Blow-up directions at space infinity for solutions of semilinear heat equations

A blowing up solution of the semilinear heat equation u t = ∆ u + f ( u ) with f satisfying lim inf f ( u ) =u p > 0 for some p > 1 is considered when initial data u 0 satis(cid:12)es u 0 (cid:20) M , u 0 ̸(cid:17) M and lim m !1 inf x 2 B m u 0 ( x ) = M with sequence of ball f B m g whose radius diverging to in(cid:12)nity. It is shown that the solution blows up only at space in(cid:12)nity. A notion of blow-up direction is introduced. A characteriza-tion for blow-up direction is also established.


Introduction and main theorems
We are interested in solutions of semilinear heat equations which blow up at space infinity.
In [8] we considered a nonnegative blowing up solution of u t = ∆u + u p x ∈ R n , t > 0 with initial data u 0 satisfying where p > 1 and M > 0 is a constant. We proved in [8] that the solution u blows up exactly at the blow-up time for the spatially constant solution with initial data M . We moreover proved that u blows up only at the space infinity. In this paper we would like to generalize this result in following two directions.
(i) (Initial data) We consider more general initial data u 0 which may not converge to M for some direction of x, for example u 0 → M as |x| → ∞ only for x in some sector. It is convenient to introduce a notion of blow up direction at the space infinity, We are able to give necessary and sufficient conditions so that particular direction is a blow-up direction.
(ii) (Nonlinear term) We extend a class of nonlinear term. It includes e u and u p + u q for p, q > 1. We consider solutions of the initial value problem for the equation The nonlinear term f is assumed to be locally Lipschitz in R with the properly that We take two constants M and N satisfying M + N > 0 and The initial data u 0 is assumed to be a measureable function in R n satisfying −N ≤ u 0 ≤ M a.e. and u 0 ̸ ≡ M a.e.
We are interested in initial data such that u 0 → M as |x| → ∞ for x in some sector of R n . We assume that essinf x∈Bm (u 0 (x) − M m ) ≥ 0 for m = 1, 2, . . . , where and nonlinear term f let T * = T * (u 0 , f ) be the maximal existence time of the solution. If T * = ∞, the solution exists globally in time. If T * < ∞, we say that the solution blows up in finite time. It is well known that lim sup where ∥ · ∥ ∞ denotes the L ∞ -norm in space variables.
In this paper, we are interested in behavior of a blowing up solution near space infinity as well as location of blow-up directions defined below. A point then we say that the solution blows up to ±∞ at space infinity.
We consider the solution v(t) of an ordinary differential equation Let T v = T * (M, f ) be the maximal existence time of solutions of (9), i. e., We are now in position to state our main results. (2) and (3). Let u 0 be a continuous function satisfying (4) and (5), and

Theorem 1. Assume that f is locally Lipschitz in R and satisfies
The convergence is uniform in every compact subset of {t : 0 ≤ t < T v }. Moreover, the solution blows up at T v .

Remark.
Our assumption T v ≤ T * (−N, f ) says that the solution does not blow up to minus infinity before it blows up to plus infinity. From the condition (4), it follows that lim m→∞ |x m | = ∞.
This result in particular implies that When we set f (u) = |u| p−1 u, such a blow-up rate estimate is known for subcritical p; see e.g. [4], [6], [7] for general bounded initial data without assuming (4) and (5). Such a blow-up estimate is very fundamental to analyze the behavior of solution near blow-up point as noted in [3]. However, for supercritical p such a blow-up rate estimate (10) may not hold in general; see e.g. [1], [9]. If one considers only radial solutions of (1) for supercritical p less than 1 + 4/(n − 4 − 2(n − 1) 1/2 ) or n ≤ 10, then the estimate (10) holds [11]. We would like to emphasize that Theorem 1 requires no restriction on p.
Our second main result is on the location of blow-up points. There is a huge literature on location of blow-up points since the work of Weissler [13] and Friedman-McLeod [2]. (We do not intend to list references exhaustively in this paper.) However, most results consider either bounded domains or solutions decaying at space infinity; such a solution does not blow up at space infinity [5].
As far as the authors know, before the result of [8] the only paper discussing blow-up at space infinity is the work of Lacey [10]. He considered the Dirichlet problem in a half line. He studied various nonlinear terms and proved that a solution blows up only at space infinity.
In particular, his result implies that the solution of   blows up only at space infinity, where u 0 satisfies 0 ≤ u 0 ≤ M with M > 1, and f (s) = s p and e s . His method is based on construction of suitable subsolutions and supersolutions. However, the construction heavily depends on the Dirichlet condition at x = 0 and does not apply to the Cauchy problem even for the case n = 1.
As previously described, the authors [8] proved the statement of Theorems 1 and 2 assuming that lim |x|→∞ u 0 (x) = M for positive solutions of u t = ∆u + u p . Later, Simozyo [12] had the same results as in [8] by relaxing the assumptions of initial data u 0 ≥ 0 which is similar to that in the present paper. His approach is a construction of a suitable supersolution which implies that a ∈ R n is not a blow-up point. Although he restricted himself for f (s) = s p , his idea works our f under slightly strong assumption on u 0 . Here we give a different approach.
By Simozyo's results [12] it is natural to consider a problem of "blow-up direction" defined in (8). We next study this "blow-up direction" for the value +∞. Our third result is on this blow-up direction. It is convenient to introduce the function A m defined by for a given sequence {y m } ∞ m=1 . This A m (s) represents the mean value of u 0 over the ball B s (y m ).
This characterizes blow up directions by profiles of initial data. This is a new result even if f (u) = |u| p−1 u or n = 1.
Here are main ideas of the proofs . To prove Theorem 1 we construct a suitable subsolution. To prove Theorem 2 we derive a non blow-up criterion. We do not appeal any energy arguments for rescaled function as is done in our previous paper [8]. Our argument consists of two parts. First we observe that near a point a ∈ R n with some δ ∈ (0, 1) when t is close to blow-up time. By a bootstrap argument we derive that u is actually bounded near a when t is close to the blow up time. To prove Theorem 3 we use comparison argument as in Theorems 1,2 and non blow-up criterion which is established in the proof of Theorem 2. We also note that there is no situation which is not covered by assumption of (i) and (ii) of Theorem 3.
This paper is organized as follows. In section 2 we prove Theorem 1 by using the Green kernel of the heat equation. The proof of Theorem 2 is given in section 3 by a priori estimate. In section 4 we show Theorem 3 using Theorems 1 and 2.

