A probabilistic proof of non-explosion of a non-linear PDE system

J. Alfredo López Mimbela^{∗} Anton Wakolbinger^{†}^{‡}

Abstract

Using a representation in terms of a two-type branching particle system, we prove that positive solutions of the system u̇ = Au + uv, v̇ = Bv + uv remain bounded for suitable bounded initial conditions, provided A and B generate processes with independent increments and one of the processes is transient with a uniform power decay of its semigroup. For the case of symmetric stable processes on ℝ^{1}, this answers a question raised in [LM-W].

1 Introduction and result

Consider the system

\frac{∂u}{∂t} = Δ_{α_{1}}u + uv, u_{0}(x) = φ_{1}(x) ≥ 0, x ∈ ℝ^{d},
\frac{∂v}{∂t} = Δ_{α_{2}}v + uv, v_{0}(x) = φ_{2}(x) ≥ 0, x ∈ ℝ^{d}, ℝ^{d},  (1.1)

where Δ_{α} := −(−Δ^{α/2}), 0 < α ≤ 2, stands for the α-Laplacian. In [LM-W] we showed that, for d = 1, (1.1) exhibits blow-up if min(α_{1}, α_{2}) > 1, and we interpreted this fact in terms of the probabilistic representation of (1.1) by means of a two type branching particle system (which we will recall below): if both motions generated by Δ_{α_{1}} and Δ_{α_{2}} are “lazy enough,” then the solution of (1.1) grows to infinity in a finite time (provided φ_{i} ≥ c1_{D} for some c > 0 and some nonempty interval D).

In [LM] it was shown that, for suitably bounded φ_{1}, φ_{2}, (1.1) admits a uniformly bounded solution if max(α_{1}/d, α_{2}/d) < 1, i.e. if both motions are “mobile enough.” It remained an open question what happens if, for d = 1, min(α_{1}, α_{2}) < 1 < max(α_{1}, α_{2}). The result of the present note answers this question in a somewhat more general framework, revealing that it is the “most mobile type” only which is responsible for blow-up resp. stability of the system.

∗Centro de Investigaci ón en Matemáticas, Apartado Postal No. 402, Guanajuato 36000, Mexico.

jalfredo@cimat.mx

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

wakolbinger@math.uni-frankfurt.de

‡Partially supported by CONACyT and DAAD

Instead of (1.1) we will consider the system

\frac{∂u}{∂t} = Au + uv, u_{0}(x) = φ_{1}(x) ≥ 0, x ∈ ℝ^{d},
\frac{∂v}{∂t} = Bv + uv, v_{0}(x) = φ_{2}(x) ≥ 0, x ∈ ℝ^{d},  (1.2)

where A and B are the generators of two Markov processes (W_{t}^{A}) and (W_{t}^{B}) on ℝ^{d}, having the semigroups (S_{t}) and (T_{t}), respectively.

Theorem 1.1 Assume there exists some γ > 0 such that for all bounded D ⊂ ℝ^{d},

S_{t}T_{s}1_{D}(x) ≤ c_{D}t^{−(1+γ)}, x ∈ ℝ^{d}, t > 0, s ≥ 0,  (1.3)

where c_{D} > 0 may depend on D but not on x, s and t. Then (1.2) admits a bounded solution, provided

φ_{i} ≤ cS_{1}1_{D}, i = 1, 2  (1.4)

for some bounded D ⊂ ℝ^{d} and some sufficiently small c > 0.

Remark 1.2 Condition (1.3) obviously is valid if

S_{t}1_{D}(x) ≤ c_{D}t^{−(1+γ)}, x ∈ ℝ^{d}, t > 0,  (1.5)

and if (W_{t}^{A}) and (W_{t}^{B}) both have independent increments (since then the two semigroups commute).

Corollary 1.3 The system (1.1) admits a bounded solution if α_{1}/d < 1 and if (1.4) is satisfied.

