Global and non global solutions for a class of coupled parabolic systems

which models a broad variety of physical phenomena called reaction-di usion equations, motivated by neuroscience, surface chemistry, gas dynamics or predator-prey interactions [3, 10, 11, 16]. Here and hereafter uj : R+ ×RN → R for j ∈ [1,m], δ ∈ {0, 1}, μ = ±1 and ajk = akj are positive real numbers. A solution u := (u1, ..., um) to (1.1) formally satis es some decay of the energy Eδ(u(t)) := 1 2 m ∑


Introduction
This paper is concerned with the Cauchy problem for a coupled heat system with power-type non-linearities which models a broad variety of physical phenomena called reaction-di usion equations, motivated by neuroscience, surface chemistry, gas dynamics or predator-prey interactions [3,10,11,16]. Here and hereafter u j : R+ × R N → R for j ∈ [ , m], δ ∈ { , }, µ = ± and a jk = a kj are positive real numbers. A solution u := (u , ..., um) to (1.1) formally satis es some decay of the energy The particular case a jk = δ k j gives some classical scalar semi-linear heat equations. Thus, before going further, let us recall some historic facts about the semi-linear parabolic equation. The model case given by a pure power non-linearity is of particular interest. The question of well-posedness in the energy space of the following heat problem (NLH)pu − ∆u ± |u| p− u = , p > , u : R × R N → R, was widely investigated. This equation satis es a scaling invariance. Indeed, if u is a solution to (NLH)p with datum u , then u λ := λ p− u(λ . , λ . ) is a solution to (NLH)p with data λ p− u (λ . ). For sc := N − p− , the spaceḢ sc whose norm is invariant under the dilatation u → u λ is relevant in this theory. The energy critical case sc = corresponds to the critical power pc := N+ N− , for N ≥ .
Local well-posedness of (NLH)p holds in the energy critical case and the local existence interval does not depend only on u H . Then, an iteration of the local well-posedness theory fails to prove global existence and a nite time blow-up of solutions may happen [9]. Now, the energy critical case of (NLH)p is known to be well-posed in some Besov spaces [15]. See [12] in the two space dimensions case and [4,24,25] in the scale of Lebesgue spaces L q (R N ).
The topic of blow-up of solutions to bi-component parabolic systems with positive data on bounded domains have been attracting great attention. There have been numerous publications in the literature in this direction and we refer the interested reader to [6,8,13,14,18,20,22] and references therein.
This paper seems to be one of few works dealing with m-component coupled semi-linear heat systems. Moreover, to the author knowledge, the stability of standing waves was not treated in the case of non-linear heat equations. The parabolic system (1.1) is a generalization of the bi-component problem considered in [26], where the global existence, long time decay and nite time blowup of solutions were investigated using the potential-well method.
It is the purpose of this manuscript to obtain global well-posedness and exponential decay of solutions to the defocusing non-linear coupled heat system (1.1), in the energy space. In the focusing sign, using the concepts of invariant sets suggested by Payne and Sattinger in [19], global and non global existence of solutions are discussed, moreover an exponential decay in the energy space holds for any global solution under the potential well. Finally, the existence of in nitely many non global solutions near the ground state is obtained.
The rest of the paper is organized as follows. The next section contains the main results and some technical tools needed in the sequel. Sections three and four are devoted to proving well-posedness of the heat system (1.1) in the energy space. In section ve, the existence of critical ground states is investigated. In section 6, global and non global existence of solutions to (1.1) are discussed. The last section contains a proof of strong instability for stationary solutions.
We de ne the product space where H (R N ) is the usual Sobolev space endowed with the complete norm We denote the real numbers and we assume here and hereafter that We mention that C will denote a constant which may vary from line to line and if A and B are non negative real numbers, A B means that A ≤ CB. For ≤ r ≤ ∞ and (s, T) ∈ [ , ∞) × ( , ∞), we denote the Lebesgue space L r := L r (R N ) with the usual norm . r := . L r , . := . and For simplicity, we denote the usual Sobolev Space W s,p := W s,p (R N ) and H s := W s, . If X is an abstract space C T (X) := C([ , T], X) stands for the set of continuous functions valued in X and X rd is the set of radial elements in X, moreover for an eventual solution to (1.1), we denote T * > its lifespan.

Main results and background
In what follows, we give the main results and some estimates needed in the sequel.

. Main results
First, local well-posedness of the heat problem (1.1) is claimed. The existence of ground states in the sub-critical case is known [21]. In the critical case, the situation is as follows.

