ON THE EFFECT OF HIGHER ORDER DERIVATIVES OF INITIAL DATA ON THE BLOW-UP SET FOR A SEMILINEAR HEAT EQUATION

. This paper concerns the blow-up problem for a semilinear heat equation where ∂ t = ∂/∂t , p > 1, N ≥ 1, Ω ⊂ R N , u 0 is a bounded continuous function in Ω. For the case u 0 ( x ) = λϕ ( x ) for some function ϕ and a suﬃciently large λ > 0, it is known that the solution blows up only near the maximum points of ϕ under suitable assumptions. Furthermore, if ϕ has several maximum points, then the blow-up set for (P) is characterized by ∆ ϕ at its maximum points. However, for initial data u 0 ( x ) = λϕ ( x ), it seems diﬃcult to obtain further information on the blow-up set such that eﬀect of higher order derivatives of initial data. In this paper, we consider another type large initial data u 0 ( x ) = λ + ϕ ( x ) and study the relationship between the blow-up set for (P) and higher order derivatives of initial data.

The location of the blow-up set for problem (1) has been studied intensively by many mathematicians. See, for example, [1,2,7,12,13], [17]- [20] and a survey [11]. In particular, the blow-up set for (1) with small diffusion is studied in [5] and [6]. (See [3] and [4] for more general nonlinearity.) More precisely, they characterized the location of the blow-up set of the solution of under the zero Dirichlet boundary condition. where > 0 is a sufficiently small parameter and ϕ is a nonnegative bounded continuous function. Here we remark that, if u solves (2) together with zero Dirichlet boundary condition, then the function u (x, t) := −1/(p−1) u(x, −1 t) solves (1) with u 0 replaced by −1/(p−1) u(x, 0). Therefore the heat equation with small diffusion is equivalent to the one with large initial data, and the results obtained in [5] and [6] also hold for (1) replaced by u 0 (x) = λϕ(x) with sufficiently large parameter λ. This relation together with the results in [5] and [6] gives the location of the blow-up set for problem (1) with u 0 (x) = λϕ(x) for a nonnegative function ϕ and sufficiently large λ. In fact, the author of this paper and Ishige proved the following results: Let λ > 0 and ϕ be a nonnegative bounded continuous function on Ω satisfying suitable assumptions, and consider problem (1) with u 0 (x) = λϕ(x). In this case, the blow-up time and the blow-up set is denoted by T (λϕ) and B(λϕ) according to the definitions, respectively. Assume that the solution u satisfies lim sup λ→∞ sup 0<t<T (λϕ) (T (λϕ) − t) 1/(p−1) u(t) L ∞ (Ω) < ∞. Then, • for any δ > 0, it holds B(λϕ) ⊂ {x ∈ Ω : ϕ(x) ≥ ϕ L ∞ (Ω) − δ} for all sufficiently large λ; • if ϕ ∈ C 2 (Ω) and a, b ∈ Ω are maximum points of ϕ satisfying there exists a positive constant δ such that B(λϕ) ∩ B(b, δ) = ∅ for all sufficiently large λ, that is, the solution u cannot blow-up near the maximum point b.
The first result implies that the solution blows up only near the maximum points of ϕ if λ is sufficiently large. Moreover, the second result implies that the location of the blow-up set is characterized by ∆ϕ at maximum points of ϕ if ϕ has several maximum points. In particular, if the set of maximum points of ϕ consists of two points a and b, then the solution blows up only near the maximum point a.
This paper concerns the case ∆u 0 (a) = ∆u 0 (b), where a and b are maximum points of ϕ. Considering the case ∆u 0 (a) = ∆u 0 (b), we try to find the next effect on the location of the blow-up set with large initial data. In order to obtain the location of the blow-up set, we study the profile of the solution near the blow-up time. We need a precise profile of the solution for the case ∆u 0 (a) = ∆u 0 (b) at the time much closer to the blow-up time T (λϕ) than the one for the case ∆u 0 (a) = ∆u 0 (b), and it seems difficult to study the case ∆u 0 (a) = ∆u 0 (b) by using the method used in [6].
In this paper we focus on another "large initial data" of the form u 0 (x) = λ+ϕ(x) with sufficiently large λ, that is, we consider   For this problem, we consider the case u 0 (x) = λ + ϕ(x) and ∆u 0 (a) = ∆u 0 (b), and study the location of the blow-up set of the solution for problem (1) for a sufficiently large λ. In particular, we characterize the location of the blow-up set by using ∆ 2 u 0 at maximum points of u 0 .
