BLOW-UP OF SEMILINEAR PDE’S AT THE CRITICAL DIMENSION. A PROBABILISTIC APPROACH

MATTHIAS BIRKNER, JOSÉ ALFREDO LÓPEZ-MIMBELA, AND ANTON WAKOLBINGER

Abstract. We present a probabilistic approach which proves blow-up of solutions of the Fujita equation ∂w/∂t = −(−Δ)^{α/2}w + w^{1+β} in the critical dimension d = α/β. By using the Feynman–Kac representation twice, we construct a subsolution which locally grows to infinity as t → ∞. In this way, we cover results proved earlier by analytic methods. Our method also applies to extend a blow-up result for systems proved for the Laplacian case by Escobedo and Levine [1] to the case of α-Laplacians with possibly different parameters α.

1. Introduction and overview

Consider the semilinear equation

(1.1)  \frac{∂w_{t}}{∂t} = Δ_{α}w_{t} + γw_{t}^{1+β},
w_{0} = φ,

in ℝ^{d}, where Δ_{α} := − (−Δ^{α/2}), 0 < α ≤ 2, denotes the α-Laplacian, β and γ are positive numbers and the initial condition φ is a nonnegative function on ℝ^{d}.

In Fujita’s pioneering work [2] it was shown (originally for the case α = 2) that d = α/β is the critical dimension for blow-up of (1.1): if d > α/β then (1.1) admits a global solution for all sufficiently small initial conditions, whereas if d < α/β, then for any non-vanishing initial condition the solution is infinite for suitably large t.

For the case d = α/β it was proved by Sugitani [10] by subtle analytic arguments that (1.1) blows up. Using different, partly probabilistic methods, this was also proved by Portnoy ([7, 8]) for the special case α = 2, β = 1. Related results on systems where the space variable is restricted to a bounded domain in ℝ^{d} can be found in the recent paper of Wang [11] and the references therein.

In this note we give a short probabilistic proof for blow-up at the critical dimension, using the Feynman–Kac representation. Here is an outline.

Consider for i = 0, 1, 2 the initial value problems

(1.2)  \frac{∂w_{t,i}}{∂t} = Δ_{α}w_{t,i} + γw_{t,i}w_{t,i−1}^{β}
w_{0,i} = φ

where w_{t,−1} = 0. Then f_{t} := w_{t,0}, g_{t} := w_{t,1} and h_{t} := w_{t,2} are all subsolutions of (1.1). Since f_{t}(y) = 𝔼_{y}[φ(W_{t})], where (W_{t}) is a symmetric α-stable process, f_{t}(y) decays like const · t^{−d/α} (see Section 2). On the other hand, by the Feynman–Kac formula (see e.g. Stroock [9] §4.3), g_{t} arises as the density of the measure

2000 Mathematics Subject Classification. 60H30, 35K57, 35B35, 60J57.

Key words and phrases. Blow-up of semilinear systems, Feynman–Kac representation, symmetric stable processes.

(1.3) ρ_{t}(B) := ∫ 𝔼_{x}[1_{B} (W_{t}) exp ∫_{0}^{t} γf_{s}(W_{s})^{β} ds] φ(x) dx, B ∈ B(ℝ^{d}).

Since “typically” f_{s}(W_{s}) should be bounded from below by const · s^{−d/α}, and also ℙ_{x} {W_{t} ∈ dy} ≥ const · t^{−d/α} dy as long as ∥y − x∥ ≤ t^{1/α}, one should expect that

g_{t}(y) ≥ ct^{−d/α} exp (const ∫_{1}^{t} s^{−dβ/α}ds)
(1.4)  = ct^{−d/α} exp (const · log t)
≥ ct^{−d/α+ε}

as long as ∥y∥ ≤ t^{1/α}. This intuition can be turned into a proof basically by applying Jensen’s inequality and scaling arguments.

After dealing in this way in Proposition 2.1 with the case i = 1, we then turn to the case i = 2 in (1.2). Like g_{t}, also h_{t} = w_{t,2} has a Feynman–Kac representation, but now with f_{s}^{β} replaced by g_{s}^{β} in the exponent of (1.3). By (1.4), the integrand g_{s}(W_{s})^{β} in this exponent should “typically” remain bounded from below by const · s^{−1+εβ}. Thus we expect that

h_{t}(y) ≥ const · t^{−d/α} exp (−c ∫_{0}^{t} s^{−1+εβ}ds) ,

and in fact we will prove this in Proposition 2.3. In particular, h_{t} thus is a subsolution of (1.1) which locally grows to infinity. This fact suffices to show blow up, as we will recall in Section 3.

