On the blow-up results for a class of strongly perturbed semilinear heat equations

We consider in this work some class of strongly perturbed for the semilinear heat equation with Sobolev sub-critical power nonlinearity. We first derive a Lyapunov functional in similarity variables and then use it to derive the blow-up rate. We also classify all possible asymptotic behaviors of the solution when it approaches to singularity. Finally, we describe precisely the blow-up profiles corresponding to these behaviors.


Introduction
We are interested in the following nonlinear parabolic equation: where u is defined for (x, t) ∈ R n × [0, T ), p is a sub-critical nonlinearity, 1 < p, (n − 2)p < n + 2.
By standard results, the problem (1) has a unique classical solution u(x, t) in L ∞ (R n ), which exists at least for small times. The solution u(x, t) may develop Email address: vtnguyen@math.univ-paris13.fr (V. T. Nguyen) singularities in some finite time. We say that a function u : R n × [0, T ) → R is a solution of (1) if u solves (1) and satisfies u, u t , ∇u, ∇ 2 u are bounded and continuous on R n × [0, τ ], ∀τ < T. (4) It is said that u(x, t) blows up in a finite time T < +∞ if u(x, t) satisfies (1), (4) and lim t→T u(t) L ∞ (R n ) = +∞.
Here we call T the blow-up time of u(x, t). In such a blow-up case, a point x 0 ∈ R n is called a blow-up point of u(x, t) if and only if there exist (x n , t n ) → (x 0 , T ) such that |u(x n , t n )| → +∞ as n → +∞.
Consider v a positive blow-up solution of the associated ODE of (1). It is clear that v is given by , v(T ) = +∞, for some T > 0. (5) Since the blow-up solution of (5) satisfies (see Lemma A. 1) it is natural to ask whether the blow-up solution u(t) of (1) has the same blow-up rate as v(t) does. More precisely, are there constants c, C > 0 such that By a simple argument based on Duhamel's formula, we can show that the lower bound in (7) is always satisfied (see [19]). For the upper blow-up rate estimate, it is much less simple and requires more work. Practically, we define for all x 0 ∈ R n (x 0 may be a blow-up point of u or not) the following similarity variables introduced in Giga and Kohn [4,5,6]: Hence w x0,T satisfies for all s ≥ − log T and for all y ∈ R n : Here, we say that w : R n × [− log T, +∞) → R is a solution of (9) if w solves (9) and satisfies w, w s , ∇w, ∇ 2 w are bounded and continuous on R n × [− log T, S], ∀S < +∞.
We can see that the study of u in the neighborhood of (x 0 , T ) is equivalent to the study of the long-time behavior of w x0,T and each result for u has an equivalent formulation in term of w x0,T . In particular, the proof of the upper bound in (7) is now equivalent to showing that there exists a timeŝ ≥ − log T large enough such that w x0,T (s) L ∞ (R n ) ≤ C, ∀s ≥ŝ.
We remark that the perturbation term added to equation (9) satisfies the following inequality, for some C 0 > 0 and s 0 > 0 (see Lemma A.2 for a proof of this fact).
When h ≡ 0, Giga and Kohn proved (12) in [5] for 1 < p < 3n+8 3n−4 or for non-negative initial data (so that the solution is positive everywhere) with subcritical p. Estimate (12) is extended for all p satisfying (2) without assuming non-negativity for initial data u 0 by Giga, Matsui and Sasayama in [7]. The proof written in [7] is strongly based on the existence of the following Lyapunov functional: Based on this functional, some energy estimates related to this structure and a bootstrap argument given in [14], the authors in [7] have established the following key integral estimate sup s≥s ′ s+1 s w x0,T (s) (p+1)q L p+1 (BR) ds ≤ C q,s ′ , ∀q ≥ 2, s ′ > − log T.
Since this estimate holds for all q ≥ 2, we obtain an upper bound for w x0,T which yields (12).
When h ≡ 0, we wonder whether a perturbation of the method of [7] would work for our problem. A key step is to find a Lyapunov functional for equation (9). Following the method introduced by Hamza and Zaag in [9,8] where γ = 8C 0 with H(z) = z 0 h(ξ)dξ. With this introduction, we derive that the functional J [w] is a decreasing function of time for equation (9), provided that s is large enough. More precisely, we have the following: Theorem 1 (Existence of a Lyapunov functional for equation (9)). Let a, p, n, M be fixed, consider w a solution of equation (9) satisfying (11). Then there existŝ 0 =ŝ 0 (a, p, n, M ) ≥ s 0 andθ 0 =θ 0 (a, p, n, M ) such that if θ ≥θ 0 , then J satisfies the following inequality, for all s 2 > s 1 ≥ max{ŝ 0 , − log T }, As mentioned above, the existence of this Lyapunov functional J is a crucial step in the derivation of the blow-up rate for equation (1). Indeed, with the functional J and some more work, we are able to adapt the analysis in [7] for equation (1) in the case h ≡ 0 and get the following result: Theorem 2 (Blow-up rate for equation (1)). Let a, p, n, M be fixed, p satisfy (2). There existsŝ 1 =ŝ 1 (a, p, n, M ) ≥ŝ 0 such that if u is a blow-up solution of equation (1) with a blow-up time T , then (i) for all s ≥ s ′ = max{ŝ 1 , − log T }, where w x0,T is defined in (8) and C is a positive constant depending only on n, p, M and a bound of w x0,T (ŝ 0 ) L ∞ .
(ii) For all t ∈ [t 1 , T ) where t 1 = T − e −s ′ , Remark 1. The proof of Theorem 2 is far from being a straightforward adaptation of [7]. Indeed, three major difficulties arise in our case and make the heart of our contribution: -the existence of a Lyapunov functional in similarity variables (see Theorem 1 above), -the control of the L 2 -norm in terms of the energy (see (ii) of Proposition 8, where we rely on a new blow-up criterion greatly simplifying the approach in [5]), -the proof of a nonlinear parabolic result (see Proposition 12 below).
The estimate obtained in Theorem 2 is a fundamental step in studying the asymptotic behavior of blow-up solutions. When h ≡ 0, Giga and Kohn in [5,6] (see also [4]) obtained the following result: For a given blow-up point x 0 , it holds that lim where κ = (p − 1) 1 p−1 , uniformly on compact subsets of R n . The result is pointwise in x 0 . Besides, for a.e. y, lim s→+∞ ∇w x0,T (y, s) = 0.
For our problem, when h ≡ 0 and h is given in (3), we also derive an analogous result on the behavior of w x0,T as s → +∞. We claim the following: Theorem 3 (Behavior of w x0,T as s → +∞). Let a, p, n, M be fixed, p satisfy (2). Consider u(t) a solution of equation (1) which blows up at time T and x 0 a blow-up point. Then holds in L 2 ρ (L 2 ρ is the weighted L 2 space associated with the weight ρ (10)), and also uniformly on each compact subset of R n .
Up to changing u 0 in −u 0 and h in −h, we may assume that w → κ in L 2 ρ as s → +∞. Let us consider φ a positive solution of the associated ordinary differential equation of equation (9 (see Lemma A.3 for a proof of the existence of φ). Let us introduce v x0,T = w x0,T − φ(s), then v x0,T (y, s) L 2 ρ → 0 as s → +∞ and v x0,T (or v for simplicity) satisfies the following equation: where L = ∆ − y 2 · ∇ + 1 and ω, F , H satisfy (see the beginning of Section 3 for the proper definitions of ω, F and G).
Since the linear part will play an important role in our analysis, let us point out its properties. The operator L is self-adjoint on L 2 ρ (R n ). Its spectrum is given by and it consists of eigenvalues. The eigenfunctions of L are derived from Hermite polynomials: -For n = 1, the eigenfunction corresponding to 1 − m 2 is -For n ≥ 2: we write the spectrum of L as For m = (m 1 , . . . , m n ) ∈ N n , the eigenfunction corresponding to 1 − |m| 2 is where h m is defined in (23).
By studying the behavior of v as s → +∞, we obtain the following result: Theorem 4 (Classification of the behavior of w as s → +∞). Consider u(t) a solution of equation (1) which blows-up at time T and x 0 a blow-up point. Let w(y, s) be a solution of equation (9). Then one of the following possibilities occurs: i) w(y, s) ≡ φ(s), ii) There exists l ∈ {1, . . . , n} such that up to an orthogonal transformation of coordinates, we have iii) There exist an integer number m ≥ 3 and constants c α not all zero such that The convergence takes place in L 2 ρ as well as in C k,γ loc for any k ≥ 1 and some γ ∈ (0, 1).
Remark 2. Applying our result to a space-independent solution of (9), we get the uniqueness of the solution of the ODE (21) that converges to κ as s → +∞.
Remark 3. Since both the perturbed (h ≡ 0) and the unperturbed (h ≡ 0) cases in equation (1) share the same convergence stated in Theorem 4, we wonder whether the perturbation h may have an influence on further terms of the expansion of w. From our result, if case (ii) occurs, we see no difference in the following term of the expansion. On the contrary, if case (i) or (iii) occurs, with which is clearly different from the unperturbed case when in case (i), we have w ≡ κ and case (iii), we have w − κ = O(e −s ), (see [10], [18]).

