Finite-time Blowup of Semilinear PDEs via the Feynman-Kac - - PowerPoint PPT Presentation

Finite-time Blowup of Semilinear PDEs via the Feynman-Kac Representation

JOS´

E ALFREDO L ´ OPEZ-MIMBELA

CENTRO DE INVESTIGACI ´

ON EN MATEM´ ATICAS

GUANAJUATO, MEXICO jalfredo@cimat.mx

Introduction and backgrownd

Consider a reaction-diffusion equation of the form ∂w ∂t = Lw + w1+β, w(0, x) = ϕ(x), x ∈ E, (1) where L is the generator of a strong Markov process in a locally compact space E, β > 0 is constant, ϕ ≥ 0 is bounded and measurable. Well-known facts:

• ∀ ϕ ≥ 0 ∃ Tϕ ∈ (0, ∞] such that (1) has a unique solution w on Rd × [0, Tϕ).
• w is bounded on Rd × [0, T] for any 0 < T < Tϕ.
• If Tϕ < ∞, then w(·, t)L∞(Rd) → ∞ as t ↑ Tϕ.

When Tϕ = ∞ we say that w is a global solution When Tϕ < ∞ we say that w blows up in finite time or that w is non-global.

1

The study of blow up properties of (1) goes back to the fundamental work of Fujita (1966): For E = Rd and L = ∆

• If d < 2/β, then any non-trivial positive solution blows up in finite time.
• If d > 2/β, then Equation (1) admits a global solution for all sufficiently small initial

values ϕ. Later on, Hayakawa (1973) and Aronson and Weinberger (1978) proved that the critical dimension d = 2 β also pertains to the finite-time blowup regime. The case of L = ∆α := −(−∆)α/2 in E = Rd was settled by Sugitani (1975), who showed that if d ≤ α/β, then for any non-vanishing initial condition the solution blows up in finite time.

2

An example with an integral power non-linearity

∂w ∂t = Lw + w2, w(0) = ϕ Blowup or stability of solutions? Recall ∂f ∂t = f 2, f = K has solution f(t) = 1

1 K − t

Thus: Blowup in the absence of motion For a (small) initial ϕ with bounded support, the motion tends to smear out u, hence counteract the blowup. Where is the border?

3

Probabilistic representation In case of integer exponents β ≥ 1 it was proved by McKean (1975) that w(t, x) = E

 

y∈Bx(t)

ϕ(y)

  ,

x ∈ E, t ≥ 0. solves the equation ∂w ∂t = Lw − w + w1+β, w(0) = ϕ Here (Bx(t))t≥0 is a branching particle system in E

• starting from an ancestor at x ∈ E,
• with exponential (mean–one) individual lifetimes,
• branching numbers 1 + β
• particle motions with generator L.

How to remove the “Feynman-Kac” term −w ?

4

FACTS:

• w is nonglobal provided that

T(t)ϕ(x) ≥

1

tβ

1/β

for some x ∈ E and t ≥ 0, where {T(t)}t≥0 is the semigroup with generator L.

• The condition

β

∞

sup

y∈E [T(s)ϕ(y)]β ds < 1

ensures sup

x∈E

w(t, x) < ∞ for all t ≥ 0, i.e. w is a global solution. (e.g. Nagasawa and Sirao (1969) and Weissler (1981))

5

Constructing Subsolutions by the Feynman-Kac Formula We now consider the equation ∂w(t) ∂t = ∆αw(t) + w(t)1+β, w(x, 0) = ϕ(x), x ∈ Rd. (2) Recall that the solution u of the IVP ∂u ∂t = ∆αu(t) + u(t)v(t), (3) u(0, x) = ϕ(x), (t, x) ∈ [0, T) × Rd with v : [0, T) × Rd → R+ locally bounded, has by the Feynman-Kac formula a proba- bilistic representation as the density of the measure

• Ex
• 1(Wt ∈ dy) exp

t

v(s)(Ws)

• ϕ(x) dx = u(t, y) dy

(4) where Ex denotes expectation with respect to the symmetric α-stable process (Wt) started at W0 = x. The representation (4) shows in particular that any solution u of (3) with v replaced by

• v ≤ v and

u0 ≤ u0 fulfills u ≤ u.