Section 4 comments shortly on the case of subcritical dimensions, and Section 5 on Portnoy’s method. In Section 6 we give some extensions. Apart from re-proving Sugitani’s result, we show that blow-up of (1.1) with a certain time-dependent nonlinearity, which was recently proved by Gedda and Kirane [3], arises as an easy corollary of our probabilistic approach.

In Section 7 we obtain conditions for blow-up of a class of semilinear systems. We are able to extend a blow-up result of Escobedo and Levine [1] and show blowup at the critical dimensions of a system which we were able to analyze before only in the case of sub– and supercritical dimensions [5, 6].

2. Constructing subsolutions by the Feynman–Kac formula

Let us now turn to the case d = α/β, and assume without loss of generality that the initial condition φ of (1.1) does not vanish a.s. on the unit ball. Let p_{t}(x) denote the transition density of the symmetric α-stable process, and write

(2.1)  f_{t}(y) := ∫ p_{t}(y − x)φ(x) dx = 𝔼_{y} [φ(W_{t})] .

For all t ≥ 1 there holds

(2.2)  f_{t}(y) ≥ c_{0}t^{−d/α}1_{B_{1}}(t^{−1/α}y) ∫_{B_{1}} φ(x) dx

for some c_{0} > 0, where B_{r} denotes the ball in ℝ^{d} with radius r centered at the origin. Indeed, let y ∈ B_{t^{1/α}}. Then we have by the scaling property of W_{t}

f_{t}(y) = 𝔼_{0} [φ(W_{t} + y)] = 𝔼_{0} [φ (t^{1/α}(W_{1} + t^{−1/α}y))]
≥ ∫_{B_{1}} p_{1}(x − t^{−1/α}y)φ(t^{1/α}x) dx ≥ c_{0} ∫_{B_{1}} φ(t^{1/α}x) dx = c_{0}t^{−d/α} ∫_{B_{t^{1/α}}}φ(x) dx.

This argument also shows that for sufficiently large t and 0 < ∫ φ(x) dx < ∞ there holds

(2.3)  f_{t}(y) ≥ c′_{0}t^{−d/α}1_{B_{1}}(t^{−1/α}y).

for some c′_{0} > 0.

2.1. The first iteration: a subsolution with a slow decay. We are going to obtain a lower bound for the solution g_{t} of

(2.4)  \frac{∂g_{t}}{∂t} = Δ_{α}g_{t} + γg_{t}f_{t}^{β},
g_{0} = φ,

where f_{t} is defined in (2.1). Since f_{t} is a subsolution of (1.1), g_{t} is a subsolution of (1.1) as well.

Proposition 2.1. There exist ε, c > 0 such that for all t ≥ 2 and all y ∈ ℝ^{d} obeying ∥y∥ ≤ t^{1/α} there holds

(2.5)  g_{t}(y) ≥ c t^{−d/α+ε}.

Proof. By the Feynman–Kac formula, g_{t} arises as the density of the measure ρ_{t} defined in (1.3). We therefore have, using (2.2) and Jensen’s inequality,

g_{t}(y) = ∫ φ(x) p_{t}(y − x) 𝔼_{x}[exp ∫_{0}^{t} γf_{s}(W_{s})^{β} ds| W_{t} = y] dx

≥ ∫ φ(x) p_{t}(y − x) 𝔼_{x}[exp ∫_{1}^{t/2} c_{2}s^{−βd/α}1_{B_{s^{1/α}}} (W_{s}) ds| W_{t} = y] dx

≥ ∫ φ(x) p_{t}(y − x) exp (c_{2} ∫_{1}^{t/2} s^{−βd/α} ℙ_{x} {W_{s} ∈ B_{s^{1/α}}| W_{t} = y} ds) dx

(2.6) ≥ c_{3}t^{−d/α} exp (c_{4} ∫_{1}^{t/2} s^{−βd/α}ds)

where the last estimate relies on Lemma 2.2 below. (Here and below c_{i}, i = 1, 2, … denote “locally defined” positive constants). The assertion now follows from our assumption d = α/β. □

The intuition behind the following assertion is clear: conditioning on some “typical” state at time t does not much affect the behavior of (W_{t}) between times 0 and t/2.

