Non-simultaneous blow-up for a quasilinear parabolic system with reaction

at the boundary

Cristina Brändle & 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 a system of two porous medium type equations in a bounded interval, coupled at the boundary in a nonlinear way. Under certain conditions, one of its components becomes unbounded in finite time while the other remains bounded, a situation that is known in the literature as non-simultaneous blow-up. We characterize completely, in the case of nondecreasing in time solutions, the set of parameters appearing in the system for which non-simultaneous blow-up indeed occurs. Moreover, we obtain the blow-up rate and the blow-up set for the component which blows up. We also prove that in the range of exponents where each of the components may blow up on its own there are special initial data such that blow-up is simultaneous. Finally, we give conditions on the exponents which lead to non-simultaneous blow-up for every initial data.

∗e-mail: cristina.brandle@uam.es, fernando.quiros@uam.es †e-mail: jrossi@dm.uba.ar AMS Subject Classification: 35B33, 35K65, 35K55.

Keywords and phrases: blow-up, parabolic systems, nonlinear diffusion, nonlinear boundary conditions.

1 Introduction and main results

We devote our attention to the formation of singularities in finite time of solutions (u, v) of a parabolic system of two porous medium type equations,

\{ \begin{matrix}u_{t} = (u^{m})_{xx}, \\ v_{t} = (v^{n})_{xx}, \end{matrix} (x, t) ∈ D_{T} = (0, L) × (0, T),  (1.1)

with a nonlinear coupling at one of the ends of the interval

\{ \begin{matrix}−(u^{m})_{x}(0, t) = u^{p_{11}}v^{p_{12}}(0, t), \\ −(v^{n})_{x}(0, t) = u^{p_{21}}v^{p_{22}}(0, t), \end{matrix} t ∈ (0, T),  (1.2)

and zero flux at the other end,

\{ \begin{matrix}(u^{m})_{x}(L, t) = 0, \\ (v^{n})_{x}(L, t) = 0, \end{matrix} t ∈ (0, T).  (1.3)

The initial data

\{ \begin{matrix}u(x, 0) = u_{0}(x), \\ v(x, 0) = v_{0}(x), \end{matrix} x ∈ (0, L),  (1.4)

are assumed to be continuous and bounded. We consider all possible parameters satisfying m, n > 0, p_{ij} ≥ 0. In this range the diffusivities may become degenerate or singular at level zero. Moreover, the reaction terms may not be Lipschitz, leading to non-uniqueness phenomena, [4]. To avoid the technicalities to which these difficulties may lead, we will assume that u_{0}, v_{0} ≥ δ > 0. Since we are interested in the behaviour of the system for large values of the solutions, this is not a significant restriction. We will also assume that the initial data are compatible with the boundary conditions, so that solutions may be (and will be) understood in a classical sense.

If we extend solutions to (1.1)–(1.4) to the interval (0, 2L) by symmetry, we get a solution to the same problem with the condition at x = L, (1.3), substituted by a condition at x = 2L,

\{ \begin{matrix}(u^{m})_{x}(2L, t) = u^{p_{11}}v^{p_{12}}(2L, t), \\ (v^{n})_{x}(2L, t) = u^{p_{21}}v^{p_{22}}(2L, t), \end{matrix} t ∈ (0, T).

Conversely, symmetric solutions to this latter problem are solutions to our original problem (1.1)–(1.4). This allows to translate many results of one of the problems to the other.

The time T denotes the maximal existence time for the solution (u, v). If it is infinite we say that the solution is global. If it is finite we say that the solution blows up.

In the last years there has been an increasing interest in the study of blow-up due to reaction at the boundary, both for scalar problems and for systems, see for example the surveys [2], [7] and the references therein.

It is known that nontrivial solutions of (1.1)–(1.4) blow up if and only if the exponents p_{ij} verify any of the following conditions,

p_{11} > min \{1, \frac{m+1}{2}\} ,
p_{22} > min \{1, \frac{n+1}{2}\} ,
p_{12}p_{21} > (min \{1, \frac{m+1}{2}\} − p_{11}) (min \{1, \frac{n+1}{2}\} − p_{22}) ,

see [18], [19]. In this case we have

lim sup_{t↗T} {∥u(·, t)∥_{∞} + ∥v(·, t)∥_{∞}} = ∞.

However, a priori there is no reason why both functions, u and v, should go to infinity simultaneously at time T. Indeed, as we will show, for certain choices of the parameters p_{ij} there are initial data for which one of the components of the system remains bounded while the other blows up. This phenomenon is known in the literature as non-simultaneous blow-up. The possibility of non-simultaneous blow-up in nonlinear parabolic systems was first mentioned in [16], and has been studied more thoroughly later in [14],

[15] and [17]. For problem (1.1)–(1.4) it was analyzed in [12] in the particular case m = n = 1.

The aim of this paper is to characterize the range of parameters for which non-simultaneous blow-up occurs under an extra condition on the monotonicity of the solution:

(H) (u, v) verifies u_{t}, v_{t} ≥ 0.

This hypothesis is satisfied, for example, if (u_{0}^{m})′′ ≥ 0 and (v_{0}^{n})′′ ≥ 0. In the sequel we assume, without further mention, that (H) holds.

Our first result gives a necessary and sufficient condition for the existence of non-simultaneous blow-up.

Theorem 1.1 (i) Let

2p_{21} < max {p_{11} − 1, 2p_{11} − (m + 1)} .  (1.5)

Then, given v_{0}, there exist initial data u_{0} such that u blows up while v remains bounded. (ii) If u blows up and v remains bounded, then (1.5) holds.

