Finite-time blowup and existence of global positive solutions of a semi-linear SPDE

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Stochastic Processes and their Applications 2 (2 767 776 www.elsevier.

1 Stochastic Processes and their Applications 2 ( Finite-time blowup and existence of global positive solutions of a semi-linear SPE Marco ozzi a,, José Alfredo López-Mimbela b a IECN, Nancy Universités, B.P. 7239, 5456, Vandoeuvre-lès-Nancy, France b Centro de Investigación en Matemáticas, Apartado Postal 42, 36 Guanajuato, Mexico Received August 29; received in revised form 27 November 29; accepted 8 ecember 29 Available online 2 January 2 Abstract We consider stochastic equations of the prototype du(t, x = ( u(t, x + u(t, x +β dt + κu(t, x dw t on a smooth domain R d, with irichlet boundary condition, where β, κ are positive constants and {W t, t } is a one-dimensional standard Wiener process. We estimate the probability of finite-time blowup of positive solutions, as well as the probability of existence of non-trivial positive global solutions. c 29 Elsevier B.V. All rights reserved. MSC: 35R6; 6H5; 74H35 Keywords: Blowup of semi-linear equations; Stochastic partial differential equations; Weak and mild solutions. Introduction Let R d be a bounded domain with smooth boundary. We consider a semi-linear equation of the form du(t, x = ( u(t, x + G(u(t, x dt + κu(t, x dw t, t >, u(, x = f (x, x, ( u(t, x =, t, x, where G : R R + is locally Lipschitz and satisfies G(z Cz +β for all z >, (2 Corresponding author. addresses: dozzi@iecn.u-nancy.fr (M. ozzi, jalfredo@cimat.mx (J.A. López-Mimbela /$ - see front matter c 29 Elsevier B.V. All rights reserved. doi:.6/j.spa

2 768 M. ozzi, J.A. López-Mimbela / Stochastic Processes and their Applications 2 ( C, β and κ are given positive numbers, {W t, t } is a standard one-dimensional Brownian motion on a stochastic basis (Ω, F, (F t, t, P, and f : R + is of class C 2 and not identically zero. We assume (2 in Sections 3 only; it is replaced by ( in Section 4. Since we do not assume G to be Lipschitz, blowup of the solution ( in finite time cannot be excluded, and our aim is to give estimates of the probability of blowup and conditions for the existence of a global solution of (. A (random time T is called blowup time of u if lim sup t T x u(t, x = + P-a.s. on {T < + }. In the classical (deterministic case where G(z = z +β and κ =, it is well known that for a nonnegative f L 2 (, the condition f (xψ(xdx > λ /β (3 already implies finite-time blowup of ( (see e.g. 9], Page. Here λ > is the first eigenvalue of the Laplacian on, and ψ the corresponding eigenfunction normalized so that ψ L =. The existence, uniqueness and trajectorial regularity of global solutions of parabolic equations perturbed by a time-homogeneous white noise have been investigated by different methods (see e.g. Chueshov and Vuillermot 4], enis et al. 6], Gyöngy and Rovira ], Krylov 2], Lototsky and Rozovskii 3], Mikulevicius and Pragarauskas 5]. Several types of solutions have been proposed (see especially the last cited reference for strong solutions, and the regularity results show that the solution is much smoother in the space variable than for equations perturbed by space-dependent white noise. Let us recall the notions of weak and mild solutions of ( we are going to use here. Let τ + be a stopping time. A continuous F t -adapted random field u = {u(t, x, t, x } is a weak solution of ( on the interval ], τ provided that, for every ϕ C 2 ( vanishing on, there holds u(t, xϕ(x dx = f (xϕ(x dx + u(s, x ϕ(x + G(u(s, xϕ(x] dx ds + κ u(t, xϕ(x dx dw s P-a.s. for all t, τ. Let {S t, t } be the semigroup of d-dimensional Brownian motion with variance parameter 2, killed at the boundary of. A continuous F t -adapted random field u = {u(t, x, t, x } is a mild solution of ( on the interval ], τ if it satisfies u(t, x = S t f (x + P-a.s. and x-a.e. in St r (G(u(r, (x dr + κ S t r (u(r, (x dw r ] for all t ], τ (see e.g. 7], Chapter IV. We refer to ] for background on existence of weak and mild solutions, and for their equivalence under local Lipschitz conditions on G. Let us note that the results in ] hold for a more general class of second order differential operators which includes the Laplacian as a special case. The positivity of the solution of ( follows from comparison theorems (see e.g. Bergé et al. 2] or Manthey and Zausinger 4].