6

Consider the initial value problems ∂ft ∂t = ∆αft, f0 = ϕ (vt ≡ 0) ∂gt ∂t = ∆αgt + ftgt, g0 = ϕ (vt ≡ ft) ∂ht ∂t = ∆αht + gtht, h0 = ϕ (vt ≡ gt). Then, by the previous remark ft ≤ gt ≤ ht ≤ wt, i.e., ft, gt and ht are subsolutions of ∂wt ∂t = ∆αwt + w1+β, w0 = ϕ.

7

Basic Estimates We denote by Br the ball in Rd with radius r centered at the origin, and write pt(x) for the transition density of (Wt). Let ft(y) :=

• pt(y − x)ϕ(x) dx = Ey [ϕ(Wt)] .

Lemma 1. For all t ≥ 1 we have the inequality ft(y) ≥ c0t−d/α1B1(t−1/αy)

• B1

ϕ(x) dx for some c0 > 0. Indeed, let y ∈ Bt1/α. Then, by the scaling property of Wt ft(y) =

Ey [ϕ(Wt)] = E0 [ϕ(Wt + y)]

=

E0

• ϕ
• t1/α(W1 + t−1/αy)
• ≥
• B1

p1(x − t−1/αy)ϕ(t1/αx) dx ≥ c0

• B1

ϕ(t1/αx) dx = c0t−d/α

• Bt1/α

ϕ(x) dx ≥ c0t−d/α1B1(t−1/αy)

• B1

ϕ(x) dx

8

This argument also shows that, for sufficiently large t, ft(y) ≥ c′

0t−d/α1B1(t−1/αy).

for some c′

0 > 0.

In the same way one can prove the following Lemma 2. There exists a c > 0 such that for all t ≥ 2, y ∈ Bt1/α, x ∈ B1 and s ∈ [1, t/2],

Px {Ws ∈ Bs1/α| Wt = y} ≥ c.

• Proof. Using self-similarity, continuity and strict positivity of stable densities, we have that

for all s ∈ [1, t/2],

• Bs1/α

ps(z − x)pt−s(y − z) pt(y − x) dz =

• Bs1/α

s−d/αp1(s−1/α(z − x))(t − s)−d/αp1((t − s)−1/α(y − z)) t−d/αp1(t−1/α(y − x)) dz ≥ s−d/α(t − s)−d/α t−d/α · (infw∈B2 p1(w))2 p1(0)

• Bs1/α

dz ≥ s−d/αVol(Bs1/α)(infw∈B2 p1(w))2 2p1(0) . 9

Let gt solve ∂gt ∂t = ∆αgt + gtf β

t ,

g0 = ϕ, (5) where ft is defined in Lemma 1. Proposition 1 Let d < α/β. Then gt grows to ∞ uniformly on the unit ball as t → ∞, i.e., lim

t→∞ inf x∈B1 gt(x) = ∞.

• Proof. From the Feynman-Kac representation we know that gt is given by

gt(y) =

• ϕ(x) pt(y − x)E x
• exp

t

fs(Ws)β ds

• Wt = y
• dx.

Using Lemma 1 and Jensen’s inequality, it follows that for y ∈ Bt1/α,

gt(y) ≥

• ϕ(x) pt(y − x)E x
• exp

t/2

1

c1s−βd/α1Bs1/α(Ws) ds

• Wt = y
• dx

≥

• ϕ(x) pt(y − x) exp
• c2

t/2

1

s−βd/αPx{Ws ∈ Bs1/α| Wt = y} ds

• dx

≥ c3t−d/α exp

• c4

t/2

1

s−βd/αds

• ,

(6)

where we have used Lemma 2 to obtain the last inequality, and where ci, i = 1, 2, 3, 4, are positive constants. The result follows from the condition d < α/β.

10

Completion of the proof of blowup

Since wt ≥ gt, from Proposition 1 it follows that K(t) := inf

x∈B1

wt(x) → ∞ as t → ∞. (7) We re-start (2) with the initial condition wt0, with a suitable choice of t0 given below. Writing ut = wt0+t, the equation becomes ∂ut ∂t = ∆αut + u1+β

t

, u0(x) = wt0(x), x ∈ Rd. Its integral form is ut(x) =

• pt(y − x)u0(y) dy +

t

ds

• pt−s(y − x)us(y)1+β dy.

