SINGLE-POINT BLOW-UP FOR A MULTI-COMPONENT REACTION-DIFFUSION SYSTEM

. In this work, we prove single-point blow-up for any positive, radially decreasing, classical and blowing-up solution of a system of m ≥ 3 heat equations in a ball of R n , which are coupled cyclically by superlinear mono-mial reaction terms. We also obtain lower pointwise estimates for the blow-up proﬁles.

In the papers [12,13], Renc lawowicz was shown that there exist blowing-up solutions (i.e. T * < ∞) of (1) for some values of the parameters p i , i = 1, ..., m. Moreover, under the equidiffusivity assumption (i.e. δ 1 = δ 2 = ... = δ m ), we know that monotone in time solutions i.e: satisfy the following blow-up rate estimates: for some positive constants c and C, where p m+k = p k for all integer k, the standard scaling exponents of system (1) (see [2,15]). We note also that (6) is still available without the equidiffusivity assumption for a very particular class of radial decreasing solutions (see [12]). The first major topic of this paper is concerned with single-point blow-up for nonnegative, radially symmetric, radially nonincreasing, classical and blowing-up solution of (1). The blow-up set of solutions of nonlinear equations and systems has drawn wide interest since the eighties, when Weissler ( [16]) has proved that blow-up occurs only at the point x = 0 for the scalar equation (i.e. m = 1) with p 1 is a large number and for a particular initial data. See [4,10] for further results in the scalar case. As an extension of the method in [4], Friedman and Giga have proved single-point blow-up for radially decreasing solutions of (1) only for n = 1, m = 2, δ 1 = δ 2 and p 1 = p 2 . More recently, Souplet ( [14]) proved that single-point blow-up occurs for a large class of radial decreasing solutions, in a ball or in the whole space for n ≥ 2 and without the restrictive condition p 1 = p 2 . However, the equidiffusivity assumption δ 1 = δ 2 is still needed and, in addition, it is required that the solution satisfies the upper type I blow-up rate estimates (6). Recently, the author, Souplet and Tayachi ( [8]) removed the previously made extra assumptions. More precisely, they improved the results in two directions: (i) without assuming the type I blow-up rate estimate (6); (ii) without assuming equidiffusivity, i.e. for any δ 1 , δ 2 > 0. The previous manuscript concerns a more general type of nonlinearity. As far as we know, the question of single-point blow-up has not been considered so far for parabolic systems with more than two components. We shall concentrate, for simplicity, on the model case (1). We obtain the following Theorem: Theorem 1.1. Let (u 1 , ..., u m ) be a blowing-up solution of (1) under the assumptions given by (2) and (3). Then blow-up occurs only at the origin, i.e.