Remark 4.
If we linearize w around κ, which is an explicit profile, we then fall in logarithmic scales µ = 1 | log ǫ| with ǫ = T − t. Further refinements in this direction should give an expansion of w − κ in terms of powers of µ, i.e in logarithmic scales of ǫ. Therefore, we can not reach significantly small error terms in the expansion of the solution w as (iii) of Theorem 4 describes. In order to escape this situation, a relevant approximation is required in order to go beyond all logarithmic scales, i.e approximations up to lower order terms such as ǫ α for some α > 0. Our idea to capture such relevant terms is to abandon the explicit profile obtained as a first order approximation, namely κ, and take an implicit profile function as a first order description of the singular behavior, namely φ(s) introduced in (21) and (22). A similar idea was used by Zaag [20] where the solution was linearized around a less explicit profile function in order to go beyond all logarithmic scales. For our problem, we particularly take φ(s) as the implicit profile function, which is a solution of the associated ODE of equation (9) in w such that φ(s) → κ as s → +∞. By linearizing the solution w around φ, we can get to error terms of polynomial order ǫ ( m 2 −1) , as stated in (iii) of Theorem 4.
If case (ii) in Theorem 4 holds, we then recover the same expansion as in the unperturbed case (h ≡ 0). On the contrary, if case (i) or (iii) occurs, then w(y, s) − κ ∼ C ′ 0 (p, q)e −λs as s → +∞. Moreover, if case (iii) in Theorem 4 holds, we have new terms in the expansion of w which was not available in the unperturbed case, namely where C k , k = 1, 2, . . . , K are some constants depending on p and q, and K ∈ N is the integer part of 1 λ m 2 − 1 . In the last section, we will extend the asymptotic behavior of w obtained in Theorem 4 to larger regions. Particularly, we claim the following: where with l the same as in ii) of Theorem 4. ii) if iii) of Theorem 4 occurs, then m ≥ 4 is even, and where with c α the same as in Theorem 4.
Let us mention briefly the structure of the paper. In Section 2, we prove the existence of Lyapunov functional for equation (9) (Theorem 1), we then get Theorem 2 and Theorem 3. In Section 3, we follow the method of [3] and [18] to prove Theorem 4. Finally, the section 4 is devoted to the proof of Theorem 5.
Acknowledgement: The author is grateful to H. Zaag for his dedicated advice, suggestions and remarks during the preparation of this paper.