Indeed, by the well-known scaling and unimodality properties of the symmetric stable densities, (1.5) holds with γ := \frac{d}{α_{1}} − 1. □

Remark 1.4 For A = B, (1.2) renders a special case of the so called Fujita equation

\frac{∂u}{∂t} = Au + u^{β},  (1.6)

which, for the case A = Δ, was studied in [Fu]. For integer β ≥ 2, Nagasawa and Sirao [N-S] obtained a probabilistic representation of the solution of (1.6), which was further developed in [LM] into the form we are going to use here (cf. [LM-W], and (2.1) below). It is instructive to compare the representation obtained in [LM] with H.P. McKean’s representation of the Kolmogorov-Petrovskii-Piskunov equation

\frac{∂u}{∂t} = Au + u^{β} − u  (1.7)

(cf. [McK]). Both are expectations of a functional of one and the same branching particle system, where the functionals differ by a factor “exponential of the tree length,” which can be interpreted as a Feynman-Kac term correcting for the difference u between (1.6) and (1.7). (We owe this observation to A. Etheridge (personal communication.))

Remark 1.5 For A = B = Δ, and u^{p_{i}}v^{q_{i}} (i = 1, 2) instead of uv in lines 1 and 2 of (1.2), respectively, Escobedo and Levine [E-L] showed, under the assumptions p_{1} > 1, p_{1}q_{2} > 0, and p_{1} + q_{2} ≤ p_{2} + q_{2}, that the system admits global solutions if 2/d < p_{1} + q_{1} − 1, and blows up otherwise. In this note we are focussing on another case, namely p_{1} = p_{2} = q_{1} = q_{2} = 1, and possibly different operators A and B.

2 The probabilistic framework

In order to recall the probabilistic solution of (1.2), let us introduce some concepts and notations.

Let T_{t} be a Yule tree (i.e. a continuous time Galton- Watson tree with offspring distribution δ_{2}) with branching rate 1, growing from one ancestor at time 0 up to time t. For our purpose, it is convenient to think of T_{t} being generated as follows: The “original” branch gives, in between times 0 and t, rise to offspring branches at rate ds, each of which, when born at time s, gives again, between times s and t, rise to offspring branches at rate dr, and so on.

For each realization τ of T_{t}, we denote by L(τ) the length of τ, i.e. the sum of the branch lengths of τ. In addition, for each realization of T_{t}, we perform a colouring of each of the branches of τ by the “colours” a and b, in such a way that an offspring branch always gets a colour different from that of its parent branch. The coloured tree τ^{(i)} ( where i = a or i = b) is thus determined by the colour i of the original branch.

For such a coloured tree τ^{(i)}, and x ∈ ℝ^{d}, let (X_{s}^{x,τ^{(i)}})_{0≤s≤t} be a two-type process indexed by τ^{(i)} which evolves as follows. An original particle starts in x and moves up to time t with A-motion if i = a and with B- motion if i = b. This particle generates offspring particles at its respective position according to the branching points of τ^{(i)}, which then move on independently according to the colouring of τ^{(i)}, and so on. For every time s ∈ [0, t], this gives rise to a random population of coloured particles on ℝ^{d}, which we denote by X_{s}^{x,τ^{(i)}}. Let

X_{s}^{x,τ^{(i)}} = X_{s,a}^{x,τ^{(i)}} + X_{s,b}^{x,τ^{(i)}}

be the decomposition of X_{s}^{x,τ^{(i)}} into its subpopulations of colours a and b.

Finally, for every counting measure ν = ∑_{n} δ_{y_{n}} and φ : ℝ^{d} → ℝ_{+}, we write

φ̂(ν) := ∏_{n} φ(y_{n}).

We now recall the probabilistic representation of the solution of (1.2), of which we include a proof for the sake of self-containedness.