Since p_{21} ≥ 0, in order to have non-simultaneous blow-up we need in particular that p_{11} > min{1, (m + 1)/2}, see [8]. Thus u can blow up by itself, without the help of v. Condition (1.5) says that p_{21} (which measures the influence of u in the equation for v) is small compared with p_{11} (which measures the capacity of u to blow up by itself); hence, when u blows up, it does not necessarily carry v along with it.

By interchanging the roles of u and v, we get that the condition in order to have solutions such that v blows up while u remains bounded reads

2p_{12} < max {p_{22} − 1, 2p_{22} − (n + 1)} .  (1.6)

It is remarkable that the non-simultaneous blow-up condition (1.5) depends strongly on the diffusivity through the parameter m. Indeed, a dramatic change occurs across the value p_{11} = m. The main reason behind this fact is that in the case of non-simultaneous blow-up the blow-up rate of u, that is, the speed at which it goes to infinity at time T, also changes across this value.

Theorem 1.2 If u blows up at time T and v remains bounded, then there exist positive constants, C_{1}, C_{2}, such that for 0 < t < T

C_{1} ≤ (T −t)^{γ}u(0, t) ≤ C_{2},  γ = 1/ max {p_{11} − 1, 2p_{11} − (m + 1)} . (1.7)

In [15] the authors study conditions for non-simultaneous blow-up for the same pair of equations defined in ℝ_{+} = (0, ∞) instead of an interval, with the same boundary conditions at the origin. In their case the nonsimultaneous blow-up condition just reads 2p_{21} < 2p_{11} − (m + 1), which is equal to the one we obtain when p_{11} > m. The reason for this coincidence is the following: in this range the blow-up set of u,

B(u) = {x : there exist x_{n} → x, t_{n} ↗ T with u(x_{n}, t_{n}) → ∞},

when blow-up is non-simultaneous, contains only the point x = 0. Hence the singularity does not see what happens away from the origin. Therefore, the boundary condition at x = L does not contribute to the blow-up behaviour of problem (1.1)–(1.4), when p_{11} > m.

Theorem 1.3 If u blows up and v remains bounded, then the blow-up set of u is given by

B(u) = \{ \begin{matrix}\{0\} & if p_{11} > m, \\ [0, L̂] ⊆ [0, L] & if p_{11} = m, \\ [0, L] & if p_{11} < m. \end{matrix}

The first case, where blow-up is localized at just one point, is known as single point blow-up. When the solution blows up at all the points where the problem is defined, we say that blow-up is global. If the blow-up set is an interval strictly contained in [0, L], blow-up is regional.

When non-simultaneous blow-up is possible, the set of initial data for which it actually occurs is expected to be rather big. We will show that it is an open set in the L^{∞}-topology. We conjecture that it should be a dense set, at least for some range of exponents.

Theorem 1.4 Let (1.5) hold. The set of initial data such that u blows up and v remains bounded is open in the L^{∞}-topology.

However, we do not exclude the possibility of exceptional solutions with simultaneous blow-up. As an example, if (1.5) and (1.6) both hold at the same time, that is, if each of the components may blow up on its own, there exist initial data for which simultaneous blow-up indeed occurs.

Theorem 1.5 Let (1.5) and (1.6) both hold at the same time. Then, there is a solution (u, v) that blows up simultaneously.

Nevertheless, sometimes blow-up is always non-simultaneous. This is the case if v cannot blow up without the help of u.

Theorem 1.6 Let (1.5) hold. If p_{22} ≤ min{1, (n + 1)/2}, then u blows up and v remains bounded for every initial data.

We think that blow-up should be also always non-simultaneous when p_{22} > min{1, (n + 1)/2} and condition (1.5) holds, but (1.6) does not. This range of exponents allows non-simultaneous blow-up and also the possibility of both components to blow-up without the help of the other one. The fact that (1.6) does not hold implies that if v blows up then u has to blow up as well. We show two partial results that may be a guide for the general case.

Theorem 1.7 Let (1.5) hold. If 2p_{12} ≥ max{p_{22} − 1, 2p_{22} − (n + 1)} and p_{22} > min{1, (n + 1)/2}, with p_{11} ≥ m and p_{22} ≥ n, then u blows up and v remains bounded for every initial data.

When m, n ≤ 1, the hypotheses p_{11} ≥ m and p_{22} ≥ n in Theorem 1.7 always hold.

Theorem 1.8 Let (1.5) hold. If 2p_{12} ≥ max{p_{22} − 1, 2p_{22} − (n + 1)} and p_{22} > min{1, (n + 1)/2}, with p_{21} = 0 and m ≥ 1, then u blows up and v remains bounded for every initial data.

Consider again the problem studied in [15]. As we have seen, it behaves similarly as (1.1)–(1.4) when p_{11} > m. As another example of this fact, the same proof of Theorem 1.7 can be used to show a new result for the problem posed in ℝ_{+}.

Theorem 1.9 Let 2p_{21} < 2p_{11} − (m + 1). If p_{22} > min{1, (n + 1)/2} and 2p_{12} ≥ max{p_{22} − 1, 2p_{22} − (n + 1)}, then solutions of (1.1)–(1.2) and (1.4) in ℝ_{+} verify that u blows up and v remains bounded for every initial data.

Observe that the restrictions p_{11} ≥ m and p_{22} ≥ n follow directly from the conditions 2p_{21} < 2p_{11} − (m + 1) and p_{22} > min{1, (n + 1)/2}.

