COEXISTENCE OF SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP IN A SEMILINEAR

PARABOLIC SYSTEM

JULIO D. ROSSI & PHILIPPE SOUPLET

Abstract. We study simultaneous and non-simultaneous blow-up for solutions of the following system

\{ \begin{matrix}u_{t} = Δu + u^{r} + v^{p}, \\ v_{t} = Δv + v^{s} + u^{q}, \end{matrix} in Ω × (0, T),

with Dirichlet boundary conditions. We show that, in the range of exponents where either component may blow up alone, there also exist initial data for which both components blow up simultaneously.

The proof is based on a continuity argument, which requires upper and lower blow-up estimates, independent of initial data, and continuous dependence of the existence time.

In turn, we prove a result of continuous dependence of the existence time, under the assumption of uniform upper blow-up estimates, in the framework of general abstract semiflows.

1. Introduction.

We study the blow-up problem for the following system

(1.1)  \{ \begin{matrix}u_{t} = Δu + u^{r} + v^{p}, in Ω × (0, T), \\ v_{t} = Δv + u^{q} + v^{s}, in Ω × (0, T), \\ u = v = 0, on ∂Ω × (0, T). \end{matrix}

Here p, q, r, s > 1 and Ω stands for a C^{2} smooth bounded domain in ℝ^{N}.

In what follows, we fix some ε ∈ (0, 1) and we shall always assume that the initial data satisfy u_{0}, v_{0} ∈ C_{0}(Ω), u_{0}, v_{0} ≥ 0, (u_{0}, v_{0}) ≢ (0, 0), along with the following assumptions:

(1.2)  \{ \begin{matrix}u_{0}, v_{0} ∈ C^{2}(\overline{Ω}), Δu_{0} = Δv_{0} = 0 on ∂Ω, \\ Δu_{0} + (1 − ε)(u_{0}^{r} + v_{0}^{p}) ≥ 0 in Ω, \\ Δv_{0} + (1 − ε)(v_{0}^{s} + u_{0}^{q}) ≥ 0 in Ω \end{matrix}

(such initial data can be constructed easily – see Remark 4.2). Problem (1.1) admits a unique, maximal, classical solution u, v ≥ 0. We shall denote by T = T (u_{0}, v_{0}) ∈ (0, ∞] its maximal existence time.

Key words and phrases. Blow-up, parabolic systems, continuous dependence. 2000 Mathematics Subject Classification. 35K55, 35B40, 35K45. JDR supported by ANPCyT PICT No. 03-05009 and UBA. JDR is a member of CONICET.

For parabolic systems like (1.1), it can happen that solutions become unbounded in finite time, that is:

T < ∞ and lim sup_{t→T} (∥u(t)∥_{∞} + ∥v(t)∥_{∞}) = ∞.

This is for instance the case under assumption (1.2) (see Proposition 2.1 below). This phenomenon is known as blow-up and has been widely studied, see for example [6, 11, 14, 19] and references therein. The system (1.1) has been considered from the point of view of blow-up and global existence in [20, 21] and a similar system with reaction and absorption terms was studied in [3].

In what follows, we shall say that u (resp., v) blows up if

T < ∞ and lim sup_{t→T} ∥u(t)∥_{∞} = ∞ (resp., lim sup_{t→T} ∥v(t)∥_{∞} = ∞).

We note that a priori there is no reason for both components of the system to blow up simultaneously. In fact it could happen that one of the components blows up as t → T, while the other remains bounded on [0, T). This phenomenon is called non-simultaneous blow-up. For parabolic systems, this phenomenon seems to have been first suggested in [19] performing some numerical experiments. Further mathematical study was carried out in [16, 17, 21], where the range of exponents for which non-simultaneous blow-up is possible was characterized in several parabolic systems.