SINGLE-POINT BLOW-UP 211
The proof of Theorem 1.1 is long and technical and requires many intermediate steps. As mentioned above, we follow the general strategy developed in [8] for systems of two equations. However, in order to treat systems with an arbitrary number of components, some nontrivial modifications are required. In particular, Steps 5-6 of the proof of Proposition 3 lead to a weaker estimate on solutions than the one which was obtained in [8] in (essentially) the same steps of the same proposition for a two-component system. To exclude blow-up, an additional Proposition 4 is needed. There are a couple of additional differences between [8] and the present manuscript: the proof of (22) by induction and the use of (29) to obtain the upper bound required by Proposition 2.
Remark 1. The result of Theorem 1.1 remains true for the Cauchy problem (that is, (1) with R = ∞ and ∂Ω = ∅) provided u 1, 0 , ..., u m, 0 are not all constant. This follows from straightforward modifications of the proof.
Note that the existence of a nonnegative, radially symmetric, radially nonincreasing and classical solution of (1) such that T * < ∞ and ∂ t u i ≥ 0, i = 1, ..., m, can be obtained for initial data (λu 1, 0 , ..., λu m, 0 ), with λ > 0 large enough, whenever (λu 1, 0 , ..., λu m, 0 ) satisfies (3) and The layout of the remaining parts of this paper is as follows. In Section 2, we prove Theorem 1.1 by a contradiction argument and assuming asymptotic comparison properties of all two components in a row near blow-up points (Proposition 1). Sections 3-6 are next devoted to proving these properties. In Section 3, we establish upper blow-up estimates away from the origin (Proposition 2). To do so, we take advantage of some useful linear algebraic properties related with the m system (1), obtained in [2,15]. In Section 4, we prove a local criterion for excluding blow-up (Proposition 3). The proof of this criterion presents additional difficulties as compared with the case m = 2 in [8]. Namely, we need to combine the analysis in similarity variables with a weaker, preliminary nondegeneracy property (Proposition 4) in original variables. In Section 5, we show the ODE behavior for rescaled solutions and the local interdependence of the components for the ODE system. In Section 6, we then prove the asymptotic comparison properties by using a contradiction argument and the results of Sections 3-5. Finally, in Section 7, we establish the pointwise lower bounds on the blow-up profiles (Theorem 1.2).
Proof. Let F i = u γi i . By differentiation of (13), we obtain
With Proposition 1 and Lemma 2.1 at hand, we can now conclude the proof of Theorem 1.1.
Proof of Theorem 1.1. Let (u 1 , ..., u m ) be a solution of system (1) satisfying the hypotheses of Theorem 1.1 and assume for contradiction that there exists ρ 0 ∈ (0, R) such that Set Taking ε > 0 sufficiently small and using (27) . By integration, we obtain for all T 1 ≤ t < T * . It follows that u i (t, ρ 2 ) is bounded for i = 1, ..., m and T 1 ≤ t < T * . Since ∂ ρ u i ≤ 0, for all i = 1, ..., m, this leads to a contradiction with (26) and proves the theorem.
3. Upper type I estimates away from the origin. In this section, we prove the upper type I estimates away from the origin for nonnegative, radially symmetric, radially nonincreasing and classical solutions of (1).

Proposition 2.
Under the hypotheses of Theorem 1.1. Then, there exists a constant C 0 > 0 (depending only on n, p i , δ i , R, T * ) such that for all t ∈ [0, T * ) and 0 < ρ ≤ R.
Proof. Since our domain is bounded, then we denote by λ 1 the first eigenvalue of −∆ in H 1 0 (B(0, R)) and ϕ 1 the corresponding eigenfunction such that ϕ 1 > 0 and B(0, R) ϕ 1 (x)dx = 1. Let t ∈ (0, T * ) and i ∈ {1; ...; m}, multiplying the equation i in (1) by ϕ 1 and integrating by parts, we obtain .., m. Using Jensen's inequality, we obtain Here and in the rest of the proof, M denotes a positive constant depending only on T * , δ i , p i , n, R and which may vary from line to line. By [15,Lemma 4,p Using (30) and similarly as in [15, pp. 580-582], we obtain where α i are given by (7). Therefore, .., m, are radially symmetric and radially nonincreasing, we deduce that The case when R/2 < ρ < R then follows from the radial nonincreasing property. This completes the proof.