A Lyapunov functional
This section is divided in four subsections: we first prove the existence of a Lyapunov functional for equation (9) (Theorem 1); after that, we derive a blowup criterion for equation (9) and some energy estimates based on this Lyapunov functional. Following the method of [7], we prove the boundedness of solution in similarity variables which determines the blow-up rate for solution of (1) (Theorem 2). Finally, we derive the limit of w as s → +∞, which concludes Theorem 3.
In what follows, we denote by C i , i = 0, 1, . . . positive constants depending only on a, n, p, M , and by L q ρ (Ω) the weighted L q (Ω) space endowed with the norm and by H 1 ρ (Ω) the space of function ϕ ∈ L 2 ρ (Ω) satisfying ∇ϕ ∈ L 2 ρ (Ω), endowed with the norm We denote by B R (x) the open ball in R n with center x and radius R, and set B R := B R (0).

Existence of a Lyapunov function
In this part, we aim at proving that the functional J defined in (16) is a Lyapunov functional for equation (9). Note that that functional is far from being trivial and it is our main contribution. We first claim the following lemma: Lemma 6. Let a, p, n, M be fixed and w be solution of equation (9) satisfying (11). There existss 0 =s 0 (a, p, n, M ) ≥ s 0 such that the functional of E defined in (17) satisfies the following inequality, for all s ≥ max{s 0 , − log T }, , C 0 is introduced in (13) and C is a positive constant depending only on a, p, n, M .
Let us first derive Theorem 1 from Lemma 6 which will be proved later.
Choosing θ large enough such that Ce This implies inequality (18) and concludes the proof of Theorem 1, assuming that Lemma 6 holds.
It remains to prove Lemma 6 in order to conclude the proof of Theorem 1.
Proof of Lemma 6 . Multiplying equation (9) with w s ρ and integrating by parts: For the last term of the above expression, denoting H(z) = z 0 h(ξ)dξ, we write in the following: This yields From the definition of the functional E given in (17), we derive a first identity in the following: A second identity is obtained by multiplying equation (9) with wρ and integrating by parts: Using again the definition of E given in (17), we derive the second identity in the following: From (28), we estimate From (13) and using the fact that |w| ≤ |w| p+1 + 1, we obtain for all s ≥ s 0 , where Using the fact that |w s w| ≤ ǫ(|w s | 2 + |w| p+1 ) + C 2 (ǫ) for all ǫ > 0 and (30), we obtain with C 4 = C3 2 + 1 8 . Substituting (32) into (31) yields (27) withs 0 = max{s 0 , s 1 }. This concludes the proof of Lemma 6. Since we have already showed that Theorem 1 is a direct consequence of Lemma 6, this is also the conclusion of Theorem 1.

A blow-up criterion for the equation in similarity variables
In this part, we give a new blow-up criterion for equation (9). Then, we will use it to control the L 2 -norm in terms of the energy (see (ii) of Proposition 8). We claim the following: Lemma 7. Let a, p, n, M be fixed and w be solution of equation (9) satisfying (11). If there existss 1 =s 1 (a, p, n, M ) then w is not defined for all (y, s) ∈ R n × [s, +∞).
Proof. We proceed by contradiction and suppose that w is defined for all s ∈ [s, +∞). From definition of J in (16) and from (29), (30), we have for all s ≥ s 0 , We take s 1 large enough such that Then, using Jensen's inequality and noting that e − γ a−1 s 1−a ≤ 1 for all s > 0, we get from (34) the following: for all s ≥ max{0, s 0 , s 1 }, Setting f (s) = R n |w(y, s)| 2 ρdy, A = −4J [w](s) and B = p−1 p+1 , then using the fact that J is decreasing in time to get that 2 , ∀s ≥s.

The hypothesis reads
By a direct integration, we obtain which is a contradiction and Lemma 7 is proved.
As a consequence of Theorem 1 and Lemma 7, we obtain the following estimates which will be useful for getting Theorem 2: Proposition 8. Let w be solution of equation (9) satisfying (11), it holds that Proof. The upper and lower bounds of E, (i) and (ii) obviously follow from Theorem 1 and Lemma 7 (in fact, since w is defined for all s ≥s 1 , condition (33) is never satisfied). (iii) By definition of E given in (17) and (30), we get for all s ≥ max{s 0 , − log T }, Let s 1 large enough such that for all s ≥ s 1 , C 0 s −a ≤ 1 2(p−1) , then for all s ≥ max{s 0 , s 1 , − log T }, This follows that for all s ≥ max{s 0 , s 1 , − log T }, (17), (29) and (30), we have ∀s ≥ max{s 0 , − log T }, Using (iii), we have for all s ≥s 3 , Let s 2 large enough such that 4C0J2(p+1) for all s ≥ s 2 and noting that E(s) is bounded from above, we obtain for all s ≥ max{s 2 ,s 2 }, which follows (iv). Since (v) and (vi) follows directly from (i) and (iii), (iv), we end the proof of Proposition 8.