Lemma 2.2. There exists a c > 0 such that for all t ≥ 2, y ∈ B_{t^{1/α}}, x ∈ B_{1} and s ∈ [1, t/2],

(2.7)  ℙ_{x} {W_{s} ∈ B_{s^{1/α}}| W_{t} = y} ≥ c.

Proof. Obviously it suffices to show

(2.8)  ∫_{B_{s^{1/α}}} p_{s}(z − x)p_{t−s}(y − z) dz ≥ c_{5}p_{t}(y − x).

To see this, let us state the following facts, which are an easy consequence of the scaling property of (W_{t}):

(i) For all z ∈ B_{s^{1/α}} and r := t − s

p_{r}(y − z) dz = ℙ_{0} {r^{1/α}W_{1} + y ∈ dz} ≥ inf_{a∈B_{2^{1/α}}}ℙ_{0} {W_{1} ∈ 2^{1/α}t^{−1/α} dz − a}
≥ c_{5}t^{−d/α}dz.

(ii) For all z ∈ B_{s^{1/α}}, p_{s}(z − x) ≥ c_{6}s^{−d/α}.

Combining (i) and (ii) we see that the LHS of (2.8) is bounded from below by c_{7}t^{−d/α}, which proves the claim. □

2.2. The second iteration: a subsolution growing to infinity. We are now aiming at a lower estimate for the solution h_{t} of

(2.9)  \frac{∂h_{t}}{∂t} = Δ_{α}h_{t} + h_{t}g_{t}^{β},
h_{0} = φ

where g_{t} is the subsolution of (1.1) constructed in the previous subsection. Clearly, also h_{t} is a subsolution of (1.1).

Proposition 2.3. inf {h_{t}(y) | ∥y∥ ≤ 1} → ∞ as t → ∞.

Proof. We proceed similarly as in the proof of Proposition 2.1. First we note that the Feynman–Kac formula renders

(2.10)  h_{t}(y) = ∫ φ(x)p_{t}(y − x) 𝔼_{x}[exp ∫_{0}^{t} γg_{s}(W_{s})^{β}ds| W_{t} = y] dx.

Using Jensen’s inequality and (2.5), we see that the RHS of (2.10) is bounded from below by

∫ φ(x)p_{t}(y − x) exp (γ ∫_{2}^{t/2} 𝔼_{x}[g_{s}(W_{s})^{β}| W_{t} = y] ds) dx

≥ ∫_{B_{1}} φ(x)p_{t}(y − x)

· exp (γ ∫_{2}^{t/2} cs^{−βd/α+εβ} ℙ_{x} {W_{s} ∈ B_{s^{1/α}}| W_{t} = y} ds) dx (2.11)

(2.12)  ≥ c_{8}t^{−d/α} exp(c_{9}t^{εβ}).

Here, we used Lemma 2.2 and the assumption d = α/β in the last inequality. □

3. Completion of the proof of blow-up

From Proposition 2.3 we know that

(3.1)  K(t) := inf_{x∈B_{1}} w_{t}(x) → ∞ as t → ∞

where B_{1} denotes the unit ball. In fact this is enough to guarantee blow-up. Here is an easy argument which is borrowed from [4] §4, and which we include for convenience.

We are going to re-start (1.1) with the initial condition w_{t_{0}}, with a suitable choice of t_{0} given below. Writing u_{t} := w_{t_{0}+t} we first recall the integral form of (1.1)

(3.2)  u_{t}(x) = ∫ p_{t}(y − x)u_{0}(y) dy + ∫_{0}^{t} γ ds ∫ p_{t−s}(y − x)u_{s}(y)^{1+β} dy.

Noting that ζ := min_{x∈B_{1}} min_{0≤s≤t} ℙ_{x} {W_{s} ∈ B} is strictly positive, we obtain for all t ∈ [0, 1] from (3.1) the estimate

(3.3)  min_{x∈B_{1}} u_{t}(x) ≥ ζK(t_{0}) + γζ ∫_{0}^{t} ( min_{y∈B_{1}} u_{s}(y))^{1+β} ds.