3 M. ozzi, J.A. López-Mimbela / Stochastic Processes and their Applications 2 ( Our aim in this communication is to study the blowup behaviour of u by means of a related random partial differential equation (see (4 below. In Section 3 we describe the blowup behaviour of the solution v of this random partial differential equation in terms of the first eigenvalue and the first eigenfunction of the Laplace operator on. This is done by solving explicitly a stochastic equation in the time variable which is obtained from the weak form of (4. The solution of this differential equation can be written in terms of integrals of exponential Brownian motion with drift. The results of ufresne 7] and Yor 8] on the law of these integrals easily imply estimates for the probability of existence of a global solution, or of blowup in finite time of u and v. In Section 4 sufficient conditions for v to be a global solution are given in terms of the semigroup of the Laplace operator using recent sharp results on its transition density. These conditions show in particular that the initial condition f has to be small enough in order to avoid for a given G the blowup of v, and that the presence of noise may help to prevent blowup. The results of Section 4 can be used to investigate the blowup behaviour of u by means of conditions (2 and (. 2. A related random partial differential equation In this section we investigate the random partial differential equation v κ2 (t, x = v(t, x t 2 v(t, x + e κw t G(e κw t v(t, x, t >, x, v(, x = f (x, x, v(t, x =, x. This equation is understood trajectorywise and classical results for partial differential equations of parabolic type apply to show existence, uniqueness and positivity of a solution up to eventual blowup (see e.g. Friedman 8] Chapter 7, Theorem 9. Proposition. Let u be a weak solution of (. Then the function v defined by solves (4. v(t, x = e κw t u(t, x, t, x. Remark. Proposition implies in particular that ( possesses a strong local solution. Proof. Recall that Itô s formula states that {e κw t, t } is the semimartingale given by e κw t = κ dw s + κ2 2 ds. Let us write u(t, ϕ u(t, xϕ(x dx. Then a weak solution of ( can be written as u(t, ϕ = u(, ϕ + u(s, ϕ ds + G(u(s, ϕ ds + κ u(s, ϕ dw s. Therefore, for ϕ fixed, {u(t, ϕ,τ (t, t } is again a semimartingale. By applying the integration by parts formula (see e.g. Klebaner ], Ch. 8 we get v(t, ϕ := v(t, xϕ(x dx (4

4 77 M. ozzi, J.A. López-Mimbela / Stochastic Processes and their Applications 2 ( = v(, ϕ + + e κw, u(, ϕ du(s, ϕ + ] (t, where the quadratic variation is given by e κw, ] u(, ϕ (t = κ 2 u(s, ϕ ds, t. Therefore, v(t, ϕ = v(, ϕ + u(s, ϕ ( κ dw s + κ2 2 e κw s ds (u(s, ϕ + G(u(s, ϕ ds + κ u(s, ϕ dw s κ u(s, ϕ dw s + κ2 t u(s, ϕ ds κ 2 u(s, ϕ ds 2 ] = v(, ϕ + v(s, ϕ + G(e κw v(s, ϕ κ2 v(s, ϕ ds. 2 The last equality implies that v is a weak solution to (4 which is differentiable with respect to t up to eventual blowup. By uniqueness of weak solutions, v solves (4 and is in C 2 ( (as is u. Moreover, by self-adjointness of the Laplacian, and the fact that ϕ(x = for x, v(s, ϕ = v(s, x ϕ(x dx = v(s, xϕ(x dx = v(s, ϕ. 3. An estimate of the probability of blowup Notice that G := G/C satisfies G (z z +β for all z >. Hence, without loss of generality we can (and will assume that C = in (2. Let ψ be the eigenfunction corresponding to the first eigenvalue λ of the Laplacian on, normalized by ψ(x dx =. It is well known that ψ is strictly positive on (see e.g. 5], Corollary ue to Proposition we have that ] v(t, ψ = v(, ψ + v(s, ψ κ2 t v(s, ψ ds + G(e κw v(s, ψ ds. 2 Moreover, v(s, ψ = λ v(s, ψ, (5 and, due to (2, G(e κw s v(s, xψ(x dx e κβw s v(s, x +β ψ(x dx. (6 By Jensen s inequality v(s, x +β ψ(x dx +β v(s, xψ(x dx] = v(s, ψ +β, (7