For the particular problem (1.1), we know the following:

(1) if r < q + 1 (resp., s < p + 1), then v (resp., u) blows up whenever T < ∞. Consequently, if r < q+1 and s < p+1, then blow-up is always simultaneous (see [21, Proposition 2.1] for Ω = ℝ^{N}; the argument there can be modified to cover the case of bounded domains);

(2) if r > q+1 (resp., s > p+1), then there exist initial data such that u (resp., v) blows up in finite time and v (resp., u) remains bounded. Consequently, in the range of exponents

(1.3)  r > q + 1 and s > p + 1,

either component may have non-simultaneous blow-up (see [21, Proposition 2.2] and Proposition 4.1 below).

None of the previous works deal with the question whether, for fixed values of the parameters p, q, r, s, it is possible to have both simultaneous and non-simultaneous blow-up, depending only on the initial data. In particular, in the range (1.3), a natural question is to determine whether simultaneous blow-up may occur. The aim of this paper is to answer this question positively.

Our main result is the following (see Theorem 4.1 for a more detailed statement).

Theorem 1.1. Assume (1.3). Then there exist initial data (u_{0}, v_{0}) such that both components of the solution of system (1.1) blow up simultaneously.

We believe that simultaneous blow-up in the non-simultaneous range (1.3) occurs for very exceptional initial data. Namely, for each fixed u_{0}, this should happen with the initial data (u_{0}, μv_{0}) for at most one value of μ > 0. However we are presently unable to prove this.

On the other hand, the possibility of simultaneous blow-up in the ”half-simultan-eous” range r > q + 1 and s ≤ p + 1 does not seem to be known. This and related questions will be investigated elsewhere.

Next, we focus on the blow-up rates for system (1.1). We have obtained the following (see Propositions 2.1 and 2.2 for more precise results).

Theorem 1.2. Assume (1.2). Suppose that blow-up is non-simultaneous, or that blow-up is simultaneous and (1.3) holds. Then there exist constants C_{1}, C_{2} > 0 such that

(1.4)

C_{1} ≤ (T − t)^{1/(r−1)}∥u(t)∥_{∞} ≤ C_{2}, 0 < t < T,  if u blows up,
C_{1} ≤ (T − t)^{1/(s−1)}∥v(t)∥_{∞} ≤ C_{2}, 0 < t < T,  if v blows up.

Estimates (1.4) in the non-simultaneous case were proved in [21] for Ω = ℝ^{N} and r, s < (N +2)/(N −2) if N ≥ 3. As for the simultaneous case, no blow-up rates were given in [21] in the range (1.3), whereas in the (disjoint) range r < p(q + 1)/(p + 1), s < q(p + 1)/(q + 1), the slower rates

(1.5)  \begin{matrix}C_{1} ≤ (T − t)^{(p+1)/(pq−1)}∥u(t)∥_{∞} ≤ C_{2}, 0 < t < T, \\ C_{1} ≤ (T − t)^{(q+1)/(pq−1)}∥v(t)∥_{∞} ≤ C_{2}, 0 < t < T, \end{matrix}

were found.

The proof of Theorem 1.1 is based on a continuity argument which can be sketched as follows.

On each vertical line {(λu_{0}, μv_{0}); μ > 0} in the plane of initial data, with λ large, we show that v (resp., u) blows up alone if μ >> 1 (resp., μ << 1); cf. Proposition 4.1. Consequently, there is a limiting value μ^{∗} such that (λu_{0}, μ^{∗}v_{0}) lies on the borderline of initial data yielding both types of behaviors.

Now assume that T depends continuously on initial data. If we know that any blowing up component is estimated below according to (1.4), uniformly with respect to initial data, then we may pass to the limit along some sequences μ_{n} → μ^{∗} to deduce that both components blow up for the initial data (λu_{0}, μ^{∗}v_{0}).

A key step is thus to establish uniform lower estimates for blowing up components (Proposition 2.1). These estimate are in fact consequences of the corresponding upper blow-up estimates in (1.4), also with uniform constants (Proposition 2.2). These uniform upper estimates, in turn, also imply the continuity of the existence time T. This is shown in Section 3, as a particular case of a result on general abstract semiflows. The more detailed version of Theorem 1.1 is stated and proved in Section 4.