4.
A local criterion for excluding blow-up. In this section, we prove a sufficient, local smallness condition, at any given time sufficiently close to T * , for excluding blow-up at a given point a 0 = 0.
Proposition 3. Under the hypotheses of Theorem 1.1. Let a 0 , a 1 be such that then a 0 is not a blow-up point of (u 1 , ..., u m ), i.e. (u 1 , ..., u m ) is uniformly bounded in the neighborhood of (T * , a 0 ).
The proposition will be proved in Subsection 4.2. As in [8], the proof uses similarity variables, delayed smoothing effects (see Subsection 4.1 below), and upper type I estimates away from the origin (Proposition 3). However, a new difficulty arises, caused by the fact that some algebraic properties for the system of 2-equations may fail for general systems. For this reason, we prove a preliminary result (Proposition 4 below) adapted from [11,Theorem 25.3,p. 195], which gives a sufficient smallness condition on nonnegative solution of (1) at a neighborhood of (T * , a) for excluding blow-up at a given point a ∈ R n .
then (u 1 , ..., u m ) is uniformly bounded in the neighborhood of (T, a).
Since α αi − 1 ≥ 0, and using (1), we obtain If p i ≥ α αi+1 , by (32) and using the fact that α i + 1 = p i α i+1 , we obtain If p i < α αi+1 , then α−α i −1 = α−p i α i+1 > 0 and (34) can be see as the following inequality By (32) and Young's inequality, we obtain Therefore, it follows that where K denotes a positive constant which may vary from line to line and It is easy to see that γ i > 0. Taking γ = min i γ i > 0.
Then, we put
Step 1. Definition of suitably modified solutions. By (28) in Proposition 2, we know that with N 0 = C 0 a −n 1 . We shall thus truncate the radial domain and consider suitably controlled extensions of the solution to the real line. We first define the following extensions u i ≥ 0 of u i , i = 1, ..., m, by setting: for any t ∈ [0, T * ).

5.
Convergence of rescaled solutions to solutions of a system of ordinary differential equalities. We note that Proposition 3 is not yet sufficient to establish the lower type I estimates of Proposition 1. Indeed, Proposition 3 shows no blowup at a given point, assuming that all components violate the type I estimate. Therefore, in order to improve from Proposition 3 to Proposition 1, we need to show a suitable interdependence of the components (under the assumption of blowup at a point different from the origin). In order to do so, for given ρ 1 ∈ (0, R), we again switch to similarity variables around (T * , ρ 1 ), already used in the previous section. Namely, we set: and we consider the rescaled solution (W 1 , ..., W m ) = (W 1,ρ1 , ..., W m,ρ1 ) associated with (u 1 , ..., u m ): defined for σ ∈ [σ, ∞) withσ = − log T * and θ ∈ (−ρ 1 e σ/2 , (R − ρ 1 )e σ/2 ). The goal of this section is to show that any such rescaled solution (W 1 , ..., W m ) behaves, in a suitable sense as σ → ∞ and θ → ∞, like a (distribution) solution of the following system of ordinary differential equalities: on the whole real line (−∞, ∞) (however, we shall eventually only use the fact that (φ 1 , ..., φ m ) solves (93) on some bounded open interval). Moreover, we single out a simple but crucial property of local interdependence of components for solutions of (93). (i) Then, for all sequence σ j → ∞, there exists a subsequence (not relabeled) such that, for each σ ∈ R, i = 1, ..., m and W m+1 = W 1 .
Since w i , i = 1, ..., m, are bounded and nonincreasing, we may define which proves assertion (i).
(ii) We first observe that the properties of the sequence obtained in the previous paragraph allow us to pass to the limit in the distribution sense in (96), it follows in particular that (w 1 , ..., w m ) is a (continuous bounded) solution of (98) We can then obtain (93) by a standard argument based on multiplication by test functions. However, in order to avoid dealing with the terms θ 2 ∂ θ w i , i = 1, ..., m, in the passage to the limit, it is convenient not to work in the current similarity variables. Thus put for x ∈ R and −∞ < t < T * . We observe that Moreover, (U 1 , ..., U m ) solves the system

NEJIB MAHMOUDI
Fix χ ∈ D(R) and ξ ∈ D((−∞, T * )) with R χ = 1. For j ∈ N, replacing x by x + j in (101) and testing with ξ(t)χ(x), we obtain (102) Due to the boundedness of w i and (100), we may therefore apply the dominated convergence theorem on the first and last terms of (102). Taking R χ = 1 and R χ xx = 0 into account, we thus obtain It follows that U + i = U + i+1 pi for all i = 1, ..., m, in the distribution sense. Converting back to φ i , the conclusion follows.
6. Completion of proof of Proposition 1. We now turn to complete the proof of Proposition 1. It is based on a contradiction argument and the results of Sections 3-5.