Noting that ζ := min

x∈B1

min

0≤s≤1 Px {Ws ∈ B1} > 0,

we deduce that, for every t ∈ [0, 1], min

x∈B1

ut(x) ≥ ζK(t0) + ζ

t

• min

y∈B1

us(y)

1+β

ds. Moreover, min

x∈B1

ut(x) ≥ v(t), where v(t) = ζK(t0) + ζ

t

v(s)1+β ds, and v(t) blows up at time τ = 1 βζ1+βK(t0)β . Due to (7), we can choose t0 so big that τ < 1. This yields min

x∈B1

u1(x) = ∞, which shows blowup of wt0+1. 11

Remark By a second application of the Feynman-Kac formula, one can easily prove that the subsolution ht ∂ht ∂t = ∆αht + gβ

t ht,

h0 = ϕ, (where gt is the subsolution obtained above), is such that inf

x∈B1 ht(x) → ∞

as t → ∞ even if d = α/β. Hence, our equation has no positive global solutions if d ≤ α/β.

12

Other Generators Consider the one-dimensional semilinear equation ∂wt ∂t = Γwt + νtσw1+β

t

, w0(x) = ϕ(x), x ∈ R+, (8) where ν, σ, β ≥ 0, ϕ ≥ 0 and

Γf(x) =

∞

• [f(x + y) − f(x)] e−y

y dy, i.e. Γ is the generator of the standard Gamma process.

• The Γ-process enjoys no self-similarity or symmetry, nor dimensional-dependent

behavior.

• The transition densities of the Γ-motion process are explicitly given,
• The bridges of the Γ-process are beta distributed.

13

In this case, we need to impose conditions on the decay of the initial value ϕ(x) as x → ∞. By adapting the Feynman-Kac approach one can prove [LM, N. Privault]:

• Every bounded, measurable initial condition ϕ ≥ 0 satisfying

c1x−a1 ≤ ϕ(x), x > x0 for some positive constants x0, c1, a1, where a1β < 1 + σ, produces a non-global solution.

• On the other side, if ϕ fulfils

ϕ(x) ≤ c2x−a2, x > x0, where x0, c2, a2 are positive numbers and a2β > 1 + σ, then the solution wt to (8) is global and, moreover, 0 ≤ wt(x) ≤ Ct−a2, x ≥ 0, for some constant C > 0.

14

• In case of σ = 0 with

ϕ(x) ∼ cx−a as x → ∞ for some constants c > 0 and a > 0, we have – explosion in finite time if aβ < 1, – existence of global solutions if aβ > 1.

• If σ = 0 and

lim inf

x→∞ x−ε+1/βϕ(x) > 0,

for some ε > 0, then the solution of (8) blows up in finite time, whereas it is global provided that lim inf

x→∞ xε+1/βϕ(x) = 0.

15

A generator with a bounded potential We consider now the semilinear equation ∂wt ∂t (x) = ∆wt(x) − V (x)wt(x) + tζw1+β

t

(x), (9) w0(x) = ϕ(x), x ∈ Rd where β > 0, ζ > 0 ϕ ≥ 0 and V is a bounded potential. The case ζ = 0 has been studied by Souplet and Zhang. Notice that V > 0 is constant and ϕ∞ ≤ 1 ⇒ w is a global solution. This is so because, in case of a constant V , L = ∆ − V generates the semigroup T V

t

:= e−V tTt where (Tt)t≥0 is the Brownian semigroup, and therefore T V

t ϕ ≤ e−V tϕ∞

which implies β

∞

sup

x∈Rd

• T V

t ϕ(x)

β dt < 1

for all sufficiently small ϕ and any β > 0.

16

Using the Feynman-Kac approach, one can prove [P . Souplet and Q.S. Zhang], [LM, N. Privault] that

• If d ≥ 3 and

0 ≤ V (x) ≤ a 1 + |x|b, x ∈ Rd, (10) for some a > 0 and b ∈ [2, ∞), then b > 2 implies finite time blow-up of (9) for all 0 < β < 2/d.

• If b = 2, then there exists β∗(a) < 2/d such that blow-up occurs if 0 < β < β∗(a).