Now choose t_{0} so big that the blow-up time of the equation

(3.4)  v(t) = ζK(t_{0}) + γζ ∫_{0}^{t} v(s)^{1+β} ds

is smaller than 1. Then, a fortiori, min_{x∈B} u_{1}(x) = ∞, which shows blow-up of w.

4. Subcritical dimensions: one iteration suffices

In the case d < α/β, (2.6) shows that already the first subsolution g_{t} (constructed in Section 2.1) grows to infinity on the unit ball B_{1} in the sense that inf{g_{t}(y)| ∥y∥ ≤ 1} → ∞ as t → ∞. Thus, in view of the previous section, for subcritical dimensions a single application of the Feynman–Kac formula suffices to show blow-up.

5. A remark on Portnoy’s method

Portnoy [7] studies the iteration scheme

(5.1)  v_{n+1}(x) = (Π_{1}v_{n}) (x) + (Π_{1}v_{n})^{2} (x)
v_{0} = φ ≥ 0

where Π_{1} is a transition probability on ℝ^{d}. He shows that under suitable assumptions on Π_{1} (which include the case of a standard Brownian transition probability), (5.1) admits no bounded solution for d = 1 and d = 2 provided φ does not a.s. vanish.

A closer look on his proofs shows that he achieves this by analyzing subsolutions v_{n}^{(i)} of (5.1) which are given by the scheme

v_{n+1}^{(0)} = Π_{1}v_{n}^{(0)} = Π_{n+1}φ

v_{n+1}^{(i)} = Π_{1}v_{n}^{(i)} + (Π_{1}v_{n}^{(i)}) (Π_{1}v_{n}^{(i−1)}) . (5.2)

The analysis of (5.2) is carried through probabilistically in terms of random walks, which is much in the spirit of a discrete time Feynman–Kac approach.

It can be extracted from Portnoy’s arguments that, for the Brownian case, say,

(5.3)  v_{n}^{(1)} grows to infinity for d = 1,

and

(5.4)  v_{n}^{(2)} grows to infinity for d = 2.

An easy application of Jensen’s inequality plus induction shows that w_{n} is bounded from below by v_{n} (where w_{t} is the solution of (1.1) with β = 1). Indeed,

w_{n} = Π_{1}w_{n−1} + ∫_{0}^{1} Π_{s}w_{n−s}^{2} ds ≥ Π_{1}w_{n−1} + (∫_{0}^{1} Π_{s}w_{n−s} ds)^{2}

≥ Π_{1}w_{n−1} + (∫_{0}^{1} Π_{s}Π_{1−s}w_{n−1} ds)^{2} ≥ Π_{1}v_{n−1} + (Π_{1}v_{n−1})^{2} = v_{n}.

Together with the argument in Section 3 above, (5.3) and (5.4) thus imply blow-up of w for β = 1 and α = 2 in one and two dimensions. (In [8], a more complicated argument is used to show w_{n} ≥ v_{n} and the blow-up of w.)

6. Extensions

6.1. Sugitani’s condition. Sugitani [10] considers instead of (1.1) the slightly more general equation

(6.1)  \frac{∂w_{t}}{∂t} = Δ_{α}w_{t} + F (w_{t})
w_{0} = φ,

where F : ℝ_{+} → ℝ_{+} is increasing and convex, and F (u) ∼ γu^{1+β} as u → 0. This requires only slight modifications in Section 2:

In (2.4) and below, f_{t}(u)^{β} has to be replaced by F (f_{t}(u))/f_{t}(u), which by assumption can be bounded from below by cf_{t}(u)^{β}.

Similarly, in (2.9) and below, g_{t}(u)^{β} has to be replaced by F (g_{t}(u))/g_{t}(u).

6.2. A time dependent nonlinearity. Recently, Guedda and Kirane [3] showed by analytic methods blow-up of the equation

(6.2)  \frac{∂w_{t}}{∂t} = Δ_{α}w_{t} + γt^{σ}w_{t}^{1+β}
w_{0} = φ (≥ 0, ≢ 0)

for σ ≥ βd/α − 1. This result also follows quickly from our probabilistic approach. In fact, it suffices to consider the case σ = βd/α − 1.