2. Uniform blow-up estimates

In this section we establish the upper and lower uniform blow-up estimates on (u, v) which are necessary to show the continuity of the existence time and the existence of a simultaneous blow-up solution. By the term ”uniform”, we here mean that these estimates involve constants which are independent of initial data (provided ε in (1.2) is fixed) and that they hold on the whole existence interval (or at least on an interval near T whose length is independent of initial data). This feature is essential for our purpose of proving existence of simultaneous blow-up solutions by a continuity argument (see Section 4).

Let us begin with the uniform upper blow-up estimate.

Proposition 2.1. Let (u_{0}, v_{0}) satisfy (1.2). Then T < ∞ and it holds

(2.1)  \begin{matrix}∥u(t)∥_{∞} ≤ C_{r}(ε)(T − t)^{− \frac{1}{r−1}} , 0 < t < T, \\ ∥v(t)∥_{∞} ≤ C_{s}(ε)(T − t)^{− \frac{1}{s−1}} , 0 < t < T, \end{matrix}

where C_{r}(ε) = ((r − 1)ε)^{−1/(r−1)}.

Let us stress that in the above Proposition, we do not assume that u and v both blow up. Although such an estimate might look trivial for a bounded component, this is actually not quite so, because the constant in the right hand side is uniform and thus provides an information which is not contained in the mere boundedness of this component.

Proof. The proof uses the idea of [9] (see also [5] for a similar idea for the system (1.1) without the terms u^{r}, v^{s}). Define the auxiliary functions

G = u_{t} − ε(u^{r} + v^{p}), J = v_{t} − ε(v^{s} + u^{q}).

in Q_{T} = Ω×(0, T). Since u, v > 0 in Q_{T} by the strong maximum principle, parabolic regularity implies that G, J ∈ C^{2,1}(Q_{T}), and we also have G, J ∈ C(\overline{Ω} × [0, T)), due to (1.2)_{1}. We compute

G_{t} = u_{tt} − εru^{r−1}u_{t} − εpv^{p−1}v_{t},

and

\begin{aligned}ΔG & = & Δu_{t} − εru^{r−1}Δu − εpv^{p−1}Δv \\ & & −εr(r − 1)u^{r−2}|∇u|^{2} − εp(p − 1)v^{p−2}|∇v|^{2}. \end{aligned}

Using u_{t} = G + ε(u^{r} + v^{p}) and v_{t} = J + ε(v^{s} + u^{q}), it follows that

G_{t} − ΔG ≥ (u^{r} + v^{p})_{t} − εru^{r−1}(u^{r} + v^{p}) − εpv^{p−1}(v^{s} + u^{q})
≥ ru^{r−1}(G + ε(u^{r} + v^{p})) − εru^{r−1}(u^{r} + v^{p})
+pv^{p−1}(J + ε(v^{s} + u^{q})) − εpv^{p−1}(v^{s} + u^{q}).

Hence

(2.2)  G_{t} − ΔG ≥ ru^{r−1}G + pv^{p−1}J.

Similarly, we get

(2.3)  J_{t} − ΔJ ≥ sv^{s−1}J + qu^{q−1}G.

Observe that, due to u, v ≥ 0, the linear system (2.2)-(2.3) has nonnegative coefficients and therefore verifies the maximum principle. Since, on the other hand, we have u_{t} = v_{t} = 0 hence G = J = 0 on S_{T} = ∂Ω × (0, T), and J, G ≥ 0 at t = 0 due to our hypothesis (1.2), we deduce that

(2.4)  J, G ≥ 0 in Q_{T}.

This implies that, for each x ∈ Ω,

(u^{1−r})_{t} = −(r − 1)u^{−r}u_{t} ≤ −(r − 1)ε.