Behavior at space infinity
In this section, we prove Theorem 1. We may assume r m ≤ r m+1 and M m ≤ M m+1 for m ∈ N without loss of generality.
Proof of Theorem 1. Let G B R (z) (x, y, t) be the Green kernel of the Dirichlet problem of the heat equation in the domain B R (z) and G(x, y, t) be that of R n . We set It is easily seen that for any measureable function ψ(y) with any r and a satisfying 0 < r < ∞ and 0 < a < ∞.
For the m-th ball B m defined in (6), let u m be the subsolution of (1) It is easily seen that X m ≤ X m+1 for any m ∈ N. It is well known that X m satisfies the integral equation when G m (x, y, t) be the Green kernel of the Dirichlet problem of the heat equation in the domain B Rm .
We shall prove that By the monotone convergence theorem we have Thus we have Since We thus obtain that It remains to prove that u blows up at t = T v . For this purpose it suffices to prove that which yields a contradiction. We thus proved that lim m→∞ u(x m , t m ) = ∞, so that u(x, t) blows up at T v .

Non blow-up point in R n
In this section we prove Theorem 2. We may assume that f (u 0 (x)) ≥ 0 for any x ∈ R n without loss of generality.
We consider the equation where B 1 = B 1 (a) with some a ∈ R n , and v is the solution of (9) and T is maximum existence time for v. C 1 ([0, T )) blow up at t = T . Then, for any ϵ > 0 and ζ ∈ (0, 1), there exist r ∈ (0, √ T * ) depending only on the space dimension, ϵ and ζ such that

Proof. From Lemma 3.4, it follows that u(x, t) ≤ ϵv(t) in
with r defined in Lemma 3.4. We argue a kind of a local bootstrap argument for u to get a bound. Let ϕ m be a C 2 −function supported on B η m (a) such that ϕ m ≡ 1 on B η m+1 and 0 ≤ ϕ m ≤ 1. (Note that since η ∈ (0, 1), ϕ m ≥ ϕ m+1 for m ∈ N.) We consider a cutoff of u defined by w m = ϕ m u. Then this w m satisfies Since ∥e t∆ h∥ ∞ ≤ ∥h∥ ∞ and ∥e t∆ ∇h∥ From these estimates it follows that for t ∈ [T − r 2 0 , T ) we estimate L ∞ -norm of w 1 : By Gronwall's inequality (see [5,Lemma2.3]) we have . Since Since f (s) ≥ Cs q for large s, by Proposition 3.5 we have and We thus conclude that By repeating the argument we have By repeating these calculations m times, we obtain where m and ϵ satisfy −(1 + mϵ q−1 )/(q − 1) + m/2 ∈ (0, 1/2 − ϵ q−1 /(q − 1)] and m ∈ (2/(q − 1 − 2ϵ q−1 ), 2/(q − 1 − 2ϵ q−1 ) + 1]. We now conclude that with some C > 0 by repeating the procedure once more. This implies that a is not a blow-up point.
Proof of Theorem 2. Putū 0 satisfying (4), (5) and Then by comparison we may assume thatū 0 = u 0 without loss of generality. Since a ∈ R n is arbitrary in Lemma 3.6, there is no blow-up point in R n .
From Lemma 3.4, Proposition 3.5 and the proof of Lemma 3.6, we have a sufficient condition for non blow-up point.

On blow-up direction
We shall prove Theorem 3 which gives a condition for blow-up direction.
Proof of Theorem 3. We first prove the case (i). By assumption we obtain that u 0 (x) satisfies (5)  We next show the case (ii). We take the sequence {x m } ∞ m=1 satisfying lim m→∞ x m /|x m | = ψ and {r m } ∞ m=1 satisfying lim m→∞ r m = ∞. We set and consider the equation By comparison we obtain u(x, t) ≤ u m (x, t) for any m ∈ N. By assumption there exist m 0 > 0 and sequence {c m } ∞ m=m 0 satisfying 0 < c m ≤ c m+1 and lim m→∞ c m = 1/s c such that for any m ≥ m 0 , where A m (r) is defined in (11). Since the solution of (1) satisfies the integral equation Let M f , δ f and T 0 be the same as proof of Lemma 3.1. We consider the solution w of We now introduceũ = vw. From the proof of Lemma 3.1, it follows that where δ is the Dirac delta function. Thus Since |x − z| ≤ 2s c for any x ∈ B sc , it follows that We thus obtain and note that δ m ∈ (0, 1) satisfies δ m ≥ δ m+1 for m ∈ N. From Lemma 3.6 and comparison it follows that there exist the sequence Since the sequence {x m } ∞ m=1 is arbitrary, we obtain that ψ is not blow-up direction.