Organization of the paper. Section 2 deals with the blow-up rate in the case of non-simultaneous blow-up. Section 3 is devoted to the study of blowup conditions for an auxiliary problem which will play an important role in the proof of our theorems. The non-simultaneous blow-up set is described in Section 4. In Section 5 we use the results of the previous sections to characterize the range of parameters for which non-simultaneous blow-up indeed occurs. Finally, in Section 6 we obtain results about the sets of initial data for which there is simultaneous or non-simultaneous blow-up.

Throughout the paper C and c denote positive constants that may change from one line to another, or even in the same line.

2 Non-simultaneous blow-up rates

A key point in the proof of Theorem 1.1 is the blow-up rate for u when blow-up is non-simultaneous. In order to determine this rate, we consider v^{p_{12}}(0, t) as a frozen coefficient. Hence we regard u as a solution of

\{ \begin{matrix}u_{t} = (u^{m})_{xx}, & (x, t) ∈ D_{T} , \\ −(u^{m})_{x}(0, t) = u^{p_{11}}(0, t)h(t), & t ∈ (0, T), \\ (u^{m})_{x}(L, t) = 0, & t ∈ (0, T), \\ u(x, 0) = u_{0}(x), & x ∈ (0, L), \end{matrix}  (2.1)

where h is a continuous, bounded, nondecreasing and strictly positive function. When h is a constant the problem has been already studied in [8]. There, it is shown that solutions blow up if and only if

p_{11} > min{1, (m + 1)/2}.

Hence, by comparison, solutions of (2.1) blow up if and only if the same restriction on p_{11} holds.

As a consequence of (H), (u^{m})_{xx} ≥ 0 and hence

min_{x∈[0,L]}(u^{m})_{x}(x, t) = (u^{m})_{x}(0, t).  (2.2)

Moreover, u_{x} ≤ 0 and therefore

max_{x∈[0,L]} u(x, t) = u(0, t).  (2.3)

Hence, if u blows up at a finite time T, {0} belongs to the blow-up set of u.

First, we prove that when u blows up and p_{11} < m then B(u) = [0, L]. Moreover, u blows up with the same speed at all points.

Lemma 2.1 Let p_{11} > min{1, (m + 1)/2} and let u be a solution to (2.1). If p_{11} < m, then for every c < 1 there exists a value t_{0} = t_{0}(c) < T such that

c u(0, t) ≤ u(x, t) ≤ u(0, t),  x ∈ [0, L], t_{0} ≤ t ≤ T.  (2.4)

In particular B(u) = [0, L].

Proof. We follow ideas from [6]. The second inequality is just (2.3). To prove the first one we apply (2.2) and the mean value theorem to get that there is a value ξ ∈ (0, x) such that

\begin{aligned}u^{m}(x, t) − u^{m}(0, t) & = x(u^{m})_{x}(ξ, t) ≥ x(u^{m})_{x}(0, t) \\ & = −xu^{p_{11}}(0, t)h(t) ≥ −CLu^{p_{11}}(0, t). \end{aligned}

Thus, u^{m}(x, t) ≥ u^{m}(0, t) (1 − CLu^{p_{11}−m}(0, t)). Since p_{11}−m < 0 and u(0, t) blows up with u_{t}(0, t) ≥ 0, we obtain that 1 − CLu^{p_{11}−m}(0, t) ≥ c^{m} for t close to T. □

This lemma actually says more: the profile at the blow-up time is flat.

Corollary 2.2 Let p_{11} > min{1, (m+1)/2} and let u be a solution to (2.1). If p_{11} < m, then

lim_{t↗T} u(x, t)/u(0, t) = 1, x ∈ [0, L].

Remark. Conditions p_{11} > min{1, (m + 1)/2} and p_{11} < m are compatible

only if m > 1. In this case, min{1, (m + 1)/2} = 1.

We arrange the proof of Theorem 1.2 in two lemmas, according to the relation between p_{11} and m. The first one uses Lemma 2.1 to obtain the blow-up rate for u, when p_{11} < m.

Lemma 2.3 Let p_{11} > min{1, (m + 1)/2} and let u be a solution of (2.1). If p_{11} < m, then (1.7) holds for 0 < t < T.

Proof. The mass F (t) = ∫_{0}^{L} u(x, t) dx behaves as u(0, t). Indeed, from (2.3) we immediately obtain F (t) ≤ Lu(0, t). On the other hand, given c < 1, we integrate (2.4) to obtain cL u(0, t) ≤ F (t) for t_{0}(c) ≤ t < T. To extend this inequality to (0, t_{0}(c)) consider a positive constant C_{1}, small enough, and such that

F (t) = ∫_{0}^{L} u(x, t) dx ≥ ∫_{0}^{L} u(x, 0) dx ≥ \frac{C_{1}}{cL} ∫_{0}^{L} u(x, t_{0}(c)) dx

Using (2.4) we conclude that

\frac{C_{1}}{cL} ∫_{0}^{L} u(x, t_{0}(c)) dx ≥ C_{1}u(0, t).

Therefore

C_{1}u(0, t) ≤ F (t) ≤ C_{2}u(0, t),  0 < t < T.  (2.5)

From (2.5) we obtain that it is enough to prove an estimate analogous to (1.7) for F (t). For all t < T the mass satisfies the ordinary differential equation

\begin{matrix}F′(t) = & ∫_{0}^{L} u_{t}(x, t) dx = ∫_{0}^{L}(u^{m})_{xx}(x, t) dx \\ = & (u^{m})_{x}(L, t) − (u^{m})_{x}(0, t) = h(t)u^{p_{11}}(0, t). \end{matrix}

Since h is bounded from below, we get F′(t) ≥ Cu^{p_{11}}(0, t) ≥ C(F (t))^{p_{11}}. Integrating in (t, T) we get the upper estimate F (t) ≤ C(T − t)^{−1/(p_{11}−1)}. The lower estimate is obtained in a similar way. □

Lemma 2.4 Let p_{11} > min{1, (m + 1)/2} and let u be a solution of (2.1). If p_{11} ≥ m, then (1.7) holds for 0 < t < T.