Before stating our main results, we introduce some notation. For a ∈ R, let [a] denote the greatest integer that is less than or equal to a. For a multi-index We denote by BC + (Ω) the set of nonnegative bounded continuous functions. For ϕ ∈ BC + (Ω) and δ > 0, put For any ϕ ∈ BC + (Ω) ∩ C 2m (Ω) for some m ∈ N and η > 0, put Put T λ := T (λ + ϕ) and B λ := B(λ + ϕ), that is, we denote the blow-up time and the blow-up set for problem (3) by T λ and B λ , respectively. For any φ ∈ L ∞ (R N ), let e t∆ φ be the unique bounded solution of Then we have On the other hand, for any functions f = f (λ) and g = g(λ) from (0, ∞) to (0, ∞), we say that f (λ) g(λ) for all sufficiently large λ, if there exist constant λ * > 0 and C > 0 such that f (λ) ≤ Cg(λ) for any λ ≥ λ * .
We are ready to state our main results.
Let p > 1 and u be the solution of problem (3). Assume that there exists a constant λ 0 > 0 such that Then, for any η > 0, there exist constants a constant δ > 0 such that for all sufficiently large λ.
As a corollary of Theorem 1.1, we obtain the following result.
Corollary 1. In addition to the same conditions as in Theorem 1.1, assume that Then lim In general, it seems difficult to obtain further results for the case (∆ 2 ϕ)(a) takes the same value for all a ∈ M (ϕ). However, if p is close to 1, then we can refine the previous result. Let u be the solution of problem (3). Assume (4) for some λ 0 > 0. Then, for any η > 0, there exists a constant δ > 0 such that for all sufficiently large λ. Assume that the solution u of (3) satisfies (4). Then We sketch the outline of the proof of main theorems. The proof of Theorem 1.2 is similar to that of Theorem 1.1, and we only explain the outline of the proof of Theorem 1.1. For the proof of Theorem 1.1, we construct comparison functions to obtain the profile of the solution of (3) just before the blow-up time. We first study the profile of the solution at t = T λ − λ −2(p−1)−1 . See Proposition 2. Compared to the argument of [4] and [6], we need more precise profile of the solution at this time. This is one of main difficulties in the proof. In order to obtain precise behavior of the solution, we divide the interval [0, T λ − λ −2(p−1)−1 ] into small intervals. The division of time intervals plays an important role in our argument to obtain precise behavior of the solution. See Lemma 3.3. For construction of comparison functions, we follow the argument of [4] and [6]. For any σ ≥ 0 and φ ∈ BC + (R N ), put Then one can easily check that • if σ = 0, then U 0 is a subsolution of (5); Using these functions, we construct comparison functions on each small time intervals. In this way, we study the profile of the solution at t = T λ − λ −2(p−1)−1 for sufficiently large λ. Furthermore, by using the profile of the solution at t = T λ − λ −2(p−1)−1 , we study the profile of the solution at t = T λ − λ −3(p−1)−1 , and obtain the location of the maximum points of u(·, T λ − λ −3(p−1)−1 ). See Proposition 3. Once we get maximum points of u(·, T λ − λ −3(p−1)−1 ), we can prove Theorem 1.1 by applying Proposition 1, which gives the location of the blow-up set for a semilinear heat equation with small diffusion. The rest of this paper is organized as follows. In Section 2 we give some preliminary results. In particular, we study short time behavior of the solution for the heat equation and an upper bound of the blow-up time T λ for sufficiently large λ. In Section 3 we study an upper estimate of the solution just before the blow-up time by constructing a family of comparison functions. In Section 4 we prove Theorems 1.1 and 1.2 by applying the results obtained in Sections 2 and 3.

2.
Preliminaries. In this section we give some preliminary results. We first study the short time behavior of the solution for the heat equation. For this purpose, we first show a fundamental identity in Lemma 2.1. Using these results, we study an upper bound of the blow-up time T λ for sufficiently large λ. Moreover, we introduce one proposition on the location of the blow-up set for a semilinear heat equation with small diffusion.
We first prove the following fundamental lemma.
for any t > 0 and a ∈ M (ϕ).