Boundedness of the solution in similarity variables
This section is devoted to the proof of Theorem 2, which is a direct consequence of the following theorem: where C is a positive constant depending only on n, p, M, R and a bound of Let us show that Theorem 2 follows from Theorem 9.
Proof of Theorem 2 admitting Theorem 9. We have from (36) that with C independent on x 0 ∈ R n . Therefore, we get from (8) that which is the conclusion of Theorem 2, assuming that Theorem 9 holds.
Following the method in [7], the proof of Theorem 9 requests the following key integral estimate: Lemma 10 (Key integral estimate). Let a, p, n, M be fixed and w be solution of equation (9) satisfying (11). For all q ≥ 2 and R > 0, there existsŝ 2 ≥s 3 and a positive constant K q such that, where K q depends only on J 0 , Q 0 , a, n, p, q, R,ŝ 2 .
Let us first show that how Theorem 9 follows from Lemma 10, then we will prove it later. In order to derive uniform bound in Theorem 9 for all p satisfying (2), we need two following techniques. The first one is an interpolation result from Cazenave and Lions [1]: for λ < λ 0 . The positive constant C depends only on α, β, γ, δ, n and R.
The second one is an interior regularity result for a nonlinear parabolic equation: where and µ 1 , µ 2 and µ 3 are uniformly bounded in t, then there exists a positive constant C depending only on µ 1 , µ 2 , µ 3 , α ′ , β ′ , n, R and τ ∈ (0, 1) such that Proof. Since the argument of the proof is analogous as in the corresponding part in [12], we then leave the proof to Appendix B.1.
Let us now use Lemma 10 to derive the conclusion of Theorem 9, then we will prove it later.
Let us now give the proof of Lemma 10 in order to complete the proof of Theorem 9 and Theorem 2 also. To this end, let ψ ∈ C 2 (R n ) be a bounded function, we introduce the following local functional, which is a perturbed version of the function of [7], We get the following bound on the local functional E ψ : Proposition 13. Let a, p, n, M be fixed and w be solution of equation (9) satisfying (11).
wheres 3 is given in Proposition 8 and Q ′ , K ′ depend on a, p, n, M , ψ 2 L ∞ , ∇ψ 2 L ∞ and J 0 . Proof. The proof is essentially the same as the corresponding part in [7], except for the control of the last term in (42). Since that control is a bit long and technical, we leave the proof to B.2.
Let R > 0, we fix ψ(y) so that it satisfies We claim the following: Let a, p, n, M be fixed and w be solution of equation (9) satisfying (11). Then there existss 5 ≥s 3 such that where Proof. From (30) and the definition of E ψ in (42), we have ∀s ≥ max{s 0 , s 1 }, where s 1 is large enough such that 2C 0 s −a ≤ 1 2(p+1) for all s ≥ s 1 . Thus, (45) follows from the lower bound of E ψ and the property of ψ. This ends the proof of Lemma 14.
Note from (i) and (iv) in Proposition 8 that (47) already holds in the case q = 2.
In order to derive (47) for all q ≥ 2, we need the following result: Lemma 15. Let a, p, n, M be fixed and w be solution of equation (9) satisfying (11). Then there existss 6 ≥s 3 such that where K 2 = K 2 (a, p, n, M, Q ′ , K ′ ).
Proof. Multiplying equation (9) with ψ 2 wρ, integrating over R n , using the definition of E ψ and estimate (30), we have Using (46), then taking s 2 large such that 4(p+1) 2 C0 (p−1)s a ≤ 1 2 and noting that E is bounded, we have for all s ≥ max{s 0 , s 1 , s 2 }, for a proof of this fact). Hence, we have for all s ≥ max{s 0 , s 1 , s 2 }, Thus, (48) follows from the property of ψ, and Lemma 15 is proved.
Since the estimate (47) already holds in the case q = 2, we now use a bootstrap argument in order to get (47) for all q ≥ 2.
Proof of (47) for all q ≥ 2 by a bootstrap argument. This part is the same as in [7]. We give it here for the sake of completeness. Suppose that (47) holds for some q ≥ 2, let us show that (47) holds for allq ∈ [q, q + ǫ] for some ǫ > 0 independent from q. We start with Holder's inequality, Using (37) and applying Lemma 11, we obtain Let us now bound ψw s L λ ′ ρ (B2R) . We remark that for q large then λ approaches to p + 1 and λ ′ approaches to p 1 = p+1 p . Let f = ψw s and make use Holder's inequality, From now on, we take λ ≥ 2 and fix θ = (λ−2)(p+1) (1−θ)q and use Holder's inequality in time to G, we obtain where we used (i) in Proposition 8. Let us bound G 1 . To this end, we use the L p −L q estimate for the heat equation (see Lemmas 6.3 and 6.4 in [7]) to get s+1 s ψw s Thus, inequality (47) is valid for allq ∈ [q, q + 2 p+1 ]. Repeating this argument, we would obtain that (47) holds for all q ≥ 2. This concludes the proof of Lemma 10, Theorem 9 and Theorem 2 too.

