Finite-time Blowup of Semilinear PDEs via the Feynman-Kac Representation JOSÉ ALFREDO LÓPEZ-MIMBELA CENTRO DE INVESTIGACIÓN EN MATEMÁTICAS GUANAJUATO, MEXICO jalfredo@cimat.mx
Introduction and backgrownd Consider a reaction-diffusion equation of the form where w t = Lw + w1+β, w(0, x) = ϕ(x), x E, (1) 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 T ϕ (0, ] such that (1) has a unique solution w on R d [0, T ϕ ). w is bounded on R d [0, T ] for any 0 < T < T ϕ. If T ϕ <, then w(, t) L (R d ) as t T ϕ. When T ϕ = we say that w is a global solution When T ϕ < we say that w blows up in finite time or that w is non-global.
The study of blow up properties of (1) goes back to the fundamental work of Fujita (1966): For E = R d 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 = R d 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.
An example with an integral power non-linearity w t = Lw + w2, w(0) = ϕ Blowup or stability of solutions? Recall has solution f(t) = 1 1 K t Thus: Blowup in the absence of motion f t = f 2, f = K For a (small) initial ϕ with bounded support, the motion tends to smear out u, hence counteract the blowup. Where is the border?
Probabilistic representation In case of integer exponents β 1 it was proved by McKean (1975) that w(t, x) = E ϕ(y), x E, t 0. y B x (t) solves the equation w t = Lw w + w1+β, w(0) = ϕ Here (B x (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?
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 ensures i.e. w is a global solution. β 0 sup x E sup y E [T (s)ϕ(y)] β ds < 1 w(t, x) < for all t 0, (e.g. Nagasawa and Sirao (1969) and Weissler (1981))
Constructing Subsolutions by the Feynman-Kac Formula We now consider the equation w(t) t = α w(t) + w(t) 1+β, w(x, 0) = ϕ(x), x R d. (2) Recall that the solution u of the IVP u t = α u(t) + u(t)v(t), (3) u(0, x) = ϕ(x), (t, x) [0, T ) R d with v : [0, T ) R d R + locally bounded, has by the Feynman-Kac formula a probabilistic representation as the density of the measure t ] E x [1(W t dy) exp v(s)(w s ) ϕ(x) dx = u(t, y) dy (4) 0 where E x denotes expectation with respect to the symmetric α-stable process (W t ) started at W 0 = x. The representation (4) shows in particular that any solution ũ of (3) with v replaced by ṽ v and ũ 0 u 0 fulfills ũ u.
Consider the initial value problems f t = α f t, f 0 = ϕ (v t 0) t g t = α g t + f t g t, g 0 = ϕ (v t f t ) t h t = α h t + g t h t, h 0 = ϕ (v t g t ). t Then, by the previous remark i.e., f t, g t and h t are subsolutions of f t g t h t w t, w t t = αw t + w 1+β, w 0 = ϕ.
Basic Estimates We denote by B r the ball in R d with radius r centered at the origin, and write p t (x) for the transition density of (W t ). Let f t (y) := p t (y x)ϕ(x) dx = E y [ϕ(w t )]. Lemma 1. For all t 1 we have the inequality f t (y) c 0 t d/α 1 B1 (t 1/α y) ϕ(x) dx B 1 for some c 0 > 0. Indeed, let y B t 1/α. Then, by the scaling property of W t f t (y) = E y [ϕ(w t )] = E 0 [ϕ(w t + y)] ( )] = E 0 t 1/α (W 1 + t 1/α y) p 1 (x t 1/α y)ϕ(t 1/α x) dx B 1 c 0 ϕ(t B 1/α x) dx 1 = c 0 t d/α ϕ(x) dx B t 1/α c 0 t d/α 1 B1 (t 1/α y) ϕ(x) dx B 1
This argument also shows that, for sufficiently large t, for some c 0 > 0. f t (y) c 0 t d/α 1 B1 (t 1/α y). In the same way one can prove the following Lemma 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], P x {W s B s 1/α W t = y} c. Proof. Using self-similarity, continuity and strict positivity of stable densities, we have that for all s [1, t/2], B s 1/α = p s (z x)p t s (y z) dz p t (y x) B s 1/α s d/α p 1 (s 1/α (z x))(t s) d/α p 1 ((t s) 1/α (y z)) dz t d/α p 1 (t 1/α (y x)) s d/α (t s) d/α (inf w B 2 p 1 (w)) 2 t d/α p 1 (0) s d/α Vol(B s 1/α) (inf w B 2 p 1 (w)) 2. 2p 1 (0) B s 1/α dz