Proof. We follow a technique from [10]. Define M(t) = max u(·, t) and

ϕ_{M}(y, s) = \frac{1}{M(t)}u(ay, bs + t),

where a = M^{m−p_{11}}, b = M^{m+1−2p_{11}}. As t ↗ T, M ↗ ∞. Under the mentioned hypotheses, 2p_{11} > m + 1; hence b ↘ 0. Moreover, if p_{11} > m then a ↘ 0; on the other hand, a = 1 if p_{11} = m.

We claim that there exist two positive constants c and C such that

c ≤ (ϕ_{M})_{s}(0, 0) ≤ C.  (2.6)

If we rewrite these inequalities in terms of M(t), we get

c ≤ M^{m−2p_{11}}(t)M′(t) ≤ C.

Integrating in (t, T) and taking into account that M(t) = u(0, t), we obtain the desired result.

The proof of the claim (2.6) relies strongly on {ϕ_{M}} being a family of uniformly bounded solutions of equations of porous medium type,

\{ \begin{matrix}(ϕ_{M})_{s} = (ϕ_{M}^{m})_{yy}, & (y, s) ∈ (0, \frac{L}{a}) × (−\frac{t}{b}, 0), \\ −(ϕ_{M}^{m})_{y}(0, s) = ϕ_{M}^{p_{11}}(0, s)h(bs + t), & s ∈ (−\frac{t}{b}, 0), \\ (ϕ_{M}^{m})_{y}(\frac{L}{a}, s) = 0, & s ∈ (−\frac{t}{b}, 0). \end{matrix}

The uniform bound, 0 ≤ ϕ_{M} ≤ 1, is a a consequence of u_{t} ≥ 0. Uniformly bounded solutions to porous medium type equations turn out to be equicontinuous in compact subsets of their common domain, cf. [5], [20]. Observe that for any S < 0, the domain of ϕ_{M} contains the compact set [0, L] × [−S, 0], if M is large enough.

Given {ϕ_{M_{j}}}, there is a continuous function Φ and a subsequence, which we denote again by ϕ_{M_{j}}, such that ϕ_{M_{j}} → Φ as M_{j} → ∞, uniformly on [0, L] × [−S, 0]. Moreover, Φ(0, 0) = 1. Therefore, there exists a neighbourhood of (0, 0), U, such that Φ > 1/2 in U. Since we have uniform convergence in \overline{U} (we can assume that \overline{U} is compact), for j large enough we have that 1/4 ≤ ϕ_{M_{j}} ≤ 1 in \overline{U}. Thus, the functions ϕ_{M_{j}} are solutions of uniformly parabolic equations in \overline{U}. Since they are uniformly bounded we get, using well known Schauder estimates, [11],

∥ϕ_{M_{j}}∥_{C^{2+α,1+α/2}} ≤ C in \overline{U}.  (2.7)

The upper bound in (2.6) follows immediately. To obtain the lower estimate assume, for contradiction, that there exists a sequence {ϕ_{M_{j}}} such that (ϕ_{M_{j}})_{s}(0, 0) → 0. Estimates (2.7) imply that w = Φ_{s} satisfies w_{s} = (mΦ^{m−1}w)_{yy} in the positivity set of Φ. Moreover, −(mΦ^{m−1}w)_{y}(0, s) = p_{11}Φ^{p_{11}−1}w(0, s)h(T). On the other hand, (ϕ_{M_{j}})_{s}(0, 0) → Φ_{s}(0, 0), which implies Φ_{s}(0, 0) = 0. Since u_{t} ≥ 0, w = Φ_{s} ≥ 0; hence w has a minimum at (0, 0) and, by Hopf’s Lemma, w ≡ 0. We get that Φ is a nonnegative stationary solution of the porous medium equation, Φ^{m}(y) = c_{1}y + c_{2}. Besides, −(Φ^{m})′(0) ≥ Φ^{p_{11}}(0)h(T) = h(T); thus Φ^{m}(y) ≤ 1 − h(T)y. Hence, if p_{11} > m, Φ becomes negative, a contradiction. If p_{11} = m, either Φ becomes negative or it does not verify (Φ^{m})′(L) = 0, again a contradiction. □

3 Blow-up results for an auxiliary scalar problem

If u blows up while v remains bounded, then (1.7) holds. Thus v is a supersolution to

\{ \begin{matrix}v_{t} = (v^{n})_{xx}, & (x, t) ∈ D_{T_{0}}, \\ −(v^{n})_{x}(0, t) = cv^{p_{22}}(0, t)(T − t)^{−γp_{21}}, & t ∈ (0, T_{0}), \\ (v^{n})_{x}(L, t) = 0, & t ∈ (0, T_{0}), \\ v(x, 0) = v_{0}(x), & x ∈ (0, L), \end{matrix}  (3.1)

with c > 0, and T_{0} the maximal existence time for v. The key point is that, in order to have v bounded up to T, γp_{21} has to be smaller than 1/2.

Theorem 3.1 (i) If γp_{21} ≥ 1/2, then every nonnegative, nontrivial solution of (3.1) blows up at a finite time T_{0} ≤ T. (ii) If γp_{21} < 1/2, then given ε > 0 and v_{0} there exists T small enough such that the solution of (3.1) verifies

sup_{0<t<T} ∥v(·, t)∥_{∞} ≤ ∥v_{0}∥_{∞} + ε.