1. Concerning the subsolution g_{t}, all what happens is that a factor s^{σ} enters into the exponentials in the Feynman–Kac representation (1.3) and in the RHS of (2.6). Since s^{−βd/α} in the RHS of (2.6) cancels against s^{σ}, the lower bound (2.6) remains unchanged, and so does the estimate (2.5).

2. Concerning the subsolution h_{t}, again a factor s^{σ} enters into the exponentials in (2.10) and (2.11). Since again (s^{−d/α})^{β} cancels against s^{σ}, the lower bound (2.12) remains unchanged, and so does the assertion in Proposition 2.3.

3. Concerning the argument in Section 3, from the special time-inhomogeneity in (6.2) a factor (t_{0} + t)^{σ} enters in front of the integral in (3.3). Still, since (2.12) guarantees a super-algebraic growth of K(t), we can choose t_{0} so big that the blow-up time of the equation

v(t) = ζK(t_{0}) + γζ(t_{0} + 1)^{σ} ∫_{0}^{t} v(s)^{1+β}ds

is smaller than 1, so that the argument of Section 3 remains valid.

7. Blow-up of systems

In this section we apply our probabilistic approach to extend a blow-up result of Escobedo and Levine [1] (Theorem 7.1 and Remark 7.2). In Theorem 7.3 we show that a system which we investigated in [6] in high dimensions blows up at the critical dimension.

Theorem 7.1. Assume that (u, v) solves

\frac{∂u_{t}}{∂t} = Δ_{α_{1}}u_{t} + u_{t}^{1+β_{1}}v_{t}^{β_{2}}
(7.1)  \frac{∂v_{t}}{∂t} = Δ_{α_{2}}v_{t} + F (u_{t}, v_{t})
u_{0} = φ_{1}, v_{0} = φ_{2},

where α_{1}, α_{2} ∈ (0, 2], β_{1} > 0, β_{2} ≥ 0, F ≥ 0, φ_{1} ≥ 0, φ_{2} ≥ 0 and both φ_{1} and φ_{2} do not a.s. vanish. Then u blows up if

(7.2)  α_{2} ≤ α_{1} and d ≤ (\frac{β_{1}}{α_{1}} + \frac{β_{2}}{α_{2}} )^{−1} .

Remark 7.2. For α_{1} = α_{2} =: α, (7.2) turns into the condition d ≤ α/(β_{1} + β_{2}), which notably is also the condition for blow-up of the partial differential equation

\frac{∂u}{∂t} = Δ_{α}u + u^{1+β_{1}+β_{2}}.

For α = 2, this specializes to one of the main results in Escobedo and Levine’s paper [1]. They investigate by analytic tools the system

\frac{∂u}{∂t} = Δu + u^{1+β_{1}}v^{β_{2}}, \frac{∂v}{∂t} = Δv + u^{θ_{1}}v^{θ_{2}}

and prove blow-up under the condition d ≤ 2/(β_{1} + β_{2}).

Proof of Theorem 7.1. Let f_{t,j}(y) := ∫ φ_{j}(x)p_{t,j}(y − x) dx, j = 1, 2, where p_{t,j} denotes the symmetric α_{j}-stable transition density. Obviously, (f_{t,1}, f_{t,2}) is a subsolution of (7.1), and from (2.2) we have for t ≥ 1

(7.3)  f_{t,1}(y) ≥ Ct^{−d/α_{1}}1_{B_{1}}(t^{−1/α_{1}}y)

and

(7.4)  f_{t,2}(y) ≥ Ct^{−d/α_{2}}1_{B_{1}}(t^{−1/α_{1}}y),

where we used the assumption α_{2} ≤ α_{1} to obtain (7.4). Thus we can proceed as in the proof of Proposition 2.1 to get the estimate

(7.5)  u_{t}(y) ≥ Ct^{−d/α_{1}+ε}

uniformly for ∥y∥ ≤ t^{1/α_{1}}. Combining (7.4) and (7.5) we get by the same argument as in the proof of Proposition 2.3 a lower bound for u_{t} which locally grows to infinity at a super-algebraic speed. By the argument at the end of Section 6.2 this is sufficient for blow-up of u_{t} even if F ≡ 0 (in which case v_{t} decays algebraically). □

Theorem 7.3. Assume that (u, v) solves

\frac{∂u_{t}}{∂t} = Δ_{α_{1}}u_{t} + u_{t}v_{t}
(7.6)  \frac{∂v_{t}}{∂t} = Δ_{α_{2}}v_{t} + u_{t}v_{t}
u_{0} = φ_{1}, v_{0} = φ_{2},

where α_{1}, α_{2} ∈ (0, 2], φ_{1} ≥ 0, φ_{2} ≥ 0 and both φ_{1} and φ_{2} do not a.s. vanish. Then (u, v) blows up if d ≤ min(α_{1}, α_{2}).