Proposition 2.1. ([LM]) The solution of (1.2) is given by u_{t}(x) = w_{t}(x, a), v_{t}(x) = w_{t}(x, b), where

w_{t}(x, i) = 𝔼[φ̂_{1} (X_{t,a}^{x,T_{t}^{(i)}}) φ̂_{2} (X_{t,b}^{x,T_{t}^{(i)}}) e^{L(T_{t})}] , i = 1, 2.  (2.1)

Proof Conditioning on the length l of the original branch of T_{t} renders

w_{t}(x, i) = e^{−t} 𝔼[φ̂_{1} (X_{t,a}^{x,T_{t}^{(i)}}) φ̂_{2} (X_{t,b}^{x,T_{t}^{(i)}}) e^{L(T_{t})}| l ≥ t]

+ ∫_{0}^{t} dr e^{−r} 𝔼[φ̂_{1} (X_{t,a}^{x,T_{t}^{(i)}}) φ̂_{2} (X_{t,b}^{x,T_{t}^{(i)}}) e^{L(T_{t})}| l = r] .

Writing W_{s}^{(x,i)} for the position of the original particle at time s ≤ l, it follows that

w_{t}(x, i) = e^{−t}e^{t} 𝔼[φ_{i} (W_{t}^{(x,i)})] + ∫_{0}^{t} dr e^{−r}e^{r} 𝔼[w_{t−r} (W_{r}^{(x,i)}, a) w_{t−r} (W_{r}^{(x,i)}, b)]

= \{ \begin{matrix}S_{t}φ_{1}(x) + ∫_{0}^{t} dr S_{r} (w_{t−r} (·, a) w_{t−r} (·, b)) (x) & for i = a, \\ T_{t}φ_{2}(x) + ∫_{0}^{t} dr T_{r} (w_{t−r} (·, a) w_{t−r} (·, b)) (x) & for i = b. \end{matrix}

□

3 Proof of Theorem 1.1

Let T_{t} be the set of trees which arise as realizations of T_{t} (as described in Section 2). On T_{t} we define the measure μ_{t} which arises by reweighing the distribution of T_{t} by e^{L(τ)}:

μ_{t}(dτ) := ℙ[T_{t} ∈ dτ] e^{L(τ)}.  (3.1)

In view of (2.1) and (3.1), we are going to analyse u_{t}(x) = w_{t}(x, a) and v_{t}(x) = w_{t}(x, b), where

w_{t}(x, i) := ∫ 𝔼[φ̂_{1} (X_{t,a}^{x,τ_{t}^{(i)}}) φ̂_{2} (X_{t,b}^{x,τ_{t}^{(i)}})] μ_{t}(dτ).

For τ ∈ T_{t}, let K(τ) denote the number of inner nodes of τ. Let us write

μ_{t}^{(≥1)}(dτ) := μ_{t}(dτ)1_{[K(τ)≥1]} and μ_{t}^{k}(dτ) := μ_{t}(dτ)1_{[K(τ)=k]}.

For example, if ρ denotes the tree in T_{t} which consists of one single branch, then K(ρ) = 0, and μ^{(0)}({ρ}) = e^{−t}e^{t} = 1. If σ(r), 0 ≤ r ≤ t, denotes the tree with one single branching point at time r, then μ^{(1)}(d(σ(r))) = 1_{[0,t]}(r)e^{−r}e^{r}e^{−2(t−r)}e^{2(t−r)} dr = 1_{[0,t]}(r) dr.

Definition 3.1 For τ ∈ T_{t} with K(τ) ≥ 1, we denote by r = r(τ) the time of its first branching, and by τ′ and τ′′ (∈ T_{t−r}) its two subtrees originating from there. Let us also introduce the notation

M_{t}(dτ) := ℙ[T_{t} ∈ dτ] ,
M_{t}^{(≥1)}(dτ) := ℙ[T_{t} ∈ dτ] 1_{[K(τ)≥1]},
M_{t}^{(k)}(dτ) := ℙ[T_{t} ∈ dτ] 1_{[K(τ)=k]}.

Lemma 3.2 (a) M_{t}^{(≥1)}(dτ) = 1_{[0,t]}(r)e^{−r}M_{t−r}(dτ′) M_{t−r}(dτ′′) dr.

