A NOTE ON GLOBAL WELL-POSEDNESS AND BLOW-UP OF SOME SEMILINEAR EVOLUTION EQUATIONS

. We investigate the initial value problems for some semilinear wave, heat and Schr¨odinger equations in two space dimensions, with exponential non- linearities. Using the potential well method based on the concepts of invariant sets, we prove either global well-posedness or ﬁnite time blow-up.

1. Introduction. In this paper, we study global well-posedness and finite time blow-up of three different evolution equations. First, we consider the Cauchy problem for the nonlinear Klein-Gordon equation ü − ∆u + u = f (u); (u,u) |t=0 = (u 0 , u 1 ) ∈ H 1 × L 2 . (1.1) Second, we treat the initial value problem for the semilinear heat equation Finally, we are interested on the Cauchy problem associated to the semilinear Schrödinger equation iu + ∆u + f (u) = 0; u |t=0 = u 0 ∈ H 1 . (1.3) Here and hereafter u := u(t, x) is a function of the variable (t, x) ∈ R + × R 2 , valued in R for the case of (1.1) or (1.2) and in C for the Schrödinger case. The nonlinearity f is a regular real function satisfying the focusing sign xf (x) > 0 for any real number x = 0 and an exponential growth at infinity to precise later. In the Schrödinger context (1.3), we assume that f takes the Hamiltonian form f (x) = xg(x 2 ) for some positive real regular function g and we extend f to C by f (z) := z |z| f (|z|). The above wave problem has various applications in the areas of nonlinear optics, plasma physics and fluid mechanics [34].
The heat problem models diffusion or heat transfer in a system out of equilibrium. The function u(t, x) might represent temperature or the concentration of some substance, a quantity which may vary with time [39].
Semilinear Schrödinger equations with various nonlinearities arise as models for diverse physical phenomena, including Bose-Einstein condensates [9,24] and as a description of the envelope dynamics of a general dispersive wave in a weakly nonlinear medium [37].
Any solution to (1.1) formally satisfies conservation of the wave energy E(t) = E(u(t),u(t)) : A solution to (1.2) formally verifies decay of the heat energy Any solution to (1.3) formally satisfies respectively conservation of the Hamiltonian, the mass and the Virial identity [16], ; Before going further, we recall a few historic facts about the previous problems. In the monomial case f (u) = u|u| p−1 , local well-posedness in the energy space holds for any 1 < p < ∞. We refer to [8] in the wave case, [7,5] in the Schrödinger context and [19] for the heat problem. So it's natural to consider problems with exponential nonlinearities, which have several applications, as for example the self trapped beams in plasma [14].
Indeed, Nakamura and Ozawa [21] proved global well-posedness and scattering for some defocusing wave equation with exponential type nonlinearity and small Cauchy data. Later on, A. Atallah [3] showed a local existence result to the defocusing 2D wave equation for 0 < α < 4π and with radially symmetric initial data (0, u 1 ) having compact support. Recently, Ibrahim-Majdoub-Masmoudi [10] obtained global well-posedness of (E 4π ) for data in the unit ball of the energy space. The author proved unconditional well-posedness for some weaker exponential growth nonlinearity [17,18]. See [35,36] for classical solutions.
For the Schrödinger problem, Nakamura and Ozawa [22] proved global wellposedness and scattering for some defocusing equation with exponential type nonlinearity and small Cauchy data. More recently, global well-posedness in the unit energy ball and scattering hold [6,4], for the equation Global well-posedness and scattering are established for some lower type exponential nonlinearity [26,29,27,28,32,30].
In the case of semilinear heat equation with exponential type nonlinearity, global well-posedness in some Orlicz space with small data is now proved [13]. Moreover, global well-posedness in the energy space holds for the defocusing sign [12].
In the focusing case, local solution to one of the above evolution equations may exist globally or blow up in finite time [15,31,33]. It is the goal of this manuscript to prove global and non global existence of solutions to the problems (1.1), (1.2) and (1.3), when the energy is under the ground state one. It is worth pointing out that the present study uses the potential well method based on the concepts of invariant sets suggested by Payne and Sattinger in [23].
The rest of the paper is organized as follows. Section two contains the main results and some tools needed in the sequel. The third section is devoted to prove the existence of a ground state solution to the associated stationary problem. In the three last sections, we prove results about global well-posedness and finite time blow up of solutions to the evolution problems (1.1), (1.2) and (1.3).