Limit of w as s → +∞
This section is devoted to the proof of Theorem 3. Note in the unperturbed case (h ≡ 0) that Theorem 3 was proved in [6] (see also [4], [5]). The proof is divided into two steps. The first step is to show that the limit of solution in similarity variables exists and belongs to the set of solutions of the following equation, Then, by using a nondegeneracy result (Lemma 19), the blow-up criterion (Lemma 7) and suitable energy arguments, we shall show that the possibility of w a → 0 as s → +∞ is excluded if a is a blow-up point. Let us restate Theorem 3 in below: Proposition 16 (Limit of w as s → +∞). Let a, p, n, M be fixed, p be a sub-critical non-linearity given in (2). Consider u(t) a solution of equation (1) which blows up at time T and a a blow-up point. Then Before going into the proof of Proposition 16, let us first derive some elementary results. The first one concerns the stationary solutions in R n of equation (50). Particularly, we have the following: Lemma 17 (Stationary solutions, Giga and Kohn [4]). Let p satisfy (2), then all bounded solutions of (50) are constants: w ≡ 0 or w ≡ ±κ.
Proof. The proof is given in Proposition 2 of [4]. For the reader's interest, we mention that the proof relies on a clever use of multiplying factors, together with a Pohozaev technique, resulting in the following identity: From (51) and the fact that p is Sopolev subcritical, it follows that n p+1 − 2−n 2 > 0 and 1 2 − 1 p+1 > 0, hence ∇w ≡ 0. This implies that w is actually a constant. This concludes the proof of Lemma 17.
Let us now give the proof of Proposition 16.
Proof of Proposition 16. Consider a a blow-up point and write w instead of w a for simplicity. By Lemma 18 and equation (9), we see that |w s (y, s)| ≤ C(|y| + 1) for some C > 0. Therefore, w, ∇w, ∇ 2 w and w s are bounded for all |y| ≤ R and s ≥ s ′ for some R > 0 and s ′ ∈ R. Let {s j } be a sequence tending to +∞ and w j (y, s) = w(y, s + s j ). By the Arzela-Ascoli theorem, there is a subsequence of s j (still denoted s j ) such that w j converges uniformly on compact sets to some w ∞ , ∇w j → ∇w ∞ , ∆w j → ∆w ∞ and w js → w ∞ s . On the other hand, by (i) and (vi) of Proposition 8, we see that as j → +∞, This implies that w ∞ s = 0 and w ∞ satisfies (50). Hence, by Lemma 17, w ∞ ≡ 0 or w ∞ ≡ ±κ. It remains to show that w(·, s j ) 0 as j → +∞. We proceed by contradiction. Let us assume that w(·, s j ) → 0 as j → +∞. We observer that if w(·, s j ) → 0, then by the definition of J given in (16), the bound of w and ∇w and dominated convergence, then J [w](s j ) → 0. Since J is a Lyapunov functional, it follows that the whole sequence Let b ∈ R n , then by (53), we have w b (y, s) and ∇w b (y, s) are bounded for all y ∈ R n and s ≥ s ′ . We now use the interpolation inequality which reads where θ ∈ (0, 2 n+2 ) if n ≥ 2 and θ = 1/2 if n = 1. By Lemma 7, we see that w b (s) L 2 (BR) ≤ C(p) J [w b ](s) 1 p+1 for all s ≥s 1 . Hence, , ∀s ≥s 1 .
Consider some ǫ > 0 small. From (56), there is s ′ (ǫ) such that J [w](s) ≤ ǫ for all s ≥ s ′ (ǫ). Therefore, by continuity depending of J [w b ](s) on b and the monotonicity of J [w b ](s) in time s, we infer that J [w b ](s) ≤ 2ǫ for all s ≥ s ′ and |b − a| small. This implies that |w b (0, s)| ≤ ǫ ′′ for all s ≥ s ′ , or |u(b, t)| ≤ ǫ ′′ (T − t) − 1 p−1 for (b, t) close to (a, T ), where ǫ ′′ = ǫ ′′ (ǫ) → 0 as ǫ → 0. Thus, a is not a blow-up point by Lemma 19, and this is a contradiction. Therefore, this concludes the proof of Proposition 16 and the proof of Theorem 3 also.
3. Classification of the behavior of w as s → +∞ in L 2 ρ This section is devoted to the proof of Theorem 4. Consider a a blow-up point and write w instead of w a for simplicity. From Theorem 3 and up to changing the signs of w and h, we may assume that w(y, s) − κ L 2 ρ → 0 as s → +∞, uniformly on compact subsets of R n . As mentioned in the introduction, by setting v(y, s) = w(y, s)−φ(s) (φ is a positive solution of (21) such that φ(s) → κ as s → +∞), we see that v(y, s) L 2 ρ → 0 as s → +∞ and v solves the following equation: where L = ∆ − y 2 · ∇ + 1 and ω, F , H are given by We remark from (22) and (13) that Let us introduce for all y ∈ R n , for all s ∈ [− log T, +∞), (note that β(s) → 1 as s → +∞). By multiplying equation (57) to β(s), we find the following equation satisfied by V : Since w(s) L ∞ ≤ C from Theorem 2, we may use a Taylor expansion, (13), (22) and the fact that β(s) = 1 + O 1 s a−1 as s → +∞ to write (see Lemma C.1 for the proof of (62), and note that (61) follows from (62)).
Since the eigenfunctions of L constitute a total orthonormal family of L 2 ρ , we can expand V as follows: where π k (V ) is the orthogonal projector of v on the eigenspace associated to where H 2 (y) = (H 2,ij , i ≤ j), with H 2,ii = h 2 (y i ) and We claim that Theorem 4 is a direct consequence of the following: Proposition 20 (Classification of the behavior of V as s → +∞). One of the following possibilities occurs: i) V (y, s) ≡ 0, ii) There exists l ∈ {1, . . . , n} such that up to an orthogonal transformation of coordinates, we have iii) There exist an integer number m ≥ 3 and constants c α not all zero such that The convergence takes place in L 2 ρ as well as in C k,γ loc for any k ≥ 1 and γ ∈ (0, 1).