Moreover,

• If for some a > 0 and 0 ≤ b < 2,

V (x) ≥ a 1 + |x|b, x ∈ Rd, (11) then (9) admits a global solution for all β > 0 and all non-negative initial values satisfying ϕ(x) ≤ c/(1 + |x|σ) for a sufficiently small constant c > 0 and all σ obeying σ ≥ b/β.

• Two critical exponents β∗(a), β∗(a) were found for the quadratic decay case

V (x) ∼+∞ a(1 + |x|2)−1, a > 0, with 0 < β∗(a) ≤ β∗(a) < 2/d and such that – any nontrivial positive solution is nonglobal provided 0 < β < β∗(a), – if β∗(a) < β, then nontrivial positive global solutions may exist. 17

QUOTED REFERENCES Aronson, D. G.; Weinberger, H. F. Multidimensional nonlinear diffusion arising in population genetics. Adv. in Math. 30 (1978), no. 1, 33–76. Birkner, M., L´

• pez-Mimbela, J. A. and Walkonbinger, A., Blow-up of Semilinear PDE’s at the Critical Dimen-
• sion. A Probabilistic Approach, Proc. Amer. Math. Soc. 130 (2002), 2431-2442.

Hayakawa, Kantaro, On nonexistence of global solutions of some semilinear parabolic differential equations.

• Proc. Japan Acad. 49 (1973), 503–505.

Fujita, H., On the Blowing up of Solutions of the Cauchy Problem for ut = ∆u + u1+α, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109-124. L´

• pez-Mimbela, J. A., A Probabilistic Approach to Existence of Global Solutions of a System of Nonlinear

Differential Equations, Aportaciones Mat. Notas Investigaci´

• n 12, 147–155, Soc. Mat. Mexicana, M´

exico, 1996. L´

• pez-Mimbela, J. A. and Privault, N., Blow-up and Stability of Semilinear PDEs with Gamma Generators,
• J. Math. Anal. Appl. 307 (2005), 181-205.

L´

• pez-Mimbela, J. A.and Privault, N., Critical exponents for semilinear PDEs with bounded potentials. Sem-

inar on Stochastic Analysis, Random Fields and Applications V, 243–259, Progr. Probab., 59, Birkh¨ auser, Basel, 2008. L´

• pez-Mimbela, J. A. and Wakolbinger, A., Length of Galton-Watson Trees and Blow-up of Semilinear

Systems, J. Appl. Probab. 35 (1998), 802-811. McKean, H. P . Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl. Math. 28 (1975), no. 3, 323–331. Nagasawa, M. and Sirao, T., Probabilistic treatment of the blowing up of solutions for a nonlinear integral

• equation. Trans. Amer. Math. Soc. 139 (1969) 301–310.

Souplet, Ph. and Zhang, Qi S., Stability for semilinear parabolic equations with decaying potentials in Rn and dynamical approach to the existence of ground states. Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire 19 (2002), no. 5, 683–703. Sugitani, S., On Nonexistence of Global Solutions for some Nonlinear Integral Equations, Osaka J. Math. 12 (1975), 45-51. Weissler, F. B., Existence and Nonexistence of Global Solutions for a Semilinear Heat Equation, Israel J.

• Math. 38 (1981), 29-40.

18

BOOKS AND REVIEWS Bandle, C. and Brunner, H., Blowup in Diffusion Equations: A Survey, J. Comput. Appl. Math. 97 (1998),

• no. 1-2, 3–22.

Bebernes, J. and Eberly, D., Mathematical Problems from Combustion Theory, Springer-Verlag, New York, 1989. Levine, H. A., The Role of Critical Exponents in Blowup Theorems, SIAM Review 32 (1990), 262-288. Deng, K. and Levine, H. A., The role of critical exponents in blow-up theorems: the sequel. J. Math. Anal.

• Appl. 243 (2000), no. 1, 85–126.

Pao, C. V., Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. Samarskii, A.A., Galaktionov, V.A et al, Blow-up in Quasilinear Parabolic Equations, The Gruyter Expositions in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1995. Quittner, P ., Souplet, P ., Superlinear parabolic problems. Blow-up, global existence and steady states. Birkh¨ auser Advanced Texts 2007. 19