Remark 7.4. It was shown in [6] that (7.6) admits global solutions if d > min(α_{1}, α_{2}) and φ_{1} and φ_{2} are sufficiently small.

Before proving Theorem 7.3, we prepare a lemma which is an easy generalization of Lemma 2.2. Here and below, (W_{t}^{(i)}) denotes the symmetric stable process with index α_{i} and p_{t,i}(x) its transition density, i = 1, 2.

Lemma 7.5. Assume that α := α_{2} ≤ α_{1}. There exists a c > 0 such that for all t ≥ 2, y ∈ B_{t^{1/α}}, x ∈ B_{1} and s ∈ [1, t/2],

ℙ_{x} {W_{s}^{(2)} ∈ B_{s^{1/α_{1}}} | W_{s}^{(2)} = y} ≥ cs^{d/α_{1}−d/α_{2}}

Proof. It suffices to show (2.8) with cs^{d/α_{1}−d/α_{2}} instead of c_{5} and p_{t,2} instead of p_{t}.

Again we have (i) and (ii) from the proof of Lemma 2.2, now with (W_{t}^{(2)}) instead of (W_{t}). Integrating the bound s^{−d/α_{2}} over B_{s^{1/α_{1}}} then gives the factor const · s^{d/α_{1}−d/α_{2}}. □

Proof of theorem 7.3. From (2.3) we have

(7.7)  u_{t} ≥ c_{1}t^{−d/α_{1}}1_{B_{t^{1/α_{1}}}}

and

(7.8)  v_{t} ≥ c_{2}t^{−d/α_{2}}1_{B_{t^{1/α_{2}}}}

for sufficiently large t. Let us now assume without loss of generality that α_{2} ≤ α_{1}. By the Feynman–Kac formula we have

u_{t}(y) = ∫ φ_{1}(x)p_{t,1}(y − x) 𝔼_{x}[exp ∫_{0}^{t} v_{s}(W_{s}^{(1)}) ds| W_{t}^{(1)} = y] dx.

By Jensen’s inequality and (7.8), this can be bounded from below by

∫ φ_{1}(x)p_{t,1}(y − x) exp (∫_{t_{0}}^{t/2} c_{2}s^{−d/α_{2}} ℙ_{x} {W_{s}^{(1)} ∈ B_{s^{1/α_{2}}} | W_{t}^{(1)} = y} ds) dx.

Noting that B_{s^{1/α_{2}}} ⊇ B_{s^{1/α_{1}}} and using Lemma 2.2, we thus arrive at the lower bound

(7.9)  c_{3}t^{−d/α_{1}} exp (c_{4} ∫_{t_{0}}^{t/2} s^{−d/α_{2}}ds) .

If d < α_{2}, then this lower bound grows super-algebraically, and we infer blow-up of u like in Section 3, combined with (7.8) and the argument in Paragraph 6.2.3.

Let us now assume d = α_{2}. Then (7.9) turns into the lower bound

(7.10)  u_{t}(y) ≥ c_{5}t^{−d/α_{1}+ε}

(uniformly in y ∈ B_{t^{1/α_{1}}} for t sufficiently large). Another application of the Feynman–Kac formula gives

(7.11)  v_{t}(y) = ∫ φ_{2}(x)p_{t,2}(y − x) 𝔼_{x}[exp ∫_{0}^{t} u_{s}(W_{s}^{(2)}) ds| W_{t}^{(2)} = y] dx.

Using Jensen’s inequality and (7.10), we can bound this from below by

∫ φ_{2}(x)p_{t,2}(y − x) exp ∫_{t_{0}}^{t/2} c_{1}s^{−d/α_{1}+ε} ℙ_{x} {W_{s}^{(2)} ∈ B_{s^{1/α_{1}}} | W_{t}^{(2)} = y} ds dx.