Remark 8.
Let us insist on the fact that the linearizing of w around κ would generate some terms of the size 1 s a , and prevent us from reaching exponentially small terms.
Let us first derive Theorem 4 assuming Proposition 20 and then we will prove it later.
Proof of Theorem 4 assuming that Proposition 20 holds. By the definition (59) of V , we see that i) of Proposition 20 directly follows that v(y, s) ≡ φ(s) which is i) of Theorem 4. Using ii) of Proposition 20 and the fact that β(s) = 1 + O( 1 s a−1 ) as s → +∞, we see that as s → +∞, This concludes the proof of Theorem 4 assuming that Proposition 20 holds.
The proof of Proposition 20 will be very close to that in [3] and [18], thanks to (61) and (62). It happens that the proofs written in Filippas, Kohn, Liu, Herrero and Velázquez [2], [3], [10], [18] in the unperturbed case (h ≡ 0) hold for equation (60) under the general assumptions (61) and (62). For that reason, we only give the sketch of the proof below and refer to these papers for details of the proofs. Following [3] and [18], we divide the proof into 3 steps which are given in separated subsections: -Step 1: deriving the fact that either V + (s) for some µ > 0. -Step 2: assuming that V (y, s) L 2 ρ ∼ V null (y, s) L 2 ρ , we find an equation satisfied by V null (s) as s → +∞. Solving this equation, we find that V (s) L 2 ρ behaves like 1 s as s → +∞. Using this information, we can get a more accurate equation for V null (s) as s → +∞ and then ii) of Proposition 20 follows. -Step 3: assuming V (s) L 2 ρ = O(e −µs ) for some µ > 0 as s → +∞, we derive i) or iii) of Proposition 20.

Finite dimension reduction of the problem.
We claim the following proposition: Proposition 21 (Competition between V + , V − and V null ). As s → +∞, Proof. Let us denote then the following lemma is claimed: Lemma 22. Let ǫ > 0, there exists s * = s * (ǫ) ∈ R such that for all s ≥ s * , whereȲ (s) = Y (s) + r(s) with r(s) = |y| Proof. From the fact that |F (V, s)| ≤ CV 2 for s large, the proof is the same as the proof of Theorem A, pages 842-847 in Filippas and Kohn [2].
Proof. The original proof is due to Filippas and Kohn [2]. For this particular statement, see Lemma A.1, page 3425 [13] for the proof.
Since V (s) L ∞ loc → 0 as s → +∞, we have X(s),Ȳ (s), Z(s) → 0 as s → +∞. Thus, Lemma 23 applies to X(s),Ȳ (s), and Z(s) and yields the desired result (use the remark after the statement). This ends the proof of Proposition 21.

Deriving conclusion ii) of Proposition 20
In this part, we recall from Filippas and Liu the proof of ii) of Proposition 20. We focus on the case ii) of Proposition 21, namely that and show that it leads to case ii) of Proposition 20. We first claim the following proposition: .., n} and as s → +∞, ii) There exist a symmetric n × n matrix A(s) such that for all s ∈ R, and where c 1 , c 2 are some positive constant and A stands for any norm on the space of n × n symmetric matrices. Moreover, Proof. Let us remark that ii) follows directly from i). Here, one has to use (62) which is more accurate than (61), in order to isolate the O(V 2 ) term in the nonlinear term. Using properties of Hermites polynomials, we may project that term and obtain (69).
In the next step, we show that although we can not derive directly from (69) the asymptotic behavior of V null (s), we can use it to show that V (s) L 2 ρ decays like 1 s as s → +∞. More precisely, we have the following proposition: for some positive constants c 1 and c 2 .
Since the proof of (73) is totally given in Section 3 of Filippas and Liu [3], we just give its steps of the proof below. The following Lemma asserts that A(s) has continuously differential eigenvalues: Lemma 26 ( [16,11]). Suppose that A(s) is a n×n symmetric and continuously differentiable matrix-function in some interval I, then there exists continuously differentiable functions λ 1 (s), . . . , λ n (s) in I such that for all i ∈ {1, . . . , n}, for some orthonormal system of vector-functions Φ (1) (s), . . . , Φ (n) (s).
Using the fact that V (s) L 2 ρ decays like 1 s , we will show that V − (s)) L 2 and not only o( V null (s)) L 2 ρ ). This new estimate will be used then to derive a more accurate equation satisfied by V null . and where A(s) is given in (70).
Proof. The proof corresponds to Section 4 in [3]. Let us mention that the proof relies on the following priori estimate of solutions of (60) shown by Herrero and Velázquez in [10]. Although they proved their result in the case N = 1, their proof holds in higher dimensions under the general assumption (61).
Let us now use 28 to derive conclusion ii) of Proposition 20. Using Lemma 26, we get from (77) that the eigenvalues of A(s) satisfy Therefore, Proposition 5.1 in [3] yields the existence of l ∈ {1, . . . , n} and a n × n orthonormal matrix Q such that Combining this with (70), it yields the behavior of V null (y, s) and V (y, s) announced in ii) of Proposition 20. The convergence in C k,γ loc follows from standard parabolic regularity (see section 5 in [3] for a brief demonstration). This completes the proof of ii) of Proposition 20.

Deriving conclusions i) and iii) of Proposition 20
In this part, we recall the proof given by Velázquez [18]. We focus on the case i) of Proposition 21, namely V (s) L 2 ρ = O(e −µs ) for some µ > 0, and we will show that it leads to either i) or iii) of Proposition 20. Let us start the first step. From equation (60), we write V (y, s) in the integration form where S L (s) is the linear semigroup corresponding to the heat-type equation ∂V = LV given by Let us fix a integer k 0 > 2 such that k0 2 − 1 < 2µ < k0+1 2 − 1 and write V (y, s) as follow: By a direct computation, we find that Using (82), we can bound the last term of the above expression by Ce −2µs . Hence, V (y, s) = |α|≤k0 (a α + β α )e (1− |α| 2 )s H α (y) + Q(y, s), where Q(y, s) L 2 ρ = O(e −2µs ). Since V (y, s) L 2 ρ = O(e −µs ), it requires a α + β α = 0 for |α| ≤ 2. Thus, we have two possibilities: if there exists an integer m ∈ [3, k 0 ] such that a α + β α = 0 for |α| = m and a α +β α = 0 for all |α| < m, then we obtain iii) of Proposition 20 for some m ∈ [3, k 0 ]. If this is not the case, we get V (y, s) L 2 ρ = O(e −2µs ). Using this new estimate and repeating the process in a finite number of steps, we may obtain either iii) of Proposition 20 for some m ≥ 3 or V (y, s) L 2 ρ = O(e −Rs ) for any R > 0. For the second case, we use the following nondegeneracy result from Herrero and Velázquez [10] in order to conclude that V (y, s) ≡ 0, which is i) of Proposition 20, Lemma 30 (Herrero and Velázquez [10]). Let V be a solution to equation (60). Assume that |V (y, s)| is bounded, and that for any R > 0 there exists Proof. Since the proof written in [10] holds under general assumption (61), we then refer the reader to Lemma 3.5, page 144 of [10] for detail of the proof.
Since the convergence in C k,γ loc for any k ≥ 1 and γ ∈ (0, 1) follows from a standard parabolic regularity, we end the proof of Proposition 20 here. This also concludes the proof of Theorem 4.