Let g t solve g t t = αg t + g t f β t, g 0 = ϕ, (5) where f t is defined in Lemma 1. Proposition 1 Let d < α/β. Then g t grows to uniformly on the unit ball as t, i.e., lim inf g t (x) =. t x B 1 Proof. From the Feynman-Kac representation we know that g t is given by t ] g t (y) = ϕ(x) p t (y x)e x [exp f s (W s ) β ds W t = y dx. Using Lemma 1 and Jensen s inequality, it follows that for y B t 1/α, g t (y) ϕ(x) p t (y x)e x [exp t/2 1 0 c 1 s βd/α 1 (W Bs 1/α s) ds W t = y t/2 ϕ(x) p t (y x) exp (c 2 s βd/α P x {W s B s 1/α W t = y} ds t/2 c 3 t d/α exp (c 4 s βd/α ds 1 1 ) ] dx ) dx, (6) where we have used Lemma 2 to obtain the last inequality, and where c i, i = 1, 2, 3, 4, are positive constants. The result follows from the condition d < α/β.
Since w t g t, from Proposition 1 it follows that Completion of the proof of blowup K(t) := inf x B 1 w t (x) as t. (7) We re-start (2) with the initial condition w t0, with a suitable choice of t 0 given below. Writing u t = w t0 +t, the equation becomes Its integral form is Noting that Moreover, u t (x) = ζ := min x B 1 min u t t = αu t + u 1+β t, u 0 (x) = w t0 (x), x R d. 0 s 1 min u t (x) v(t), x B 1 p t (y x)u 0 (y) dy + t 0 ds p t s (y x)u s (y) 1+β dy. P x {W s B 1 } > 0, we deduce that, for every t [0, 1], min u t (x) ζk(t 0 ) + ζ x B 1 where t 0 ( min u s (y) y B 1 ) 1+β ds. and v(t) blows up at time v(t) = ζk(t 0 ) + ζ τ = t 0 1 βζ 1+β K(t 0 ) β. v(s) 1+β ds, Due to (7), we can choose t 0 so big that τ < 1. This yields min x B 1 u 1 (x) =, which shows blowup of w t0 +1.
Remark By a second application of the Feynman-Kac formula, one can easily prove that the subsolution h t h t t = αh t + g β t h t, h 0 = ϕ, (where g t is the subsolution obtained above), is such that inf h t (x) as t x B 1 even if d = α/β. Hence, our equation has no positive global solutions if d α/β.
Other Generators Consider the one-dimensional semilinear equation where and w t t = Γ w t + νt σ w 1+β t, w 0 (x) = ϕ(x), x R +, (8) Γ f(x) = ν, σ, β 0, ϕ 0 0 [f(x + y) f(x)] e y y 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. dy,
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 c 1 x a 1 ϕ(x), x > x 0 for some positive constants x 0, c 1, a 1, where a 1 β < 1 + σ, produces a non-global solution. On the other side, if ϕ fulfils ϕ(x) c 2 x a 2, x > x 0, where x 0, c 2, a 2 are positive numbers and a 2 β > 1 + σ, then the solution w t to (8) is global and, moreover, for some constant C > 0. 0 w t (x) Ct a 2, x 0,
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/β ϕ(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/β ϕ(x) = 0.
We consider now the semilinear equation A generator with a bounded potential w t t (x) = w t(x) V (x)w t (x) + t ζ w 1+β t (x), (9) w 0 (x) = ϕ(x), x R d 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 where (T t ) t 0 is the Brownian semigroup, and therefore which implies T V t ϕ e V t ϕ ( β sup T V t ϕ(x) ) β dt < 1 0 x R d for all sufficiently small ϕ and any β > 0. := e V t T t
Using the Feynman-Kac approach, one can prove [P. Souplet and Q.S. Zhang], [LM, N. Privault] that If d 3 and a 0 V (x) 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 and such that 0 < β (a) β (a) < 2/d any nontrivial positive solution is nonglobal provided 0 < β < β (a), if β (a) < β, then nontrivial positive global solutions may exist.
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