In this manuscript, we are interested in the two space dimensions case, so, here and hereafter, we denote . dx := R 2 . dx. For p ≥ 1, L p := L p (R 2 ) is the Lebesgue space endowed with the norm . p := . L p , . := . 2 and H 1 is the usual Sobolev space endowed with the norm .
We mention that C denotes an absolute positive constant which may vary from line to line and if A and B are nonnegative real numbers, A B means that Finally, we define the derivative operator (Df )(x) := xf (x).
2. Background material. In this section we give the main results and some technical tools needed in the sequel. First, let us fix the set of nonlinearities considered along this paper. 1. Behavior on zero (2.6)
We give two explicit examples.
The proof is achieved.
We introduce several notations related to the evolution equations to be considered in this note. Here and hereafter, for α, β ∈ R, v ∈ H 1 (R 2 , R) and w ∈ H 1 (R 2 , C), we denote the action The following quantity will be called constraint The quadratic and nonlinear parts of the constraint are Take the minimizing problem under constraint + , it is known [28] that m := m α,β > 0 is independent of (α, β) and is the energy of some solution to the stationary problem associated to one of the above evolution equations. Such a solution is called ground state. ∆φ Finally, we denote some stable sets under the flows of the evolution equations, they will play an essential role in our study. The following sets are adapted to study the wave problem.
The next sets will be considered when treating the heat or Schrödinger problem.
. Results proved in this paper are listed in the following subsection.
. Results listed in this subsection deal with discussing if the maximal solution is global or not.
The study of the Schrödinger problem (1.3) is related to the Virial identity (1.4), which involves the quantity K 1,−1 . This quantity arises also when investigating the wave problem (1.1). So, we are reduced to establish the existence of ground state in the case (α, β) = (1, −1), because the existence of a ground state is known [28] for nonnegative α and β.
The last result is about the Schrödinger problem (1.3).
Theorem 2.8. Assume that f takes the Hamiltonian form and verifies (2.5), (2.7) and [(2.8) or (2.9)]. Let u 0 ∈ H 1 and u ∈ C T * (H 1 ) be the maximal solution to We collect in what follows some auxiliary results.

2.2.
Tools. This subsection is devoted to give some standard estimates needed along this paper. When α and β are nonnegative, the existence of a ground state solution to (2.10) is known [28].
The fact that m α,β is independent of (α, β) implies that some sets are also independent of (α, β).
. By the previous result, the reunion A +δ α,β ∪ A −δ α,β is independent of (α, β). So, it is sufficient to prove that A +δ α,β is independent of (α, β). If S(v) < m and K α,β (v) = 0, then v = 0. So, A +δ α,β is open. The rescaling v λ := e αλ v(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 definition, A −δ α,β is open and 0 ∈ A +δ α,β ∩ A +δ α ,β , using a connectivity argument, we have A +δ α,β = A +δ α ,β . The proof is similar in the case of I + α,β and I − α,β . Remark 2.11. The independence of m α,β on the real couple (α, β) is based on the existence of ground state, which holds only for some particular couples. For this reason, we keep dependence of the above sets on the parameter (α, β).
Let us recall the so-called generalized Pohozaev identity [28].
In order to control an exponential type nonlinearity in the energy space, we will use the following Moser-Trudinger inequality [1,20,38]. ∈ (0, 4π), a constant C α exists such that for all u ∈ H 1 satisfying ∇u ≤ 1, we have Moreover, the previous inequality is false if α ≥ 4π. The limit case α = 4π becomes admissible if we take u H 1 ≤ 1 rather than ∇u ≤ 1. In this case and this is false for α > 4π. See [25] for more details.
Denoting H 1 rd the set of radial functions in H 1 , we give some useful Sobolev embeddings [2].
We close this subsection with a classical result about ordinary differential equation.
Proposition 2.15. Let ε > 0. There is no real function G ∈ C 2 (R + ) satisfying Proof. Assume with contradiction, the existence of such a function. Then This is a Riccati inequality with blow-up time T < 1 ε G(0) G (0) . This contradiction achieves the proof.