Integrating between t and T leads to

u^{1−r}(t, x) ≥ u^{1−r}(t, x) − u^{1−r}(T, x) ≥ (r − 1)ε(T − t),

hence T < ∞ and u(t, x) ≤ C_{r}(ε)(T − t)^{−1/(r−1)}. The estimate for v is obtained similarly. □

With the uniform upper blow-up estimates at hand, we are now able to establish the uniform lower blow-up estimates.

Proposition 2.2. Let (u_{0}, v_{0}) satisfy (1.2). Assume s > p + 1 and u blows up (resp. r > q + 1 and v blows up). Then it holds

(2.5)  ∥u(t)∥_{∞} ≥ C′_{r}(T − t)^{− \frac{1}{r−1}} , 0 < T − t < τ_{1},

resp.

(2.6)  ∥v(t)∥_{∞} ≥ C′_{s}(T − t)^{− \frac{1}{s−1}} , 0 < T − t < τ_{2},

with C′_{r} = 2^{−(r+1)/(r−1)}, τ_{1} = τ_{1}(p, r, s, ε) > 0 and τ_{2} = τ_{2}(q, r, s, ε) > 0.

Proof. Assume for instance that s > p + 1 and u blows up and let 0 < t < T. We first claim that

(2.7)  ∥u(t)∥_{∞} ≤ (T − t)(2∥u(t)∥_{∞})^{r} + c_{1}(T − t)^{1−p/(s−1)}.

To prove (2.7), we modify an argument of [22]. Using the variation-of-constants formula, (2.1) and s > p + 1, we have, for all t < t̃< T,

\begin{aligned}∥u(t̃)∥_{∞} & ≤ & ∥u(t)∥_{∞} + ∫ t̃ ; t∥u(σ)∥_{∞}^{r} dσ + C_{s}^{p}(ε) ∫ t̃ ; t(T − σ)^{−p/(s−1)} dσ \\ & ≤ & ∥u(t)∥_{∞} + ∫ t̃ ; t∥u(σ)∥_{∞}^{r} dσ + c_{1}(T − t)^{1−p/(s−1)}, \end{aligned}

with

c_{1} = \frac{C_{s}^{p}(ε)}{1 − p/(s − 1)}.

Since u blows up, there is a first t̃∈ [t, T) such that ∥u(t̃)∥_{∞} = 2∥u(t)∥_{∞}. It follows that

(2.8) ∥u(t̃)∥_{∞} = 2∥u(t)∥_{∞} ≤ ∥u(t)∥_{∞} + (t̃− t)(2∥u(t)∥_{∞})^{r} + c_{1}(T − t)^{1−p/(s−1)}

hence (2.7). Next, due to (2.7), we have either

(2.9)  (T − t)(2∥u(t)∥_{∞})^{r} ≤ c_{1}(T − t)^{1−p/(s−1)}

so that

(2.10)  ∥u(t)∥_{∞} ≤ 2c_{1}(T − t)^{1−p/(s−1)},

or

(2.11)  (T − t)(2∥u(t)∥_{∞})^{r} ≥ c_{1}(T − t)^{1−p/(s−1)},

so that

(T − t)(2∥u(t)∥_{∞})^{r} ≥ (1/2)∥u(t)∥_{∞},

hence

(2.12)  ∥u(t)∥_{∞} ≥ 2^{−(r+1)/(r−1)}(T − t)^{−1/(r−1)}.

Since u blows up, (2.10) and hence (2.9) cannot hold for t close to T. If (2.11) and hence (2.12) hold on [0, T), then we are done. Otherwise there is a largest t = t_{1} such that (2.9) holds and we then get

(T − t_{1})(2∥u(t_{1})∥_{∞})^{r} = c_{1}(T − t_{1})^{1−p/(s−1)},

hence

2^{−(r+1)/(r−1)}(T − t_{1})^{−1/(r−1)} ≤ ∥u(t_{1})∥_{∞} ≤ 2c_{1}(T − t_{1})^{1−p/(s−1)},

which implies

(T − t_{1})^{r/(r−1)−p/(s−1)} ≥ c_{1}^{−1}2^{−2r/(r−1)}.