In particular, v is bounded.

If v remains bounded up to time t = T, we expect it to behave as a solution of

\{ \begin{matrix}v_{t} = (v^{n})_{xx}, & (x, t) ∈ D_{T} , \\ −(v^{n})_{x}(0, t) = c(T − t)^{−γp_{21}}, & t ∈ (0, T), \\ (v^{n})_{x}(L, t) = 0, & t ∈ (0, T), \\ v(x, 0) = v_{0}(x), & x ∈ (0, L). \end{matrix}  (3.2)

A result analogous to Theorem 3.1 holds for problem (3.2).

Theorem 3.2 (i) If γp_{21} ≥ 1/2, then every nonnegative, nontrivial solution of (3.2) blows up at time T. (ii) If γp_{21} < 1/2, then given ε > 0 and v_{0} there exists T small enough such that the solution of (3.2) verifies

sup_{0<t<T} ∥v(·, t)∥_{∞} ≤ ∥v_{0}∥_{∞} + ε.  (3.3)

In particular, v is bounded.

Remark. The same result holds if there is a constant C > 0 in front of v_{t}.

Theorem 3.1 is a consequence of this latter result.

Proof of Theorem 3.1. (i) As we are considering strictly positive initial data, we have that v is a supersolution of (3.2) with γp_{21} ≥ 1/2. Since solutions of this latter problem blow up, v also blows up. (ii) Given ε > 0 and v_{0}, let z be a solution of (3.2) with initial datum z(x, 0) = v_{0}(x) and c big enough. If we choose T small, z has the property (3.3). Hence it is a supersolution of (3.1). We conclude that v must satisfy (3.3). □

We come now the proof of Theorem 3.2. We begin by showing that problem (3.2) has a contraction property in L^{∞}.

Lemma 3.3 Let v_{1} and v_{2} be two solutions of (3.2), with ordered initial data, v_{1}(x, 0) ≥ v_{2}(x, 0). Then

∥v_{1}(·, t_{2}) − v_{2}(·, t_{2})∥_{∞} ≤ ∥v_{1}(·, t_{1}) − v_{2}(·, t_{1})∥_{∞},  0 ≤ t_{1} < t_{2} < T. (3.4)

Proof. If v_{1}(x, 0) ≥ v_{2}(x, 0), comparison shows that v_{1} ≥ v_{2}. Defining

Φ(x, t) = \{ \begin{matrix}\frac{v_{1}^{n}(x, t) − v_{2}^{n}(x, t)}{v_{1}(x, t) − v_{2}(x, t)} & if v_{1}(x, t) ≠ v_{2}(x, t), \\ nv_{1}^{n−1}(x, t) & if v_{1}(x, t) = v_{2}(x, t), \end{matrix}

we get that w = v_{1} − v_{2} ≥ 0 is a solution of the parabolic equation

\{ \begin{matrix}w_{t} = (Φ(x, t)w)_{xx}, & x ∈ (0, L), t ∈ (t_{1}, t_{2}), \\ −(Φ(x, t)w)_{x}(0, t) = 0, & t ∈ (t_{1}, t_{2}), \\ (Φ(x, t)w)_{x}(L, t) = 0, & t ∈ (t_{1}, t_{2}), \\ w(x, t_{1}) = v_{1}(x, t_{1}) − v_{2}(x, t_{1}), & x ∈ (0, L). \end{matrix}

Hence, the maximum of w in [t_{1}, t_{2}] × [0, L] lies on the parabolic boundary. If the maximum is achieved at t = t_{1}, the result follows. If it is attained at x = 0 or x = L, Hopf’s Lemma (observe that the equation is uniformly parabolic in a neighbourhood of the maximum point, since v_{1} > 0 there) implies that w ≡ 0, and the monotonicity of ∥w(·, t)∥_{∞} becomes trivial. □

This contraction property implies that in order to prove Theorem 3.2 (and hence Theorem 3.1), it is enough to consider any initial data.

Corollary 3.4 Either every solution to (3.2) is bounded up to time T or every solution blows up at time T.

Proof. Let v_{1} be any solution of (3.2) and v_{2} be the solution with initial data v_{2}(x, 0) = 0. Then, by comparison, v_{1}(x, t) ≥ v_{2}(x, t). Since the difference ∥v_{1}(·, t) − v_{2}(·, t)∥_{∞} does not increase with time, see (3.4), any solution v_{1} blows up at time t = T if and only if v_{2} does so. □

Lemma 3.5 Theorem 3.2 holds true in the linear case, n = 1.

Proof. (i) We use ideas from [15]. Thanks to the previous lemma, we can restrict to initial data v_{0} = 0. Let G be Green’s function for the problem. We have, G(x, y, t, s) = Γ(x−y, t−s)+H(x, y, t, s), where Γ is the fundamental solution for the heat equation,

Γ(x, t) = \frac{1}{(4πt)^{1/2}} exp (−\frac{x^{2}}{4t} ) ,

and H(x, y, t, s) is the solution of

\{ \begin{matrix}H_{s}(x, y, t, s) + H_{yy}(x, y, t, s) = 0, & (y, s) ∈ D_{T} , \\ H_{y}(x, 0, t, s) = Γ_{x}(x, t − s), & s ∈ (0, T), \\ H_{y}(x, L, t, s) = Γ_{x}(x − L, t − s), & s ∈ (0, T), \\ H(x, y, t, t) = 0, & y ∈ (0, L). \end{matrix}