Bow-up profile for equation (1) in extended spaces regions
We give the proof of Theorem 5 in this section. Note that the derivation of Theorem 5 from Theorem 4 in the unperturbed case (h ≡ 0) was done by Velázquez in [17]. The idea to extend the convergence up to sets of the type s } is to estimate the effect of the convective term − y 2 · ∇w in the equation (9) in L q ρ spaces with q > 1. Since the proof of Theorem 5 is actually in spirit by the method given in [17], all that we need to do is to control the strong perturbation term in equation (9). We therefore give the main steps of the proof and focus only on the new arguments. The proof will be separated into two parts: the first part concerns case ii) in Theorem 4 and gives the asymptotic behavior of w in the y √ s variable, and the second part concerns case iii) in Theorem 4 and gives the asymptotic behavior of w in the ye −( 1 2 − 1 m )s variable. In Part 1, we stick to the method of Velázquez [17], whereas, in Part 2, where we work in the scale e −µs for µ > 0, we need new ideas to get rid of the term in the scale 1 s coming from the strong perturbation. Proposition 31 (Asymptotic behavior in the y √ s variable). Assume that w is a solution of equation (9) which satisfies ii) of Theorem 4. Then, for all Proof. Following the method in [17], we define q = w − ϕ, where Using Taylor's formula in (83) and ii) of Theorem 4, we find that Straightforward calculations based on equation (9) yield where ω(y, s) = p(ϕ p−1 − κ p−1 ) + e −s h ′ e s p−1 ϕ , Let K 0 > 0 be fixed, we consider first the case |y| ≥ 2K 0 √ s and then |y| ≤ 2K 0 √ s and make a Taylor expansion for ξ = y √ s bounded. Simultaneously, noticing from (13), we then obtain for all s ≥ s 0 , ω(y, s) ≤ C 1 s , Let Q = |q|, we then use the above estimates and Kato's inequality, i.e ∆f · sign(f ) ≤ ∆(|f |), to derive from equation (85) the following: for all K 0 > 0 fixed, there are C * = C * (K 0 , M 0 ) > 0 and a time s ′ > 0 large enough such that for all s ≥ s * = max{s ′ , − log T }, (86) Since the conclusion of Proposition 31 follows if we show that Let us now focus on the proof of (87) in order to conclude Proposition 31. For this purpose, we introduce the following norm: for r ≥ 0, q > 1 and f ∈ L q loc (R n ), Following the idea in [17], we shall make estimates on solution of (86) in the L 2,r(τ ) ρ norm where r(τ ) = K 0 e τ −s 2 ≤ K 0 √ τ . Particularly, we have the following: Lemma 32. Let s be large enough ands is defined by e s−s = s. Then for all τ ∈ [s, s] and for all K 0 > 0, it holds that C 0 = C 0 (C * , M 0 , K 0 ) and z + = max{z, 0}.
Proof. Multiplying (86) by α(τ ) = e τ s C * t dt, then we write Q(y, τ ) for all (y, τ ) ∈ R n × [s, s] in the integration form: where S L is the linear semigroup corresponding to the operator L.
Next, we take the L 2,r(K0,τ,s) ρ -norms both sides in order to get the following: Proposition 2.3 in [17] (with a slight modification for the estimate of J 3 ) yields Putting together the estimates on J i , i = 1, 2, 3, 4, we conclude the proof of Lemma 32.
We now use the following Gronwall lemma from Velázquez [17]: Lemma 33 (Velázquez [17]). Let ǫ, C, R and δ be positive constants, δ ∈ (0, 1). Assume that H(τ ) is a family of continuous functions satisfying Then there exist θ = θ(δ, C, R) and ǫ 0 = ǫ 0 (δ, C, R) such that for all ǫ ∈ (0, ǫ 0 ) and any τ for which ǫe τ ≤ θ, we have Applying Lemma 33 with H ≡ g, we see from Lemma 32 that for s large enough, If τ = s, then e s−s = s, r = K 0 √ s and By using the regularizing effects of the semigroup S L (see Proposition 2.3 in [17]), we then obtain as s → +∞, which concludes the proof of Proposition 31. Let us restate ii) of Theorem 5 in the following proposition: Proposition 34 (Asymptotic behavior in the ye −( 1 2 − 1 m )s variable). Assume that w is a solution of equation (9) and satisfies iii) of Theorem 4. Then, for all K > 0, , and m ≥ 4 is an even integer.
Proof. Note that the proof will proceed in the same way as for the proof of Proposition 31. Let us introduce q = w − ϕ, where with Note that Velázquez [17] takes ϕ = J, and if we do the same, we will obtain some terms in the scale of 1 s , much stronger than the e −µs scale that we intended to work in.
Let Q = |q| and use Kato's inequality, we obtain from (92) and from the above estimates that: for all K 0 > 0 fixed, there are C * = C * (K 0 , M 0 ) > 0 and a time s ′ > 0 large enough such that for all s ≥ s * = max{s ′ , − log T }, (93) We claim the following: Lemma 35. Let s be large enough ands = 2s m . Then for all τ ∈ [s, s], τ −s ≥ 2 and for all K 0 > 0, it holds that Proof. Proceeding as in the proof of Lemma (32), we write One can show that fors large enough (see Proposition 2.4 in [17]), It remains to show that m is even. Indeed, from (88), we can see that if m is not even, there would exist ξ 0 ∈ R n such that w ξ 0 e ( 1 2 − 1 m )s , s → ψ m (ξ 0 ) → +∞ as s → +∞, which contradicts the fact that w is bounded as stated in (19). Therefore, m must be even. This concludes the proof of Proposition 34 and Theorem 5 too.