We conclude that there exists τ_{1}(p, r, s, ε) > 0, such that for all 0 < T − t < τ_{1}, (2.11) and hence (2.12) hold. We have thus proved (2.5). □

Proof of Theorem 1.2. The upper inequalities are given by Proposition 2.1. Under hypotheses of simultaneous blow-up and (1.3), the lower inequalities follow from Proposition 2.2. In case of non-simultaneous blow-up of u, an obvious modification of the proof of (2.7) yields

(2.13)  ∥u(t)∥_{∞} ≤ (T − t)(2∥u(t)∥_{∞})^{r} + C,

for some C > 0 (depending on (u, v)). For t close to T, we have ∥u(t)∥_{∞} ≥ 2C, so that (2.13) implies (2.12). The lower inequality follows. □

3. Continuity of the existence time

In this section we prove the continuity of the existence time with respect to initial data for solutions of (1.1). More precisely, let E = C_{0}(Ω) × C_{0}(Ω), equipped with the L^{∞} × L^{∞} norm, and let

X := {(u_{0}, v_{0}) ∈ E; u_{0}, v_{0} ≥ 0 and (1.2) is satisfied}.

We have:

Theorem 3.1. The existence time T of solutions of problem (1.1) is continuous as a function of the initial data (u_{0}, v_{0}) from X to (0, ∞].

Note that Theorem 3.1 gives the continuity of the existence time regardless the blow-up is simultaneous or not. Also, the result of Theorem 3.1 remains valid if the terms u^{r}, v^{s} are dropped in system (1.1) (and in assumption (1.2) of course).

The continuous dependence of the existence time has been studied in [2, 12, 13, 15, 18], for scalar superlinear parabolic problems, typically of the form u_{t}−Δu = u^{p}, e. g. with Dirichlet boundary conditions. Recall that continuity [18], or even Hölder continuity [12], is true in general for subcritical p, i.e. (N − 2)p < (N + 2) and that it may fail for some supercritical p (see [11]).

Let us remark that in the above references, the continuity of T relies on energy properties, or on some uniform estimates of the blow-up rate, like the ones obtained in the previous section for system (1.1). Although there is a number of results on blow-up rates for systems (see for instance [1, 4, 5, 7, 8, 21] for system (1.1) without the terms u^{r}, v^{s}), the question of the continuity of the existence time for systems has not been addressed before, as far as we know.

Theorem 3.1 wil be proved as a particular case of a result on general abstract semiflows. Although the underlying argument is close to those in, e.g., [12, 18], we think it may be useful in view of future application to state the result in this more general and natural context.

Let E be a Banach space, with norm ∥ · ∥, and let X be a subset of E. By a (local) semiflow (S(t))_{t≥0} on X, we mean the following. For each U_{0} ∈ X, we are given an existence time T (U_{0}) ∈ (0, ∞] and a trajectory

U(·; U_{0}) ∈ C([0, T (U_{0})); X), with U(0; U_{0}) = U_{0}.

We then set S(t)U_{0} := U(t; U_{0}) for each U_{0} ∈ X and each t ∈ [0, T (U_{0})). Finally, we denote

X_{t} := C([0, t]; X), ∥U∥_{X_{t}} := sup_{s∈[0,t]} ∥U(s)∥.

On the semiflow we impose the following hypotheses (where we put U(t) = S(t)U_{0}, V (t) = S(t)V_{0}).

Continuation property:

(H1)  lim sup_{t→T (U_{0})} ∥U(t)∥ = ∞, for each U_{0} ∈ X such that T (U_{0}) < ∞.

Continuous dependence:

(H2)

∥U − V ∥_{X_{t}} ≤ F (U_{0}, t, ∥U_{0} − V_{0}∥, ∥U∥_{X_{t}} + ∥V ∥_{X_{t}}),
0 < t < min(T (U_{0}), T (V_{0})), U_{0}, V_{0} ∈ X,
where, for each U_{0} ∈ X, lim_{h→0} F (U_{0}, t, h, M) = 0,

uniformly for t, M bounded,

and F (U_{0}, t, h, M) is nondecreasing in the last variable.