3. The stationary problem. It is the aim of this section to prove that (2.10) has a ground state solution in the particular case (α, β) = (1, −1), in the meaning that it has a nontrivial positive radial solution which minimizes the action S when K 1,−1 vanishes. Precisely, we establish Theorem 2.5.
Remark 3.1. We treat the real case, the complex one follows similarly taking account of the Hamiltonian form of the function f .
In this section we denote, for v ∈ H 1 (R 2 , R) and α, β ∈ R, the quantities  1. We extend the operator L α,β as follows, if A : With a direct computation, we get K α,β = L α,β S, so T = (2 − L)S.
The proof of Theorem 2.5 is based on the following intermediate result. Then, there exists n 0 ∈ N such that K(φ n ) > 0 for all n ≥ n 0 .
Proof. There exists some p > 2 satisfying |rf (r)| + |f (r)| r p (e α0r 2 − 1). In fact, by (2.8) or (2.9), the ratio is bounded at infinity and using the assumptions on zero, the ratio is bounded near zero. Thus, for any q ≥ 1, Moreover, using the interpolation inequality on R 2 , Since ∇φ n 2 = K Q (φ n ) and (φ n ) is bounded in H 1 , taking q such that p − 2 q > 2, we get for n going to infinity K(φ n ) K Q (φ n ) > 0. The proof is achieved.
Since K Q (λφ n ) → 0 as λ → 0, taking account of Lemma 3.3, there exists λ ∈ (0, 1) such that K(λφ n ) = 0. So, we can suppose that φ n is radial decreasing and satisfies (3.11). We decompose the rest of the proof to four steps.
Proof of Theorem 2.6.

TAREK SAANOUNI
Independently, since K 0,1 (u) > 0, the identity 2m > 2E = u 2 + ∇u 2 + K 0,1 (u) implies that sup Thus, A Gronwall argument gives The previous inequality implies that the L 2 norm of u does not explode in finite time. So, u is global because it is bounded in H 1 .
First, we give stable sets under the flow of the heat problem.
We discuss two cases.
(a) First case. S(u 0 ) > 0. By Lemmas 2.10 and 5.1 via (4.12), we get for any λ > 0, Thus, for any ε > 0, Taking account of the identity S(u) = 1 Taking λ := aε for some a > 1 and γ := m − S(u 0 ), we get On the other hand, Since 0 < γ < m, we choose ε > 0 small enough such that Then, there exists a > 1 satisfying 2ε This choice, via (2.6), implies that the terms (I) and (III) are non negative. Thus, Thanks to Cauchy-Schwarz inequality, it follows that In fact, if L(t) = 0 for some positive time, we get K 1,0 (u(t)) = 0, which contradicts Lemma 5.1. Thus Taking account of Propostion 2.15, for some finite time T > 0, Thus, T * < ∞ and u is not global. This ends the proof. (b) Second case. S(u 0 ) ≤ 0. Using (2.6), we compute So, thanks to the identityṠ(u) = − u 2 , we get Now, the proof goes by contradiction assuming that T * = ∞.
Claim 1. There exists t 1 > 0 such that t1 0 u(s) 2 ds > 0. Indeed, otherwise u(t) = u 0 almost everywhere and solves the elliptic stationary equation −∆u + u = f (u). Therefore, u 2 H 1 = u f (u)dx and Taking account of the focusing sign of the nonlinearity, we get u 0 = 0 which contradicts the fact that K 0,1 (u 0 ) < 0.
Thanks to Proposition 2.15, this ordinary differential inequality blows up in finite time and contradicts our assumption that the solution is global. This ends the proof. 2. By Lemmas 2.10 and 5.1, u(t) ∈ A + 1,1 for any t ∈ [0, T * ). So, thanks to the assumption (2.6), we have Thus, u(t) is bounded inḢ 1 . Precisely sup t∈(0,T * ) Moreover, since ∂ t ( u(t) 2 ) = −K 1,0 (u) < 0, the L 2 norm of u is decreasing and so u(t) ≤ u 0 . Thus sup t∈(0,T * ) Then, u is global. Let us start with some auxiliary results.
Proof of Theorem 2.8.
This contradiction finishes the proof of the claim. Thus, Q < −8δ. Integrating twice, Q becomes negative for some positive time. This absurdity closes the proof.
Moreover, since the L 2 norm of u is conserved, we have sup t∈(0,T * ) Thus, u is global. This ends the proof.