A. Appendix A
The following lemma shows the asymptotic behavior of the solution of the associated ODE of equation 1: Let v be a blow-up solution of the following ordinary differential equation: Proof. Divide (A.1) by v p and note that h(v) v p → 0 as v → +∞, we see that for all ε > 0, there exists a number δ = δ(ǫ) > 0 such that Solving (A.2) with noting that v(T ) = +∞ yields This concludes the proof of Lemma A.1.
The following lemma gives us an estimation of the perturbation term in equation (9): Let h be the function defined in (3), then it holds that where C = C(a, p, µ, M ) > 0 andŝ =ŝ(a, p) > 0 such that log s s ≤ p a(p−1) for all s ≥ŝ.
≤ s −a for all s ≥ŝ, we have the conclusion. This ends the proof of Lemma A.2.
The following lemma shows us the existence of solutions of the associated ODE of equation (9): Lemma A.3. Let φ be a positive solution of the following ordinary differential equation: Then φ(s) → κ as s → +∞ and φ(s) is given by where C 0 = µ p−1 2 a and b j = (−1) j j−1 i=0 (a + i). Proof. By the following transformation

B.1. Proof of Proposition 12
We give the proof of Proposition 12 here.
Proof. The idea of the proof is given in Ladyženskaja and al. [12]. Note that we still get interior regularity even if we know nothing about the initial or boundary data. Indeed, let τ ∈ (0, 1) and fix t 0 such that t 0 − τ > 0, we denote Q τ (t 0 ) = B R/2 × (t 0 − τ, t 0 ) ⊂ Q R , and let ϕ(x, t) be a smooth function defined in Q R such that 0 ≤ ϕ(x, t) ≤ 1 and ϕ(x, t) = 0 for all (x, t) ∈ Q R \ Q τ (t 0 ). Let Then, multiplying equation (38) by v k ϕ 2 and integrating over Q τ (t 0 ), we find that Using the assumption |F | ≤ g(|v| + 1) and some elementary inequalities with noticing that ϕ(·, t 0 − τ ) = 0, we then obtain For the last term in the right-hand side (denote by I), we use Holder's inequality and (39), which reads From pages 184 and 185 in [12], we have the following interpolation identity: where ǫ ∈ (0, 1), r ≥ 2, β > 0 are constants, Since θ k ≤ τ R β1/α1 , we can take τ small enough such that Then from (B.1), we have By Remark 6.4, page 109 and Theorem 6.2, page 103 in [12], we know that if v satisfies (B.2) for any k ≥ 1, then for all (x, Analogous arguments with the function −v would yield the same estimate. Since µ 1 , µ 2 and µ 3 are uniformly bounded in t 0 , this implies that estimate (B.3) holds for all (x, t) ∈ B R/4 ×(τ /2, +∞). This concludes the proof of Proposition 12.

B.2. Proof of Proposition 13
We prove Proposition 13 here. Let us first derive the upper bound for E ψ .
Proof of the upper bound for E ψ . Multiplying equation (9) with ψ 2 w s and integrating on R n yield We derive the following identity from the definition (42)  where C = C(a, p, n, M, ψ 2 L ∞ ). Using (iii) and (iv) of Proposition 8, we see that d ds E ψ [w](s) ≤ C 1 1 + w s L 2 ρ (R n ) , ∀s ≥s 3 , (B.5) where C 1 = C 1 a, p, n, N, J 3 , J 4 , ψ 2 L ∞ , ∇ψ 2 L ∞ and J i is introduced in Proposition 8.
From the definition of E ψ given in (42), we have This concludes the proof of the upper bound for E ψ .
It remains to prove the lower bound in order to conclude the proof of Proposition (8).
Let g(s) = 2E ψ + C 3 and f (s) = 1 2 R n ψ 2 |w| 2 ρdy. Using Jensen's inequality, we have We therefore obtain for all s ≥ S 1 = max{S,s 3 }, f ′ (s) ≥ −g(s) + C 4 f (s) h(s)ds ≤ C 6 by using (i) of Proposition 8, where C 5 , C 6 are some positive constants. We claim that the function of g is bounded from below by some constant M . Arguing by contradiction, we suppose that there exists a time s * ≥ S 1 such that g(s * ) ≤ −M . Then for all s ≥ s * , we write g(s) = g(s * ) + On the other hand, we know that the solution of the following equation Thus, we fix M = m + C 5 T * + 1 to get M − m − C 5 (s − s * ) ≥ 1 for all s ∈ [s * , s * + T * ]. Therefore, f blows up in some finite time before s * + T * . But this contradicts with the existence global of w. This follows (43) and we complete the proof of Proposition 13.

C. Appendix C
We claim the following: Lemma C.1 (Estimate onF ). For s large enough, we have where C = C(a, p, M, µ) > 0.