5 and therefore M. ozzi, J.A. López-Mimbela / Stochastic Processes and their Applications 2 ( d v(t, ψ dt (λ + κ2 2 v(t, ψ + e κβw t v(t, ψ +β. Hence v(t, ψ I (t for all t, where I ( solves d (λ dt I (t = + κ2 I (t + e κβw s I (t +β, 2 and is given by with I (t = e (λ +κ 2 /2t { τ := inf t I ( = v(, ψ, v(, ψ β β e (λ +κ 2 β /2βs+κβW s ds], t < τ, e (λ +κ 2 /2βs+κβW s ds β v(, ψ β }. (8 It follows that I exhibits finite-time blowup on the event τ < ]. Since I v(, ψ, τ is an upper bound for the blowup time of v(, ψ, and therefore for the blowup times of v and u. Remark 2. The same formula for the blowup time of a stochastic differential equation, containing a stochastic integral with respect to W, has been obtained in Bandle et al. ]. The argument based on the first eigenvalue (and the corresponding eigenfunction of the Laplace operator on is applied there directly to u, and leads to a stochastic differential inequality for u(t, ψ. The associated stochastic differential equation can again be solved explicitly by means of the Itô calculus, and the same formula as above is obtained for the blowup time of the solution of this equation. Both approaches are therefore equivalent, but the approach in ] requires a more complicated comparison theorem for stochastic differential inequalities. Let us now give an estimate for the probability of blowup in finite time of v. From (8, Pτ = + ] = P exp( (λ + κ 2 /2βs + κβw s ds < ] β v(, ψ β for all t > = P exp( (λ + κ 2 /2βs + κβw s ds ] β v(, ψ β = P exp(2 βw (µ s ds β v(, ψ β ], (9 where W s (µ := µs + W s, µ := (λ + κ 2 /2/κ, and β := κβ/2. Setting µ = µ/ β we get, by performing the time change s s( β 2, Pτ = + ] = P 4 κ 2 β 2 It follows from 8] (Chapter 6, Corollary.2 that exp(2w ( µ s ds = 2Z µ exp(2w ( µ s ds β v(, ψ β ]. (

6 772 M. ozzi, J.A. López-Mimbela / Stochastic Processes and their Applications 2 ( in distribution, where Z µ is a random variable with law Γ ( µ, i.e. P(Z µ dy = Γ ( µ e y y µ dy. We get therefore (see also formula..4 ( in 3] Pτ = + ] = β v(,ψ β h(ydy, where h(y = (κ2 β 2 y/2 (2λ +κ 2 /κ 2 β ( yγ ((2λ + κ 2 /(κ 2 β exp 2 κ 2 β 2. y In this way we have proved the following Proposition 3. The probability that the solution of ( blows up in finite time is lower bounded by + h(y dy. β v(,ψ β Remark 4.. Notice that formula..4 in 3] (pp. 264 and 645 expresses the probability density function of exp( (λ +κ 2 /2βs +κβw s ds in terms of the inverse Laplace transform of a multiple of the Whittaker function. (For an explicit definition and basic properties of the Whittaker function see 3], p. 64. Remark 4.2. By putting κ = we get v = u and, moreover, in (9 we obtain that Pτ = + ] = or according to f (xψ(x dx > λ/β or f (xψ(x dx λ/β, which is a probabilistic counterpart to (3. 4. Nonexplosion of v We consider again Eq. (4 and assume that κ and that G : R + R + satisfies G( =, G(z/z is increasing and G(z Λz +β for all z >, ( where Λ and β are certain positive numbers. Let {S t, t } again denote the semigroup of d- dimensional Brownian motion killed at the boundary of. Recall that Eq. (4 can be re-written as ( v(t, x = e κ2t/2 S t f (x + e κ2 (t r/2 S t r (e κw r G e κw r v(r, (x dr. (2 We give now a sufficient condition for the existence of a global solution of (4. Theorem 5. Assume that f satisfies Λβ e κβw r e κ2 r/2 S r f β dr <. (3 Then Eq. (4 admits a global solution v(t, x that satisfies e κ2t/2 S t f (x v(t, x ( Λβ, t. (4 t eκβw r e κ2r/2 S r f β β dr