Uniform upper blow-up estimate:

(H3)

∥U(t)∥ ≤ G(∥U_{0}∥, \frac{1}{T (U_{0}) − t}),

0 < t < T (U_{0}), for each U_{0} ∈ X such that T (U_{0}) < ∞,

where G is nondecreasing in both variables.

Theorem 3.2. Let (S(t))_{t≥0} be a semiflow satisfying (H1), (H2) and (H3). Then the existence time T is upper semicontinuous from X to (0, ∞].

Proof. Fix U_{0} such that T = T (U_{0}) is finite (otherwise there is nothing to prove). Let U_{n} be a sequence of trajectories such that U_{n}(0) → U_{0} as n → ∞, and set e_{n}(t) = ∥U_{n}(t) − U(t)∥. Since e_{n}(0) → 0, we may assume e_{n}(0) < 1. It is enough to consider n such that T_{n} > T.

Assume that for all t < T, e_{n}(t) < 1, then T_{n} ≤ T due to (H1); but this is impossible. Hence, there exists a minimal t_{n} < T such that e_{n}(t_{n}) = 1. Our hypotheses (H2), (H3) imply that

1 = e_{n}(t_{n}) ≤ F (U_{0}, t_{n}, e_{n}(0), 1 + 2G(∥U_{0}∥ + 1, \frac{1}{T_{n} − t_{n}} )).

Since

F (U_{0}, t, h, M) → 0, h → 0,

uniformly for t, M bounded, and e_{n}(0) → 0 as n → ∞, it follows that

G(∥U_{0}∥ + 1, \frac{1}{T_{n} − t_{n}} ) → ∞, n → ∞.

Consequently, T_{n} − t_{n} → 0 as n → ∞. Since T_{n} − T < T_{n} − t_{n} we get

lim sup_{n→∞} T_{n} ≤ T,

as we wanted to prove. □

As it is well known, the lower semicontinuity is just a standard consequence of continuation and continuous dependence properties. We recall the following for completeness.

Proposition 3.1. Let (S(t))_{t≥0} be a semiflow satisfying (H1) and (H2). Then the existence time T is lower semicontinuous from X to (0, ∞].

Proof. Fix U_{0} ∈ X, with T := T (U_{0}) ≤ ∞. With the notation of the previous proof, we may assume e_{n}(0) → 0, e_{n}(0) < 1 and we need only consider n such that T_{n} is finite and T_{n} < T.

By (H1), there is a first t_{n} < T_{n} such that e_{n}(t_{n}) = 1 and (H2) implies that

1 = e_{n}(t_{n}) ≤ F (U_{0}, t_{n}, e_{n}(0), 1 + 2∥U∥_{X_{tn}} ).

Due to (H2) and e_{n}(0) → 0, if T < ∞, then ∥U∥_{X_{tn}} → ∞, hence t_{n} → T; whereas if T = ∞, then t_{n} + ∥U∥_{X_{tn}} → ∞, hence t_{n} → ∞. Since T_{n} > t_{n}, it follows that

lim inf_{n→∞} T_{n} ≥ T,

as we wanted to prove. □

Proof of Theorem 3.1. Take E and X as defined at the beginning of this section and consider the semiflow defined by S(t)U_{0} = (u(t), v(t)), the solution of (1.1), where U_{0} = (u_{0}, v_{0}). Properties (H1) and (H2) are standard for parabolic systems like (1.1). Property (H3) follows from Proposition 2.1. The result is thus a consequence of Theorem 3.2 and Proposition 3.1. □

4. Simultaneous blow-up in the non simultaneous range

Our main result Theorem 1.1 is a consequence of the following more precise statement.

