just seen that two situations can appear: u quenches while v does not, or the other way round. We now show that the third possibility, simultaneous quenching, in fact occurs for some initial data. We conjecture that this behaviour is exceptional in this range of parameters, in contrast with what happens for p, q ≥ 1.

Proof of Theorem 1.2. Fix (u_{0}, v_{0}). For any λ > 0, consider problem (1.1)–(1.4) with initial data (λu_{0}, v_{0}), whose solution we denote by (u_{λ}, v_{λ}). Arguing as in the proof of Theorem 1.3, we have that

\begin{matrix}V_{λ}(t) ≥ & C_{1} − C_{2} ∫_{0}^{T_{λ}}(T_{λ} − s)^{− \frac{q}{q+1} − \frac{1}{2}} ds = C_{1} − C_{2}Tλ^{\frac{1−q}{2(1+q)}} , \\ U_{λ}(t) ≥ & C_{1}λ − C_{2} ∫_{0}^{T_{λ}}(T_{λ} − s)^{− \frac{p}{p+1} − \frac{1}{2}} ds = C_{1}λ − C_{2}Tλ^{\frac{1−p}{2(1+p)}} , \end{matrix}

where the quenching time satisfies T_{λ} ≤ C min{λ^{q}, λ}, see (3.5). Hence, if λ is small we have V_{λ}(t) ≥ C_{1} − C_{2}λ^{\frac{1−q}{2(1+q)}} > 0 and thus v_{λ} does not quench. On the other hand, U_{λ}(t) ≥ C_{1}λ − C_{2}λ^{\frac{q(1−p)}{2(1+p)}} > 0 if λ is large, and therefore u_{λ} does not quench.

We define the sets A^{+} = {λ > 0 : U_{λ}(T_{λ}) > 0} and A^{−} = {λ > 0 : V_{λ}(T_{λ}) > 0}. Theorem 2.2 shows that A^{+} and A^{−} are both open, so there exists a closed subset A ⊂ ℝ^{+} such that if λ ∈ A then quenching is simultaneous. □

Finally, we prove that there exists p_{0} ≥ 1 such that if q < 1 and p ≥ p_{0}, then quenching is always non-simultaneous. We conjecture that the result is true for p_{0} = 1.

Lemma 3.1 If q < 1 and p ≥ p_{0} = (1 + q)/(1 − q) then quenching is always nonsimultaneous.

Proof. We argue by contradiction. Assume that quenching is simultaneous and q < 1 < p. Using again the representation formula for v, for 0 < x < 1 and 0 < t < z < T, we have

\begin{aligned}v(x, z) & = ∫_{0}^{1} Γ(x − y, z − t)v(y, t) dy − ∫_{t}^{z} v(0, s)\frac{∂Γ}{∂x}(x, z − s) ds \\ & + ∫_{t}^{z} v(1, s)\frac{∂Γ}{∂x}(x − 1, z − s) ds − ∫_{t}^{z} u^{−q}(0, s)Γ(x, z − s) ds, . \end{aligned}

Taking limits as x → 0, z → T, we get

0 = ∫_{0}^{1} Γ(y, T − t)v(y, t) dy + ∫_{t}^{T} v(1, s)\frac{∂Γ}{∂x}(−1, T − s) ds − ∫_{t}^{T} U^{−q}(s)Γ(0, T − s) ds.

Therefore,

V (t) ≤ C ∫_{t}^{T} U^{−q}(s)(T − s)^{−1/2} ds.

On the other hand, the lower bound for U(t) obtained in Corollary 2.1 gives us

U(t) ≥ C(T − t)^{1/(q+1)} ⇒ V (t) ≤ C(T − t)^{\frac{1−q}{2(1+q)}} .  (3.6)

Now, we introduce this upper estimate in the representation formula for u, to obtain

0 ≤ ∫_{0}^{1} Γ(y, T)u(y, 0) dy + ∫_{0}^{T} u(1, s)\frac{∂Γ}{∂x}(−1, T − s) ds − C ∫_{0}^{T} (T − s)^{− \frac{1}{2} − \frac{p(1−q)}{2(1+q)}} ds.

The two first integrals are bounded, while the third one diverges if p ≥ (1 + q)/(1 − q). We arrive to a contradiction. □

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