7 M. ozzi, J.A. López-Mimbela / Stochastic Processes and their Applications 2 ( Proof. efining ( B(t = Λβ we get B( = and e κβw r e κ2 r/2 S r f β dr β, t, db dt (t = ΛeκβW t e κ2t/2 S t f β B +β (t, which implies B(t = + Λ e κβw r e κ2 r/2 S r f β B +β (r dr. Suppose now that (t, x V t (x is a nonnegative continuous function such that V t ( C (, t, and e κ2 t/2 S t f (x V t (x B(te κ2 t/2 S t f (x, t, x. (5 Let R(V (t, x := e κ2t/2 S t f (x + e κw r e κ2 (t r/2 S t r (G(e κw r V r ( (x dr. (6 Then, ( R(V (t, x = e κ2t/2 S t f (x + e κw r G(e e κ2 (t r/2 κw r V r ( S t r V r ( (x dr V r ( e κ2t/2 S t f (x ( t + e κw r e κ2 (t r/2 G(e κw r B(r e κ2r/2 S r f S t r V (r (x dr B(r e κ2r/2 S r f e κ2 t/2 S t f (x + Λ e κβw r B +β (r e κ2r/2 S r f β e κ2 (t r/2 S t r (e κ2r/2 S r f (x dr ] = e κ2t/2 S t f (x + Λ e κβw r B +β (r e κ2r/2 S r f β dr = e κ2 t/2 S t f (xb(t, (7 where to obtain the first inequality we used the rightmost inequality in (5 and the fact that G(z/z is increasing, and to obtain the second inequality we used (. Consequently, Let e κ2 t/2 S t f (x R(V (t, x B(te κ2 t/2 S t f (x, t, x. v t (x := e κ2 t/2 S t f (x and v n+ t (x = R(v n (t, x, n =,, 2,.... Obviously vt (x R(v (t, x = vt (x. Suppose that vn t (x vt n (x for some n and all t and x. Since by assumption G(z/z is increasing, G(e κw r v n r ( = G(eκW r v n r ( e κw r v n r ( = G(e κw r v n r (. e κw r vr n ( G(eκWr vr n ( e κw r vr n e κw r vr n ( (

8 774 M. ozzi, J.A. López-Mimbela / Stochastic Processes and their Applications 2 ( Combining the above inequality with (6 render vt n+ (x = R(v n (t, x R(v n (t, x = vt n (x, t, x, which shows that (v n n= is a monotone increasing sequence of nonnegative functions. Letting n yields, for t and x, v(t, x = lim n vn t (x B(te κ2 t/2 S t f (x e κ2t/2 S t f (x ( Λβ eκβw r e κ2r/2 S r f dr β /β. Hence, v(t, x is a global solution of (2 due to the monotone convergence theorem. Remark. If we modify (4 and (2 by replacing G(e κw r v(t, x by G(v(t, x, then a global positive solution still exists for all f small enough, even if the inequality in ( holds only for z (, C, where C is some positive constant. In fact, if f satisfies f C ( Λβ e κw r e κ2 r/2 S r f β dr β, (8 then Theorem 5 still holds if we replace the factor e κβw r in (3 and (4 by the factor e κw r. We only have to verify that assuming z (, C in ( already implies the second inequality in (7: where e κ2 t/2 S t f f B (t = This yields C ( Λβ ( Λβ e κw r e κ2r/2 S r f β β dr. e κw r e κ2r/2 S r f β β C dr = B (t for all t, B (t e κ2 t/2 S t f (, C (9 for all t, since f. Let us now proceed to derive a sufficient condition for (3 in terms of the transition kernels {p t (x, y, t > } of {S t, t } and the first eigenvalue λ and corresponding eigenfunction ψ. We recall the following sharp bounds for {p t (x, y, t > }, which we borrowed from Ouhabaz and Wang 6]. Theorem 6. Let ψ > be the first irichlet eigenfunction on a connected bounded C,α - domain R d, where α > and d, and let p t (x, y be the corresponding irichlet heat kernel. There exists a constant c > such that, for any t >, max {, c } t (d+2/2 e λ t sup x,y p t (x, y ψ(xψ(y + c( t (d+2/2 e (λ 2 λ t,

9 M. ozzi, J.A. López-Mimbela / Stochastic Processes and their Applications 2 ( where λ 2 > λ are the first two irichlet eigenvalues. This estimate is sharp for both short and long times. The above theorem is useful in verifying condition (3. Indeed, let the initial value f be chosen so that f (y K S η ψ(y, y, (2 where η is fixed and K > is a sufficiently small constant to be specified later on. Therefore S t f K S t+η ψ, and for any t >, S t f (x K p t+η (x, yψ(y dy = K e λ (t+η p t+η(x, y ψ(xψ(y e λ(t+η ψ(xψ 2 (y dy ( 2 K sup ψ(x e λ p (t+η t+η (x, y sup x x,y ψ(xψ(y e λ(t+η ψ(y dy ( 2 K sup ψ(x x ( + c( (t + η (d+2/2 e (λ 2 λ (t+η e λ(t+η ψ(y dy ( 2 ( = K sup ψ(x e λ(t+η + ce λ 2(t+η x ( 2 K ( + ce λ η sup ψ(x e λ t x ψ(y dy, ψ(y dy which is independent of x. Since the function (t, x S t f (x is uniformly bounded in x, condition (3 is satisfied provided that ( 2 ] β Λβ K ( + ce λ η sup ψ(x ψ(y dy dr e κβw r (λ +κ 2 /2βr <, x or dr e κβw r (λ +κ 2 /2βr < Λβ e λ βη ( K ( + c sup ψ(x x 2 Notice that condition (8 is satisfied if K in (2 is sufficiently small. We have proved the following ψ(y dy ] β. (2 Theorem 7. Let G satisfy (, and let be a connected, bounded C,α -domain in R d, where α >. If (2 and (2 hold for some η > and K >, then the solution of Eq. (2 is global. Remark. ( The integral on the left of (2 coincides with the corresponding integral in Sections 2 and 3. The same type of bounds as in Section 3 can therefore be applied to estimate the probability of existence of a global positive solution. By means of the law of the iterated