Theorem 4.1. Assume (1.3) and let (u_{0}, v_{0}) satisfy (1.2) and ∥u_{0}∥_{∞}, ∥v_{0}∥_{∞} ≥ 1. Then there exists λ_{0} ≥ 1 such that for all λ ≥ λ_{0} and for some μ = μ(λ) ≥ 1, the solution (u, v) of (1.1) with initial data (λu_{0}, μv_{0}) has simultaneous blowup. Moreover, (u, v) satisfies the upper and lower blow-up estimates (2.1), (2.5) and (2.6).

Remark 4.1. We may choose λ_{0} = λ_{0}(∥v_{0}∥_{∞}, q, r, s, ε).

Remark 4.2. To ensure condition (1.2) on the initial data, it is enough for 0 ≤ u_{0} ∈ C^{2}(\overline{Ω}) to satisfy Δu_{0} + (1 − ε)u_{0}^{r} ≥ 0 in Ω and Δu_{0} = 0 on ∂Ω (and similar condition for v_{0}). It is easy to check that such u_{0} (moreover satisfying ∥u_{0}∥_{∞} ≥ 1) can be found under the form u_{0} = kφ_{1}−k^{3/4}χ, where φ_{1} > 0 is a first eigenfunction of the Dirichlet Laplacian in Ω, χ is the solution to −Δχ = φ_{1}^{1/2} in Ω, with χ = 0 on ∂Ω, and the number k > 0 is large enough.

Remark 4.3. Actually, assumption (1.2) in Theorem 4.1 can be replaced by its weak formulation, namely:

∫_{Ω}\{u_{0}Δφ + (1 − ε)(u_{0}^{r} + v_{0}^{p})φ\} dx ≥ 0, 0 ≤ φ ∈ C^{2}(\overline{Ω}), φ = 0 on ∂Ω

(and similar second inequality). The only change is in the proof of Proposition 2.1, where G, J no longer belong to C(\overline{Ω} × [0, T)), but where a suitable weak (instead of classical) form of the maximum principle, using duality, can be applied. For this, we only require u_{0}, v_{0} ∈ C_{0}(Ω) to be Hölder continuous.

Before proving Theorem 4.1, we shall first establish the following result (used in the proof of Theorem 4.1), which confirms that s > p + 1 or r > q + 1 is indeed the range of non-simultaneous blow-up, in the sense that if v_{0} is much larger than u_{0} in a suitable sense, then v blows up while u remains bounded (and vice-versa). This fact was proved in [21], for Ω = ℝ^{N}, under restrictive assumptions on the initial data (implying N = 1 or u, v spatially homogeneous).

Proposition 4.1. Assume s > p + 1 and let (u_{0}, v_{0}) satisfy (1.2). There exists a constant C_{3} > 0 depending only on p, r, s, ε such that, if

(4.1)  ∥u_{0}∥_{∞} ≥ 1 and ∥v_{0}∥_{∞} > C_{3}∥u_{0}∥_{∞}^{(r−1)/(s−1)},

then v blows up and u remains bounded.

Remark 4.4. The symmetric statement holds when r > q + 1 by interchanging the roles of u, v, of p, q, and of r, s.

Proof. By the argument leading to (2.8) (applied with t = 0), if there exists a first T_{0} ∈ [0, T) such that ∥u(T_{0})∥_{∞} = 2∥u_{0}∥_{∞}, then it satisfies

(4.2)  ∥u_{0}∥_{∞} ≤ T_{0}(2∥u_{0}∥_{∞})^{r} + c_{1}T^{1−p/(s−1)}.

Assume (4.1) with C_{3} > (2c_{1})^{1/(s−1−p)}C_{s}(ε), where C_{s}(ε) is given by (2.1). Using (2.1) with t = 0, we have

T ≤ C_{s}^{s−1}(ε)∥v_{0}∥_{∞}^{1−s} ≤ (C_{s}(ε)/C_{3})^{s−1} < (2c_{1})^{(1−s)/(s−1−p)}.

Therefore

∥u_{0}∥_{∞} ≥ 1 ≥ 2c_{1}T^{1−p/(s−1)}.