(b) For k ≥ 1, M_{t}^{(k)}(dτ) = 1_{[0,t]}(r)e^{−r} dr \frac{1}{k} ∑_{j=0}^{k−1} M_{t−r}^{(j)}(dτ′) M_{t−r}^{(k−1−j)}(dτ′′).

Proof (a) is immediate from the definition of Yule trees. (b) results from (a) and the fact that the genealogy of a Yule tree is identical in law with that of a Polya urn (starting with two balls after the first branching point). Consequently, given there are k inner nodes (and therefore k +1 leaves), the total number of leaves of one of the two subtrees, say T′_{t−r}, is 1 + J, where J is uniformly distributed on {0, 1, …, k − 1}.

□

Corollary 3.3 For k = 1,

μ_{t}^{(k)}(dτ) = dr \frac{1}{k} ∑_{j=0}^{k−1} μ_{t−r}^{(j)}(dτ′) μ_{t−r}^{(k−1−j)}(dτ′′).  (3.2)

Proof This is immediate from the previous lemma and the fact that L(τ) = r + L(τ′) + L(τ′′). □

Definition and Remark 3.4 We write u_{t}^{(k)}(x) := w_{t}^{(k)}(x, a), v_{t}^{(k)}(x) := w_{t}^{(k)}(x, b), and

w_{t}^{(k)}(x, i) := ∫ 𝔼[φ̂_{1} (X_{t,a}^{x,τ_{t}^{(i)}}) φ̂_{2} (X_{t,b}^{x,τ_{t}^{(i)}})] μ_{t}^{(k)}(dτ).  (3.3)

We obviously have

u_{t}(x) = ∑_{k=0}^{∞} u_{t}^{(k)}(x), v_{t}(x) = ∑_{k=0}^{∞} v_{t}^{(k)}(x),  (3.4)

and from Corollary 3.4 and (3.3) it is clear that for k ≥ 1

u_{t}^{(k)}(x) = \frac{1}{k} ∑_{j=0}^{k−1} ∫_{0}^{t} S_{r} (u_{t−r}^{(j)} u_{t−r}^{(k−1−j)}) (x) dr,  (3.5)

with the analogous formula being valid for v_{t}^{(k)}. In order to bound u_{t}^{(k)} in a suitable manner, we are going to work with a decomposition of u_{t}^{(k)} along the “second branch,” splitting off successively all the offspring of newborn type a-individuals. To write this decomposition in a neat form, consider the following stickbreaking scheme:

Let J_{1} be uniformly distributed on {0, 1, …, k − 1}; given J_{1} let J_{2} be uniformly distributed on {0, 1, …, k − J_{1} − 2}; given (J_{1}, J_{2}), let J_{3} be uniformly distributed on {0, 1, …, k − J_{1} − J_{2} − 3}, and so on, till k − J_{1} − J_{2} − ⋯ − J_{N} − N = 0. Iterating (3.5), we arrive at the following decomposition of u_{t}^{(k)}:

u_{t}^{(k)}(x) = 𝔼[∫_{0}^{t} dr_{1} S_{r_{1}} (u_{t−r_{1}}^{(J_{1})} ∫_{0}^{t−r_{1}}dr_{2} T_{r_{2}} (u_{t−r_{1}−r_{2}}^{(J_{2})}  (3.6)

⋯ ∫_{0}^{t−r_{1}−···−r_{N−1}}dr_{N} T_{r_{N}} (u_{t−r_{1}−···−r_{N}} T_{t−r_{1}−···−r_{N}} φ_{2}) ⋯)) (x)] .