10 776 M. ozzi, J.A. López-Mimbela / Stochastic Processes and their Applications 2 ( logarithm for W we see from (2 that the presence of a noise may help to prevent blowup in finite time. (2 If G(z = Λz +β, the results of this section can be applied to the solution u of Eq. ( since v(t, x = e κw t u(t, x, t, x. Acknowledgements M. ozzi would like to thank Prof. Z. Brzeźniak for discussions on blowup of PEs and SPEs and acknowledges the European Commission for partial support by the grant PIRSES 2384 Multifractionality. J.A. López-Mimbela acknowledges the hospitality of Institut Elie Cartan, Nancy University, where part of this work was done. References ] C. Bandle, M. ozzi, R. Schott, Blow up behaviour of a stochastic partial differential equation of reaction diffusion type. Stochastic analysis: random fields and measure-valued processes (Ramat Gan, 993/995, Israel Math. Conf. Proc. Bar-Ilan Univ., Ramat Gan ( ] B. Bergé, I.. Chueshov, P.-A. Vuillermot, On the behaviour of solutions to certain parabolic SPE s driven by Wiener processes, Stochastic Process. Appl. 92 ( ] Andrei N. Borodin, Paavo Salminen, Handbook of Brownian Motion Facts and Formulae, second edition, in: Probability and its Applications, Birkhäuser, Verlag, Basel, 22. 4] Igor. Chueshov, Pierre A. Vuillermot, Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Itô s case, Stochastic Anal. Appl. 8 (4 ( ] E.B. avies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, ] Laurent enis, Anis Matoussi, Lucretiu Stoica, L p estimates for the uniform norm of solutions of quasilinear SPE s, Probab. Theory Related Fields 33 (4 ( ] aniel ufresne, The distribution of a perpetuity, with applications to risk theory and pension funding, Scand. Actuar. J. ( 2 ( ] Avner Friedman, Partial ifferential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J, ] H. Fujita, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, in: Nonlinear Functional Analysis, Providence, R.I., 97, Proc. Sympos. Pure Math. 8 ( ( ] István Gyöngy, Carles Rovira, On L p -solutions of semilinear stochastic partial differential equations, Stochastic Process. Appl. 9 ( ( ] Fima C. Klebaner, Introduction to Stochastic Calculus with Applications, second edition, Imperial College Press, London, 25. 2] N.V. Krylov, An analytic approach to SPEs. Stochastic partial differential equations: Six perspectives, in: Math. Surveys Monogr., vol. 64, Amer. Math. Soc., Providence, RI, 999, pp ] Sergey Lototsky, Boris Rozovskii, Stochastic differential equations: A Wiener chaos approach, in: From Stochastic Calculus to Mathematical Finance, Springer, Berlin, 26, pp ] R. Manthey, T. Zausinger, Stochastic evolution equations in Lρ 2ν, Stoch. Stoch. Rep. 66 ( ] R. Mikulevicius, H. Pragarauskas, On Cauchy-irichlet problem for parabolic quasilinear SPEs, Potential Anal. 25 ( ( ] El Maati Ouhabaz, Feng-Yu Wang, Sharp estimates for intrinsic ultracontractivity on C,α -domains, Manuscripta Math. 22 (2 ( ] A. Pazy, Semigroups of Linear Operators and Applications to Partial ifferential Equations, in: Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, ] Marc Yor, Exponential Functionals of Brownian Motion and Related Processes. Springer Finance, Springer-Verlag, Berlin, 2.

Large time behavior of reaction-diffusion equations with Bessel generators

Large time behavior of reaction-diffusion equations with Bessel generators José Alfredo López-Mimbela Nicolas Privault Abstract We investigate explosion in finite time of one-dimensional semilinear equations