Since |H(0, 0, t, s)| is bounded, we get from the representation formula using Green’s function, that

v(0, t) ≥ C ∫_{0}^{t} \frac{ds}{(t − s)^{1/2}(T − s)^{γp_{21}}} − C.  (3.5)

Since γp_{21} ≥ 1/2, v blows up at time T. (ii) Consider any initial data v_{0}. Then v can be written as v = v_{1} + v_{2}, with v_{1} a solution of the heat equation with zero flux and v_{1}(x, 0) = v_{0}(x) and v_{2} a solution of (3.2) (with n = 1) with initial datum v_{0}(x) = 0. From the general theory for the heat equation we have that v_{1}(x, t) ≤ v_{0}(x). On the other hand, from (3.5) we have that v_{2}(x, t) ≤ CT^{1/2−γp_{21}}. Hence v(x, t) ≤ v_{0}(x) + CT^{1/2−γp_{21}}, from where (3.3) follows, if T is small. □

Proof of Theorem 3.2. In order to extend the proof to a general n we follow again [15]. The idea is to use that

0 < c ≤ nv^{n−1} ≤ C  (3.6)

to obtain supersolutions of the heat equation.

(i) Assume that there is a solution of (3.2), v, bounded up to time T. Let z = v^{n}. We get

\{ \begin{matrix}\frac{1}{n}z^{(1−n)/n}z_{t} = z_{xx}, & (x, t) ∈ D_{T} , \\ −z_{x}(0, t) = c(T − t)^{−γp_{21}}, & t ∈ (0, T), \\ z_{x}(L, t) = 0, & t ∈ (0, T), \\ z(x, 0) = v_{0}^{n}(x), & x ∈ (0, L). \end{matrix}

As either all solutions blow up or none of them does, we can take any initial data without loss of generality. We take z(x, 0) ≥ δ > 0 such that z_{xx}(x, 0) ≥ 0. Thus, z_{t} ≥ 0 and z(x, t) ≥ δ for all t. Hence, condition (3.6) holds, and we have that

\{ \begin{matrix}Cz_{t} ≥ z_{xx}, & (x, t) ∈ D_{T} , \\ −z_{x}(0, t) = c(T − t)^{−γp_{21}}, & t ∈ (0, T), \\ z_{x}(L, t) = 0, & t ∈ (0, T), \\ z(x, 0) ≥ δ, & x ∈ (0, L). \end{matrix}

Therefore, z is a bounded supersolution of a heat equation with a flux at the boundary given by c(T − t)^{−γp_{21}}, a contradiction. (ii) Given v_{0}, let z be a solution of (3.2) with n = 1, z_{0}(x) = v_{0}^{n}(x), and a constant k(z_{0}) = n(z_{0} + ε)^{(n−1)/n} in front of z_{t}. Let v̄ = z^{1/n}, then

\{ \begin{matrix}nv̄^{n−1}v̄_{t} = (v̄^{n})_{xx}, & (x, t) ∈ D_{T} , \\ −(v̄^{n})_{x}(0, t) = c(T − t)^{−γp_{21}}, & t ∈ (0, T), \\ (v̄^{n})_{x}(L, t) = 0, & t ∈ (0, T), \\ v̄(x, 0) = z_{0}^{1/n}(x), & x ∈ (0, L). \end{matrix}

As γp_{21} < 1/2, we can find T small enough such that z(x, t) ≤ ∥z_{0}∥_{∞} + ε. Since z is also bounded away from zero and z_{t} > 0, we have, no matter the value of n > 0, taking δ appropriately,

nv̄^{n−1} ≤ k(z_{0}).

Therefore v̄ is a supersolution of problem (3.2). Hence,

sup_{0<t<T} ∥v(·, t)∥_{∞} ≤ sup_{0<t<T} ∥v̄(·, t)∥_{∞} ≤ (∥v_{0}∥_{∞}^{n} + ε)^{1/n},

from where (3.3) follows. □

Remark. The mass ∫_{0}^{L} v satisfies

∫_{0}^{L} v(x, t) dx − ∫_{0}^{L} v_{0}(x) dx = ∫_{0}^{t} c(T − τ)^{−γp_{21}} dτ.

Hence, a solution of (3.2) remains bounded in the L^{1}-norm up to time T if and only if γp_{21} < 1. This condition differs from the condition for blow-up in the L^{∞}-norm, γp_{21} < 1/2.

4 Non-simultaneous blow-up set

The results of sections 2 and 3 allow us to obtain the blow-up set when blow-up is non-simultaneous.

Proof of Theorem 1.3. If p_{11} ≤ m, since (1.7) holds, u is a supersolution of

\{ \begin{matrix}u_{t} = (u^{m})_{xx}, & (x, t) ∈ D_{T} , \\ u(0, t) = c(T − t)^{−γ}, & t ∈ (0, T), \\ (u^{m})_{x}(L, t) = 0, & t ∈ (0, T), \\ u(x, 0) = u_{0}(x), & x ∈ (0, L). \end{matrix}  (4.1)

Hence if ũ is a subsolution of (4.1, B(ũ) ⊆ B(u). In our case, we take as subsolution, ũ, the solution of problem (4.1) defined for x ∈ ℝ_{+}. The blowup set for such solutions has been described in [3] and [9]. The authors show that if p_{11} = m then B(ũ) = [0, L̃] and if p_{11} < m then B(ũ) = ℝ_{+}.