Proof. Put
for all n ∈ N and i ∈ {1, . . . , N }, we have For any n ∈ N and i ∈ {1, . . . , N }, by integration by parts we have This implies that Here we put (2β i − 1)!! := 1 for the case where β i = 0. Thus, by (6) we have On the other hand, by the multinomial theorem we have and by (7) we obtain the desired equality. This completes the proof of Lemma 2.1.
Next, we study the short time behavior of the solution for the heat equation with the aid of Lemma 2.1.

Lemma 2.2. Assume the same conditions as in Lemma
for all a ∈ M (ϕ), λ > 0 and all sufficiently small t > 0.
Proof. Let µ > 0 and fix it. Since ϕ ∈ C 2m (Ω) ∩ BC + (Ω) and M (ϕ, 0 ) is compact in Ω, by the Taylor theorem we see that there exists a positive constant δ > 0 such that B(a, δ) ⊂ Ω for all a ∈ M (ϕ) and for all x ∈ B(a, δ) and all a ∈ M (ϕ), where Put where G(x, y; t) denotes the Green function for the heat equation in B(0, 1) under the zero Dirichlet boundary condition. Then w satisfies   Then the comparison principle together with B(a, δ) ⊂ Ω yields Since it follows from [9, Lemma 2.1] that there exist positive constants C and r ∈ (0, 1) such that for all y ∈ B(0, r) and all sufficiently small t > 0, we have for all a ∈ M (ϕ) and all sufficiently small t > 0. Then, by Lemma 2.1 and (10) we have for all sufficiently small t > 0. Here we remark that for all α ∈ (N ∪ {0}) N with |α| ≤ 2m and all sufficiently small t > 0. Thus, by (12) we obtain for all sufficiently small t > 0. This together with arbitrariness of µ > 0 implies (9), and the proof of Lemma 2.2 is complete.
Repeating the argument as in [  Then

Remark 1.
The assumption 1 < p < 1+1/(m−2) for the case m ≥ 3 is weaker than the condition 1 < p < p(m), where p(m) appears in the statement of Theorem 1.2 and will be given in the beginning of Section 3. In fact, if 1 < p < p(m) and m ≥ 3, that is, p satisfies (22), then 1 < p < 1 + 1/(m − 2) holds. Therefore the conclusion of Lemma 2.3 holds under the assumption of Theorem 1.2.
Proof. Puts λ := (λ M /2) −(p−1) /(p − 1). Then we see thats λ is the blow-up time of the solution for the ordinary differential equation Let z be the solution of (8). Since ϕ(a) = ϕ L ∞ (Ω) = M for a ∈ M (ϕ), by Lemma 2.2 we have for all a ∈ M (ϕ) and all sufficiently small t > 0. Since (∆ k ϕ)(a) takes the same value for all a ∈ M (ϕ) and k = 1, . . . , m − 1, by the arbitrariness of a ∈ M (ϕ) and the definition of d and d k , we obtain for all λ > 0 and all sufficiently small t > 0. Note that ∆ϕ(a) ≤ 0 for all a ∈ M (ϕ). Sinces λ → 0 as λ → ∞, by (13)  = (p − 1)s λ for all sufficiently large λ. Therefore there exists a constant T λ ∈ (0,s λ ) such that By the comparison principle we have z(T λ ) L ∞ (Ω) ≤ z(0) L ∞ (Ω) = λ M , so by (14) we obtain Since the function u, defined by is a sub-solution of (1), we have u(x, t) ≥ u(x, t) in Ω × (0, T λ ). In particular, by (14) we see that T λ is the blow-up time of u and obtain T λ ≤ T λ for all sufficiently large λ. Then, since for all sufficiently small s > 0, by (13) and (14) we have for all sufficiently large λ. In particular, by (15) we have for all sufficiently large λ. This together with (14) and (16) implies that for all sufficiently large λ. Note that m(p − 1) + 1 < 2(p − 1) + 2 by the assumption 1 < p < 1 + 1/(m − 2), so λ Let T and B be the blow-up time and the blow-up set of the solution u , respectively. Assume sup Furthermore, assume that there exists a family {φ } 0< Then, for any η * > 0, if 3. Upper bound of the solution just before the blow-up time. In this section we construct suitable supersolutions for problem (1), and study an upper bound of the solution just before the blow-up time. This section plays an important role in the proof of main theorems.
Hereafter, we denote · L ∞ (R N ) by · ∞ for simplicity. Assume the same conditions as in Theorem 1.1 and we use the same notation as in Lemma 2.3. Then we have d = |∆ϕ(α)| for any α ∈ M (ϕ) under the assumptions of Theorem 1.1. For the case m ≥ 3, we can take a constant p(m) > 1 such that, for all p ∈ (1, p(m)), there holds where we may assume that p(2) = ∞ without loss of generality and p can be taken arbitrarily for the case m = 2. Let δ be a constant satisfying Put Then we define the function ψ λ by Since ϕ ∈ C 4 (Ω) ∩ BC + (Ω) and M (ϕ, 0 ) is a compact set in Ω for some 0 > 0, we have sup{|∇ 2 ϕ(x)| : x ∈ M (ϕ, 0 )} < ∞, so we can apply [6, Lemma 4.1] with for all sufficiently large λ. This together with (24) and (25) implies that for all sufficiently large λ. We first prepare one lemma on the short time behavior of the solution for a heat equation.
Proof. We apply the argument as in [6,Lemma 4.2]. Assertion (i) is proved by the same argument as in [6,Lemma 4.2]. Furthermore, assertion (iii) can be easily derived from C 2m regularity of ϕ. Put Let C > 0. Since ϕ λ = ϕ in U λ and V λ ⊂ Ω, for any µ > 0, by the Taylor theorem we obtain for all x ∈ V λ and all sufficiently large λ, where K is the constant given by (11). This implies that ] and all sufficiently large λ. See also the proof of Lemma 2.2. Once we get (28), since µ > 0 is arbitrary, by similar argument as in the proof of [6,Lemma 4.2] we can obtain assertions (ii) and (iv). Therefore we complete the proof of Lemma 3.1.
In order to construct supersolutions for problem (1), we consider the problem Then, by (24) and (25) we have U (x, 0) ≥ λ + ϕ(x) in Ω, and see that the solution U of (1) is a supersolution of (1). For any σ ≥ 0, we define the function It is easy to check that, if U satisfies for some T > 0, then U is a supersolution of (29) in R N × (0, T ). Using above functions, we study the profile of U (·, t) at t = S λ and obtain an upper estimate of u, where We divide the argument into several steps. In the first step, we study the profile of the solution u at t = s For simplicity, we put α := p − 1 > 0.
Lemma 3.2. Assume the same conditions as in Theorem 1.1. Then there exists a constant C 1 > 0 such that the function ϕ 1 ∈ C 1 (R N ), defined by for all x ∈ R N and all sufficiently large λ.
Proof. Put