Remark 2.7.
It is not proved that the minimizer of (2.3) is a solution to (2.2), because of a lack of uniqueness of such a solution. Despite, we will call Ψ as ground state.
Using the potential well method [19], we discuss global existence and nite-time blow-up of solutions to the focusing problem (1.1). Let us de ne the sets For easy notation, we write The last result concerns instability by blow-up for standing waves of the heat problem (1.1). Indeed, near a ground state, there exist in nitely many data giving non global solutions to (1.1).
Theorem 2.9. Take ≤ N ≤ , ≤ p < p c and µ = δ = . Let Ψ be a ground state solution to (2.2). Then, for any ε > , there exists u ∈ H such that u − Ψ H < ε and the maximal solution to (1.1) with data u is not global.
In the next subsection, we give some standard estimates needed in the paper.

. Tools
We start with some properties of the free heat kernel [4].
Let us recall the so-called Strichartz estimate [7].
Proposition 2.12. Let two admissible pairs (q, r) and (a, b). Then, there exists a positive real number C such that for any T > , Existence of a ground state solution to (1.1) was obtained recently [21].
Proposition 2.13. Take N ≥ , p * < p < p c and two real numbers (α, β) ∈ R * + × R+ ∪ {( , − N )}. Then, 1/ m := m α,β is nonzero and independent of (α, β); 2/ there is a minimizer of (2.3), which is some nontrivial solution to (2.2); 3/ if we make the following assumptions a jj = µ j and a jk = µ for j ≠ k ∈ [ , m] then, at least two components of the minimizer are non zero if µ > is large enough; In the rest of this subsection, we collect some standard estimates independent of the parabolic problem (1.1).
Proof. Assume with contradiction, the existence of such a function. Then (G −( +ε) G ) ≥ and This is a Riccati inequality with blow-up time T < ε G( ) G ( ) . This contradiction achieves the proof. Let us gather some useful Sobolev embeddings [1].

Proposition 2.15. The continuous injections hold
The following Gagliardo-Nirenberg inequality [17] will be useful.
Proposition 2.16. Take N ≥ and < p ≤ p c . Then, for any (u , .., um)∈ H, We close this subsection with an absorption result.

Local well-posedness
This section is devoted to prove Theorem 2.1. The proof contains three steps: local existence, uniqueness and global existence in the sub-critical case. In this section, we take µ = − , indeed the sign of the non-linearity has no local e ect.

. Local existence
We use a standard xed point argument. For T > , R := C Ψ H we denote the space We prove the existence of some small T, R > such that ϕ is a contraction on the ball B T (R) with center zero and radius R. Take u, v ∈ E T , applying the Strichartz estimate (2.5), we get for small T > , To derive the contraction, consider the function With the mean value Theorem Using Hölder inequality, Sobolev embedding and denoting the quantity we compute via a symmetry argument Thus, for T > small enough, ϕ is a contraction satisfying Taking in the last inequality v = , yields Moreover, thanks to Hölder inequality and Sobolev embedding, we obtain Since ≤ p < p c , ϕ is a contraction of X T,R for some T > small enough.

. Uniqueness
In what follows, we prove uniqueness of solution to the Cauchy problem (1.1). Let T > be a positive time, u, v ∈ C T (H) two solutions to (1.1) and (w , .., wm) = w := u − v. Theṅ Applying Strichartz estimate with the admissible pair (q, r) = ( p N(p− ) , p) and denoting for simplicity L q T (L r ) the norm of (L q T (L r )) m , we get for small T > , Taking T > small enough, with a continuity argument, we may assume that max j= ,...,m u j L ∞ T (H ) ≤ .
Using previous computation with Then Uniqueness follows for small time and then for all time with a translation argument.

. Global existence in the defocusing sub-critical case
The global existence is a consequence of the energy decay and previous calculations. Let u ∈ C([ , T * ), H) be the unique maximal solution of (1.1). We prove that u is global. By contradiction, suppose that T * < ∞. Consider for < s < T * , the problem (Ps) Using the same arguments of local existence, we can prove a real τ > and a solution v = (v , ..., vm) to (Ps) on C [s, s + τ], H). Thanks to the decay of energy we see that τ does not depend on s. Thus, if we let s be close to T * such that T * < s + τ, this fact contradicts the maximality of T * .

. Exponential decay
This subsection is devoted to prove that u ∈ C(R+, H), the global solution to (1.1) for δ = −µ = and < p < p c satis es an exponential decay in the energy space.

