Finite-time Blowup of Semilinear PDEs via the Feynman-Kac Representation. CENTRO DE INVESTIGACIÓN EN MATEMÁTICAS GUANAJUATO, MEXICO

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1 Finite-time Blowup of Semilinear PDEs via the Feynman-Kac Representation JOSÉ ALFREDO LÓPEZ-MIMBELA CENTRO DE INVESTIGACIÓN EN MATEMÁTICAS GUANAJUATO, MEXICO

2 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.

3 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.

4 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?

5 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?

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8 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))

9 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.

10 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.

11 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

12 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

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14 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.

15 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.

16 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.

17 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,

18 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,

19 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.

20 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

21 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.

22 QUOTED REFERENCES Aronson, D. G.; Weinberger, H. F. Multidimensional nonlinear diffusion arising in population genetics. Adv. in Math. 30 (1978), no. 1, Birkner, M., López-Mimbela, J. A. and Walkonbinger, A., Blow-up of Semilinear PDE s at the Critical Dimension. A Probabilistic Approach, Proc. Amer. Math. Soc. 130 (2002), Hayakawa, Kantaro, On nonexistence of global solutions of some semilinear parabolic differential equations. Proc. Japan Acad. 49 (1973), Fujita, H., On the Blowing up of Solutions of the Cauchy Problem for u t = u + u 1+α, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), López-Mimbela, J. A., A Probabilistic Approach to Existence of Global Solutions of a System of Nonlinear Differential Equations, Aportaciones Mat. Notas Investigación 12, , Soc. Mat. Mexicana, México, López-Mimbela, J. A. and Privault, N., Blow-up and Stability of Semilinear PDEs with Gamma Generators, J. Math. Anal. Appl. 307 (2005), López-Mimbela, J. A.and Privault, N., Critical exponents for semilinear PDEs with bounded potentials. Seminar on Stochastic Analysis, Random Fields and Applications V, , Progr. Probab., 59, Birkhäuser, Basel, López-Mimbela, J. A. and Wakolbinger, A., Length of Galton-Watson Trees and Blow-up of Semilinear Systems, J. Appl. Probab. 35 (1998), McKean, H. P. Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl. Math. 28 (1975), no. 3, Nagasawa, M. and Sirao, T., Probabilistic treatment of the blowing up of solutions for a nonlinear integral equation. Trans. Amer. Math. Soc. 139 (1969) Souplet, Ph. and Zhang, Qi S., Stability for semilinear parabolic equations with decaying potentials in R n and dynamical approach to the existence of ground states. Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), no. 5, Sugitani, S., On Nonexistence of Global Solutions for some Nonlinear Integral Equations, Osaka J. Math. 12 (1975), Weissler, F. B., Existence and Nonexistence of Global Solutions for a Semilinear Heat Equation, Israel J. Math. 38 (1981),

23 BOOKS AND REVIEWS Bandle, C. and Brunner, H., Blowup in Diffusion Equations: A Survey, J. Comput. Appl. Math. 97 (1998), no. 1-2, Bebernes, J. and Eberly, D., Mathematical Problems from Combustion Theory, Springer-Verlag, New York, Levine, H. A., The Role of Critical Exponents in Blowup Theorems, SIAM Review 32 (1990), 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, Pao, C. V., Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 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, Quittner, P., Souplet, P., Superlinear parabolic problems. Birkhäuser Advanced Texts Blow-up, global existence and steady states.