By (26) and (29) we have
for all sufficiently large λ. Furthermore, by (26) we have for all sufficiently large λ. This together with (23) implies that for all (x, t) ∈ E 1 and all sufficiently large λ. Thus we obtain sup (x,t)∈E1 for all sufficiently large λ. Then, in view of (26) and (32), there exists a constant c 1 > 0 such that for all sufficiently large λ. Note that p = α + 1. Putting A 1 = c 1 , by (30) and (34) we see that the function U 1 is a supersolution of (29), and we can take a constant C 1 > 0 such that for all x ∈ R N and all sufficiently large λ. Then, repeating the argument as (33), for all sufficiently large λ, and by the definition of ϕ 1 and (26) we obtain for all x ∈ R N and all sufficiently large λ. Therefore we obtain the desired inequalities, and conclude the proof of Lemma 3.2.
In the second step, we construct a family of supersolutions of (29) in order to study an upper bound of the solution at t = λ −α and let U i be the solution of where ϕ i for i ≥ 2 will be constructed inductively in Lemma 3.3 and ϕ 1 is the function given in Lemma 3.2. We put U 1 := U , which is the solution of (29), for convenience. Furthermore, we define the function z i by z i (x, t) := (e t∆ z i−1 (·, s (i−1) λ ))(x) inductively, where z 1 is given in the proof of Lemma 3.2. Then we prove the following lemma. satisfies for all x ∈ R N and all sufficiently large λ.  (30) and (47) we see that the function U i is a supersolution of (36), and by (46) we have for all sufficiently large λ. Therefore we can find a constant C i > 0 such that the function ϕ i defined by (37) satisfies (38). We can easily derive (39) for ϕ i from (37) by similar calculation as in the proof of Lemma 3.2. Thus we complete the proof of Lemma 3.3.
Finally, using supersolutions constructed above, we study an upper bound of the solution of (1) at the time t = S λ . To this end, we consider the problem and prove the following lemma. Put