But (4.2) then implies that ∥u_{0}∥_{∞} ≤ 2T_{0}(2∥u_{0}∥_{∞})^{r}. Assuming further that C_{3} > 2^{(r+1)/(s−1)}C_{s}(ε), we get that

T_{0} ≥ 2^{−r−1}∥u_{0}∥_{∞}^{1−r} > 2^{−r−1}C_{3}^{s−1}∥v_{0}∥_{∞}^{1−s} ≥ T.

It follows that ∥u(t)∥_{∞} < 2∥u_{0}∥_{∞} on [0, T) and the result is proved. □

Proof of Theorem 4.1. First observe that for all λ, μ ≥ 1, (λu_{0}, μv_{0}) also satisfies (1.2). By Proposition 4.1 in the case r > q+1, there exists λ_{0} = λ_{0}(∥v_{0}∥_{∞}, q, r, s, ε) ≥ 1 such that for all λ ≥ λ_{0} and μ ∈ [1, 2], the solution of (1.1) with initial data (λu_{0}, μv_{0}) has non-simultaneous blow-up with v bounded.

Fix λ ≥ λ_{0} and denote (u_{μ}, v_{μ}) the solution of (1.1) with initial data (λu_{0}, μv_{0}) and by T_{μ} its blow-up time. By Theorem 3.1, T_{μ} is a continuous function of μ for μ ≥ 1. Applying Proposition 4.1 for s > p + 1, if μ is large enough, then (u_{μ}, v_{μ}) has non-simultaneous blow-up with u_{μ} bounded. Let then

E = {μ > 1; u_{μ} is bounded and v_{μ} blows up}.

We thus have μ^{∗} := inf E ∈ [2, ∞).

Let T_{μ^{∗}} − τ_{1} < t < T_{μ^{∗}}, where τ_{1} is defined in Proposition 2.2. By continuity of T_{μ}, we have T_{μ} − τ_{1} < t < T_{μ} for μ close to μ^{∗}. By (2.5), it follows that

(4.3)  ∥u_{μ}(t)∥_{∞} ≥ C′_{r}(T_{μ} − t)^{−1/(r−1)}, for μ < μ^{∗} close to μ^{∗}.

Letting μ → μ^{∗}− in (4.3) and using continuity of T_{μ} and of u_{μ}, we get

∥u_{μ^{∗}}(t)∥_{∞} ≥ C′_{r}(T_{μ^{∗}} − t)^{−1/(r−1)}.

It follows that u_{μ^{∗}} blows up.

Let now T_{μ^{∗}} − τ_{2} < t < T_{μ^{∗}}, where τ_{2} is defined in Proposition 2.2, and pick a sequence μ_{j} ∈ E with μ_{j} → μ^{∗}. Again by continuity of T_{μ}, we have T_{μ_{j}} − τ_{2} < t < T_{μ_{j}} for j large. By (2.6), we know that

∥v_{μ_{j}}(t)∥_{∞} ≥ C′_{s}(T_{μ_{j}} − t)^{−1/(s−1)}.

Letting j → ∞ and using the continuity of T_{μ} and of v_{μ}, we get

∥v_{μ^{∗}}(t)∥_{∞} ≥ C′_{s}(T_{μ^{∗}} − t)^{−1/(s−1)}.

It follows that v_{μ^{∗}} blows up. The proof is complete. □

Acknowledgements. This work was started during a stay of PS at the University of Buenos Aires and finished during a visit of JDR to the University of Versailles. We wish to thank for the hospitality found in those institutions.

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Departamento de Matemática, FCEyN UBA (1428) Buenos Aires, Argentina.

E-mail address: jrossi@dm.uba.ar

Département de Mathématiques, INSSET Université de Picardie, 02109 St-Quentin, France

Laboratoire de Mathématiques Appliquées, UMR CNRS 7641 Université de Versailles, 45 avenue des Etats-Unis, 78035 Versailles, France.

E-mail address: souplet@math.uvsq.fr