With the probability weights

ℙ[N = n; J_{1} = j_{1}, …, J_{n} = j_{n}] =: π(n; j_{1}, …, j_{n}),

(3.6) can be rewritten as

u_{t}^{(k)}(x) = ∑_{n} ∑_{j_{1}+···+j_{n}=0}^{k−n} π (n; j_{1}, …, j_{n}) ∫_{0}^{t} dr_{1} S_{r_{1}} (u_{t−r_{1}}^{(j_{1})} ∫_{0}^{t−r_{1}}dr_{2} T_{r_{2}} (u_{t−r_{1}−r_{2}}^{(j_{2})}

⋯ ∫_{0}^{t−r_{1}−···−r_{n−1}}dr_{n} T_{r_{n}} (u_{t−r_{1}−···−r_{n}} T_{t−r_{1}−···−r_{n}}φ_{2}) ⋯)) (x)

=: ∑_{n} ∑_{j_{1}+···+j_{n}=0}^{k−n} π (n; j_{1}, …, j_{n}) A_{t} (n; j_{1}, …, j_{n}) .  (3.7)

Proposition 3.5 Assume (S_{t}) and (T_{t}) satisfy (1.3), and assume

φ_{i} ≤ S_{1}1_{D}, i = 1, 2,  (3.8)

for some bounded D ⊂ ℝ^{d}. Then

u_{s}^{(k)}(x) ≤ (2^{2+γ})^{k} c_{D}^{k+1}γ^{−k}(s + 1)^{−(1+γ)}  (3.9)

uniformly in s ≥ 0 and x ∈ ℝ^{d}, and

v_{s}^{(k)}(x) ≤ (2^{2+γ})^{k} c_{D}^{k+1}γ^{−k}T_{s}φ_{2}(x).  (3.10)

Proof We will use induction over k.

For k = 0, u_{s}^{(0)}(x) = S_{s}φ_{1}(x) ≤ S_{s+1}1_{D}(x) ≤ c_{D}(s + 1)^{−(1+γ)} by (3.8) and (1.3).

Now assume (3.9) holds true for l = 0, …, k − 1. Since for all s > 0 there holds

∫_{0}^{s}(r + 1)^{−(1+γ)} dr = \frac{1}{γ} (1 − (s + 1)^{−γ}) ≤ \frac{1}{γ} ,

by the induction assumption the term A_{t} (n; j_{1}, …, j_{n}) in the decomposition (3.7) is bounded by

c_{D}^{j_{1}+···+j_{n}+n} (2^{2+γ})^{j_{1}+···+j_{n}} (\frac{1}{γ} )^{j_{1}+···+j_{n}+(n−1)} ∫_{0}^{t} dr_{1} (t + 1 − r_{1})^{−(1+γ)} S_{r_{1}}T_{t−r_{1}}φ_{2}(x)

≤ c_{D}^{k} (2^{2+γ} \frac{1}{γ} )^{k−1} c_{D} ∫_{0}^{t} (t + 1 − r)^{−(1+γ)} (r + 1)^{−(1+γ)} dr  (3.11)

where we again used (1.3) and (3.8) in the last inequality. The induction argument is completed by observing that

∫_{0}^{t} (t + 1 − r)^{−(1+γ)} (r + 1)^{−(1+γ)} dr ≤ 2 (\frac{t}{2} + 1)^{−(1+γ)} ∫_{0}^{\frac{t}{2}}(r + 1)^{−(1+γ)} dr

≤ 2^{2+γ} \frac{1}{γ} (t + 1)^{−(1+γ)}.

In order to prove (3.10) first observe that v_{t}^{(k)} has the same representation as u_{t}^{(k)} in (3.7), but with S_{r_{1}} replaced by T_{r_{1}}. Replacing S_{r_{1}} by T_{r_{1}} also in the LHS of (??) we obtain (3.10). □

To conclude the proof of Theorem 1.1 it suffices to remark that, if the initial conditions φ_{1} and φ_{2} both are multiplied by a factor c > 0, then a factor c^{k+1} enters into both u^{(k)} and v^{(k)}. Hence, due to (3.4), (3.9) and (3.10), u_{t}(x) and v_{t}(x) are majorized by convergent geometric series, provided (1.4) holds true with sufficiently small c > 0. □

Acknowledgement A.W. thanks CIMAT for the kind hospitality during his stay in autumn 1999.

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