In view of Lemma 7.5 we thus obtain as a lower bound for v_{t}(y) (as long as t is sufficiently large and y ∈ B_{t^{1/α_{2}}} ):

c_{6}t^{−d/α_{2}} exp ∫_{t_{0}}^{t/2} c_{7}s^{−d/α_{1}+ε}s^{d/α_{1}−d/α_{2}}ds = c_{6}t^{−d/α_{2}} exp ∫_{t_{0}}^{t/2} c_{7}s^{−d/α_{2}+ε}ds
= c_{6}t^{−d/α_{2}} exp (c_{8}t^{ε}) .

As before, the growth of this lower bound implies blow-up of v. □

Remark 7.6. Consider instead of (7.6) the more general system

\frac{∂u_{t}}{∂t} = Δ_{α_{1}}u_{t} + u_{t}v_{t}^{β_{1}}
(7.12)  \frac{∂v_{t}}{∂t} = Δ_{α_{2}}v_{t} + u_{t}^{β_{2}}v_{t}
u_{0} = φ_{1}, v_{0} = φ_{2},

where α_{1}, α_{2}, φ_{1}, φ_{2} are as in Theorem 7.3, and β_{1}, β_{2} > 0. Assume that α_{2} ≤ α_{1}. Proceeding as in the proof of Theorem 7.3 but using the simple bound (7.7) instead of (7.10) in the Feynman–Kac representation corresponding to (7.11) one obtains quickly that (7.12) blows up if

(7.13)  d < max (\frac{α_{2}}{β_{1}} , (\frac{β_{2} − 1}{α_{1}} + \frac{1}{α_{2}} )^{−1}) .

It remains an interesting question whether the RHS of (7.13) is the critical dimension for blow-up of (7.12) and whether there is blow-up at the critical dimension. We conjecture that this is the case at least for α_{1} = α_{2} =: α, in which case the RHS of (7.13) turns into α/ min(β_{1}, β_{2}). Indeed, for the special case α = 2, this was proved by Escobedo and Levine [1].

Acknowledgment The second named author appreciates the kind hospitality of Frankfurt University during his visit in summer 2000, and CONACyT (Mexico) and DAAD (Germany) for partial support.

References

[1] Escobedo, M. and Levine, H. (1995). Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations. Arch. Rational Mech. Anal. 129, 47-100.

[2] Fujita, H. (1966). On the blowing up of solutions of the Cauchy problem for u_{t} = Δu+u^{1+α}. J. Fac. Sci. Univ. Tokyo Sect. I 13, 109-124.

[3] Guedda, M., and Kirane, M. (1999). A note on nonexistence of global solutions to a nonlinear integral equation. Bull. Belg. Math. Soc. Simon Stevin 6, 491-497.

[4] Kobayashi, K., Sirao, T. and Tanaka, H. (1977). On the growing up problem for semilinear heat equations. J. Math. Soc. Japan 29, 407-424.

[5] López-Mimbela, J.A. and Wakolbinger, A. (1998). Length of Galton-Watson trees and blow-up of semilinear systems. J. Appl. Prob. 35, 802-811.

[6] López-Mimbela, J.A. and Wakolbinger, A. (2000). A probabilistic proof of non-explosion of a non-linear PDE system. Adv. Appl. Prob. (in press).

[7] Portnoy, S. (1975). Transience and solvability of a non-linear diffusion equation. Ann. Probab. 3, 465-477.

[8] Portnoy, S. (1976). On solutions to u_{t} = Δu + u^{2} in two dimensions. J. Math. Anal. Appl. 55, 291-294.

[9] Stroock, D. (1993). Probability theory, an analytic view. Cambridge University Press.

[10] Sugitani, S. (1975). On nonexistence of global solutions for some nonlinear integral equations. Osaka J. Math. 12, 45-51.

[11] Wang, L. (2000). The blow-up for weakly coupled reaction-diffusion systems. Proc. Amer. Math. Soc., electronically published.

FB Mathematik, J.W. Goethe Universität, D-60054 Frankfurt am Main, Germany.

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FB Mathematik, J.W. Goethe Universität, D-60054 Frankfurt am Main, Germany.

E-mail address: wakolbinger@math.uni-frankfurt.de