If p_{11} > m we use that u is a subsolution of (4.1) that blows up. Hence if ũ is a solution of that problem, B(u) ⊆ B(ũ). We are therefore confronted with the study of the blow-up set of ũ. This was studied for m ≥ 1 in [8], where it was proved that B(ũ) = {0}. When m < 1 we can argue as follows: consider a special solution to ũ_{t} = (ũ^{m})_{xx} for (x, t) ∈ (0, 2L)×(0, ∞) with initial datum u_{0} (extended to (0, 2L) by symmetry), and such that ũ(x, t) → ∞ as x → 0, 2L for every t (this kind of solution was constructed in [1]). This solution provides us with a maximal solution for our problem, i.e. ũ(x, t) ≥ u(x, t). As ũ is bounded in the interior of the interval (0, 2L), we conclude that B(ũ) = {0}. □

5 Conditions for non-simultaneous blow-up

We arrive to the proof of Theorem 1.1.

Proof of Theorem. 1.1. (i) Let N = ∥v_{0}∥_{∞}. Since p_{11} > min{1, (m + 1)/2}, we can choose the initial data u_{0} large enough in order to obtain a blowup solution of (1.1)–(1.4) with T small. Let t_{0} be the first time where v touches the height 2N. If such t_{0} does not exist, ∥v(·, t)∥_{∞} ≤ 2N, and the result follows. Hence we assume that t_{0} < T. For any t ∈ [0, t_{0}] we have that u is a solution of (2.1) with δ ≤ h(t) ≤ 2N. Using the nonsimultaneous blow-up rate from above, which is valid up to time t_{0}, we get that v is a subsolution of (3.1) for t ≤ t_{0}. Let v̄ be the solution of (3.1). Since γp_{21} < 1/2, v̄ does not blow up. Moreover, as T is small, v̄(0, t_{0}) < 3N/2. But then, 2N = v(0, t_{0}) ≤ v̄(0, t_{0}) < 3N/2, a contradiction. We conclude that v remains bounded up to time T. (ii) Since v(x, t) ≤ C for all 0 < t < T, u is a subsolution of (2.1). Hence we need p_{11} > min{1, (m + 1)/2} for u to blow up, [8]. Now, if we use the blow-up rate for u from below, and plug it into the equation for v, which is assumed to be bounded, we get that v is a supersolution of (3.1). By Theorem 3.1, we have γp_{21} < 1/2. □

6 Simultaneous / Non-simultaneous blow-up

Our next aim is to study the set of initial data for which blow-up is nonsimultaneous. We show that it is an open set in the L^{∞}-topology.

Proof of Theorem 1.4. Let (u, v) be a solution such that u blows up at time T and v remains bounded up to that time, say v ≤ C. As u blows up at time T, it becomes large at time T − ε. We can find a neighbourhood of (u(x, 0), v(x, 0)) in L^{∞} such that, if (û, v̂) has initial data in such neighbourhood and û_{t} ≥ 0, then û becomes large at time T − ε and v̂ ≤ C + 1 up to T − ε (this follows by continuity with respect to the initial conditions since up to T − ε we are dealing with bounded solutions). The argument in the proof of Theorem 1.1 allows to conclude that û blows up and v̂ remains bounded. □

This result suggests that the set of initial data leading to non-simul-taneous blow-up is large. However, exceptional solutions with simultaneous blow-up may exist. We show that this happens indeed if (1.5) and (1.6) both hold at the same time. Observe that in this range both components of the solution may blow-up by themselves, without the help of the other component.

Proof of Theorem 1.5. Let (u_{0}, v_{0}) be such that the solution of (1.1)–(1.4) blows up. For any λ ∈ (0, 1) we denote by (u_{λ}, v_{λ}) the solution of (1.1)–(1.4) with initial data (u_{0}/λ, v_{0}/(1 − λ)). Define

A_{λ} = {λ ∈ (0, 1) : u_{λ} blows up and v_{λ} remains bounded},
B_{λ} = {λ ∈ (0, 1) : v_{λ} blows up and u_{λ} remains bounded}.

Thanks to the previous results, we know that if λ is small enough u blows up and v remains bounded. Hence, A_{λ} ≠ ∅. Analogously, if λ is close to 1, then λ ∈ B_{λ}. On the other hand, we know from Theorem 1.4 that A_{λ} and B_{λ} are open. A connectedness argument shows that there exist λ_{0} ∈ (0, 1) such that λ_{0} ∉ A_{λ} ∪ B_{λ}. Since (u_{λ_{0}}, v_{λ_{0}}) blows up, we have constructed a solution with simultaneous blow up. □

Let us show now that, under certain conditions on p_{12} and p_{22}, blow-up is always non-simultaneous.

Proof of Theorem 1.6. Under our assumptions u must blow up. Then, there are two possibilities: either u blows up alone, or u and v blow up simultaneously. Let us exclude the second one.

By considering the solution at time T −ε as initial data, we may assume that the blow-up time is as small as pleased. Let ∥v_{0}∥_{∞} = N and let t_{0} be the first time where v(0, t_{0}) = 2N. If such t_{0} does not exist, ∥v(·, t)∥_{∞} ≤ 2N, and v is bounded.

Assume that such t_{0} exists. As v is bounded at least up to t_{0}, using the same argument as in the proof of Theorem 1.1, we conclude that v does not exceed 3N/2, a contradiction. This argument is valid for any N as long as v_{0} ≥ δ > 0. □

Proof of Theorem 1.7. Since p_{11} ≥ m, then 2p_{11} > m + 1 and condition (1.5) becomes 2p_{21} < 2p_{11} − (m + 1). Analogously, 2p_{22} > n + 1. Trivially we have that 2p_{12} ≥ 2p_{22} − (n + 1).