K(u(s)) ds E(u(t)).
Thus, for some positive real number T > , Taking account of the monotonicity of the energy, for large T > , Then, Finally, because µ = − , The proof is nished.

Global existence in the critical case
We give an auxiliary result. Using Strichartz estimate, we get .
As previously Using Hölder inequality and Sobolev embedding, yields Using Hölder inequality and Sobolev embedding, yields Then, thanks to Sobolev injections Moreover, taking in the previous inequality v = , we get for small δ > , With a classical Picard argument, for small a = δ, b > , there exists u ∈ X a,b a solution to (1.1) satisfying With Strichartz estimate and arguing as previously, the solution u ∈ C(I, H).
We are ready to prove Theorem 2.3. it su ces to prove that u Ḣ remains small on the whole interval of existence of u. Write with conservation of the energy and Sobolev's inequality

Proof of
So by Lemma 2.17, if ξ (Ψ) is su ciently small, then u stays small in theḢ norm. Global existence is established.

Invariant sets and applications
In this section, we prove Theorem 2.8 about global and non global existence of solutions to (1.1) in the energy space. We suppose in all this section that µ = . First, we give some stable sets. Proof. Let Ψ ∈ A δ,+ α,β and u ∈ C T * (H) be the maximal solution to (1.1). Assume that u(t ) ∉ A δ,+ α,β , for some time t ∈ ( , T * ). Since the energy is decreasing, we have K δ α,β (u(t )) < . So, with a continuity argument, there exists a positive time t ∈ ( , t ) such that K δ α,β (u(t )) = . This contradicts the de nition of m. The proof is similar in the case of A δ,− α,β . The fact that m α,β is independent of (α, β) implies that some sets are also independent of (α, β). Lemma 6.2. The sets A δ,+ α,β and A δ,− α,β are independent of (α, β).
Proof. Let (α, β) and (α , β ) in R * + × R+ ∪ {( , − N )}. By the Propositions 2.6-2.13, the reunion A δ,+ α,β ∪ A δ,− α,β is independent of (α, β). So, it is su cient to prove that A δ,+ α,β is independent of (α, β). If E δ (u) < m and K δ α,β (u) = , then u = . So, A δ,+ α,β is open. The rescaling u λ := e αλ u(e −βλ .) implies that a neighborhood of zero is in A δ,+ α,β . Moreover, this rescaling with λ → −∞ gives that A δ,+ α,β is contracted to zero and so it is connected. Now, write Since by the de nition, A δ,− α,β is open and ∈ A δ,+ α,β ∩ A δ,+ α ,β , using a connectivity argument, we have A δ,+ α,β = A δ,+ α ,β . Now, we prove the main result of this section. The global existence follows with classical methods since T * depends only on the quantity u H . Now, we prove an exponential decay of the solution. For small u , since sup t u(t) H , thanks to Proposition 2.16, we get On the other hand Moreover, for T > , Thus, for some positive real number T > , This implies that, for t ≥ T , Taking account of the monotonicity of the energy, for large T > , Then, Finally, The proof is nished. 2/ With a translation argument, we can assume that t = . Thus, E δ (u(t)) ≤ E δ (u ) < m. Moreover, with Lemma 6.1, u(t) ∈ A δ,− α,β for any t ∈ [ , T * ). Take  We discuss two cases. 1/ First case: E δ (u ) > . By Lemmas 6.1-6.2, we get for any λ > , Thus, for any ε > , Taking account of the identities we obtain Write On the other hand, a jk |u j u k | p dx.
Take λ = + and This choice implies that the terms (I) and (III) are non negative. Thus, Thanks to Cauchy-Schwarz inequality, it follows that Moreover, if L(t) = for some positive time, we get K , (u(t)) = , which contradicts Lemma 6.1. Thus Taking account of Propostion 2.14, for some nite time T > , So, thanks to the identityĖ δ (u) = − m j= u j , we get Now, the proof goes by contradiction assuming that T * = ∞. where we used (6.7) in the rst estimate, Cauchy-Schwarz inequality in the second and Claim 2 in the last one. Now choosing α such that < ( +ε)α := + ε , we get LL > ( + ε )L , for large time.
Thanks to Proposition 2.14, this ordinary di erential inequality blows up in nite time and contradicts our assumption that the solution is global. This ends the proof.
Proof. We have Since p > p * , a monotony argument closes the proof of ( ), ( ) and ( ). For ( ), it is su cient to compute using ( ).
Moreover, thanks to the decay of energy, it follows that for any t > ,
The proof is nished via the fact that lim λ→ Ψ λ − Ψ H = .