Assume that there is an initial data (u_{0}, v_{0}), such that u and v both blow up at time T. Following [13], define

M(t) = max u(·, t) and N(t) = max v(·, t),

and set, for t < T,

\begin{matrix}φ_{M}(y, s) = \frac{1}{M(t)}u(ay, bs + t), & y > 0, −\frac{t}{b} < s < 0, \\ ψ_{N}(y, s) = \frac{1}{N(t)}v(cy, ds + t), & y > 0, −\frac{t}{d} < s < 0, \end{matrix}

with

a = \frac{M^{m−p_{11}}}{N^{p_{12}}} , b = \frac{M^{−2p_{11}+m+1}}{N^{2p_{12}}} , c = \frac{N^{n−p_{22}}}{M^{p_{21}}} , d = \frac{N^{−2p_{22}+n+1}}{M^{2p_{21}}} .

With the same ideas of the proof of Lemma 2.4 (see also [13]) it is easy to show that

c ≤ (φ_{M})_{s}(0, 0) ≤ C,  c ≤ (ψ_{N})_{s}(0, 0) ≤ C.  (6.1)

Writing (6.1) in terms of M and N, we have

M^{m−2p_{11}}M′ ≥ cN^{2p_{12}}, N^{n−2p_{22}}N′ ≤ CM^{2p_{21}}.

Assume first that 2p_{12} > 2p_{22} −(n+1). After a straightforward computation we obtain

CM(t)^{2p_{21}−2p_{11}+m+1} + C ≥ CN(t)^{2p_{12}−2p_{22}+n+1} − C.

As 2p_{21} < 2p_{11} −(m+1) and 2p_{12} > 2p_{22} −(n+1), we obtain a contradiction with the assumption of simultaneous blow-up.

If 2p_{12} = 2p_{22} − (n + 1), we get

CM(t)^{2p_{21}−2p_{11}+m+1} + C ≥ C ln N(t) − C,

which is again a contradiction. □

Proof of Theorem 1.8. Let us first assume that m = 1. Since p_{21} = 0, we have that v is a solution of

\{ \begin{matrix}v_{t} = (v^{n})_{xx}, & (x, t) ∈ D_{T_{v}}, \\ −(v^{n})_{x}(0, t) = v^{p_{22}}(0, t), & t ∈ (0, T_{v}), \\ (v^{n})_{x}(L, t) = 0, & t ∈ (0, T_{v}), \\ v(x, 0) = v_{0}(x), & x ∈ (0, L). \end{matrix}

As p_{22} > min{1, (n + 1)/2}, v blows up at a time T_{v}. We can apply the results of Section 2 with h(t) = 1 (see also [8]) and get that v verifies the blow-up rate,

c(T_{v} − t)^{−η} ≤ v(0, t) ≤ C(T_{v} − t)^{−η}, η = 1/ max {p_{22} − 1, 2p_{22} − (n + 1)} .

This implies that u is a supersolution of

\{ \begin{matrix}u_{t} = u_{xx}, & (x, t) ∈ D_{T_{u}}, \\ −u_{x}(0, t) = Cu^{p_{11}}(T_{v} − t)^{−ηp_{12}}, & t ∈ (0, T_{u}), \\ u_{x}(L, t) = 0, & t ∈ (0, T_{u}), \\ u(x, 0) = u_{0}(x), & x ∈ (0, L), \end{matrix}  (6.2)

with ηp_{12} ≥ 1/2. From Theorem 3.1 we know that every nonnegative solution ũ of (6.2) blows up in time Tũ ≤ T_{v}. Our purpose is to show that we have a strict inequality. Hence u blows up at time T_{u} ≤ Tũ < T_{v}, which means that blow-up is non-simultaneous.

Since m = 1, from the representation formula, we get

ũ(0, t) ≥ C ∫_{0}^{t} \frac{ũ(0, s)^{p_{11}}}{(T_{v} − s)^{η}(t − s)^{1/2}} ds ≥ C ∫_{0}^{t} \frac{ũ(0, s)^{p_{11}}}{(T_{v} − s)^{η+1/2}} ds = Cw(t).

Therefore ũ(0, t)^{p_{11}}(T_{v} − t)^{−η−1/2} = w′(t) and hence

\frac{w′(t)}{w(t)^{p_{11}}} ≥ \frac{C}{(T_{v} − t)^{η+1/2}} .

Integrating this expression in (t, T_{v}) we get

\frac{c_{1}}{w(T_{v})^{p_{11}−1}} ≤ c_{2} − ∫_{t}^{T_{v}} \frac{C}{(T_{v} − s)^{η+1/2}} ds = c_{2} − I(t).

Thus,

cw(T_{v}) ≥ (c_{2} − I(t))^{−1/(p_{11}−1)} .

Since I(t) diverges as t ↗ T_{v}, there is a time T_{0} < T_{v}, such that c_{2} = I(T_{0}) and therefore w(T_{0}) ≥ ∞. Now, since ũ(0, t) ≥ w(t) we get the desired result, Tũ ≤ T_{0} < T_{v}.

To extend the proof to m > 1 we follow the ideas of the proof of Theorem 3.2. Observe that

0 < c ≤ mu^{m−1}.  (6.3)

If we define z = u^{m}, using (6.3) we get that z is a supersolution of the heat equation. Hence z blows up at T_{0} < T_{v}. So, u blows up at T_{u} ≤ T_{0} < T_{v} and blow-up is non-simultaneous.

Let us point out that this idea cannot be used when m < 1, since in this case (6.3) is false. □

Acknowledgements

C. Brändle and F. Quirós are supported by project BFM2002-04572-C02-02. J. D. Rossi is supported by ANPCyT PICT 5009, UBA X066 and Fundación Antorchas. He is a member of CONICET.

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