Blow-up profiles of solutions for the exponential reaction-diffusion equation

Transcription

1 Blow-up profiles of solutions for the exponential reaction-diffusion equation Aappo Pulkkinen Department of Mathematics and Systems Analysis Aalto University School of Science and Technology Finland 4th Euro-Japanese Workshop on Blow-up, Leiden, Sep 6-10, 2010.

2 The equation Consider the following equation u t = u+f(u), x Ω, t > 0, u = 0, x Ω, t > 0, u(x,0) = u 0 (x), x Ω, (1) where Ω = B(R) = {x R n : x < R} and u 0 C 1 (Ω). Blow-up profiles of solutions for the exponential reaction-diffusion equation p.1

3 The equation Consider the following equation u t = u+f(u), x Ω, t > 0, u = 0, x Ω, t > 0, u(x,0) = u 0 (x), x Ω, (2) where Ω = B(R) = {x R n : x < R} and u 0 C 1 (Ω). I will focus on the nonlinearities f(u) = e u and f(u) = u u p 1 with p > 1. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.1

4 Preliminaries Subcritical case: n 2 or n > 2 and 1 < p < p s = n+2 n 2 (when f(u) = u u p 1 ). Blow-up profiles of solutions for the exponential reaction-diffusion equation p.2

5 Preliminaries Subcritical case: n 2 or n > 2 and 1 < p < p s = n+2 n 2 (when f(u) = u u p 1 ). A solution is said to blow-up if sup u(x,t), as t T. x Ω Pioneering works for sufficient conditions [Kaplan 63], [Fujita 66]. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.2

6 Preliminaries Subcritical case: n 2 or n > 2 and 1 < p < p s = n+2 n 2 (when f(u) = u u p 1 ). A solution is said to blow-up if sup u(x,t), as t T. x Ω Pioneering works for sufficient conditions [Kaplan 63], [Fujita 66]. Blow-up is said to be of type I if the blow-up rate is the same as the BU rate of the ODE v = f(v). f(u) = e u : C 1 log(t t)+ u(x,t) C 2 f(u) = u u p 1 : (T t) 1/(p 1) u(x,t) C Blow-up profiles of solutions for the exponential reaction-diffusion equation p.2

7 Type I BU BU is type I in the subcritical range. [Giga, Kohn 85, 87], [Giga, Matsui, Sasayama 04] for f(u) = u u p 1. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.3

8 Type I BU BU is type I in the subcritical range. [Giga, Kohn 85, 87], [Giga, Matsui, Sasayama 04] for f(u) = u u p 1. BU is type I also in the supercritical range when u is radially symmetric and [Matano, Merle 04] f(u) = u u p 1 and p S < p < p JL = {, n 10 4 n 4 2 n 1, n > 10 [Fila, P. 08] f(u) = e u and 2 < n < 10. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.3

9 Similarity variables Assume x = 0 is the blow-up point. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.4

10 Similarity variables Assume x = 0 is the blow-up point. BU solutions are often treated with respect to similarity variables s = log(t t) and y = x/ T t. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.4

11 Similarity variables Assume x = 0 is the blow-up point. BU solutions are often treated with respect to similarity variables s = log(t t) and y = x/ T t. Rescaling for f(u) = e u : w(y,s) = log(t t)+u(x,t) w s = w y 2 w+ew 1 Blow-up profiles of solutions for the exponential reaction-diffusion equation p.4

12 Similarity variables Assume x = 0 is the blow-up point. BU solutions are often treated with respect to similarity variables s = log(t t) and y = x/ T t. Rescaling for f(u) = e u : and for f(u) = u u p 1 : w(y,s) = log(t t)+u(x,t) w s = w y 2 w+ew 1 w(y,s) = (T t) 1/(p 1) u(x,t) w s = w y 2 w 1 p 1 w+w w p 1 + appropriate boundary and initial conditions Blow-up profiles of solutions for the exponential reaction-diffusion equation p.4

13 Steady states [Giga, Kohn 85, 89] For f(u) = u p and subcritical case the only steady states of the rescaled equation are {0, κ, κ} and for u 0 0 it holds w(y,s) κ = ( 1 p 1 )1/(p 1), as s. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.5

14 Steady states [Giga, Kohn 85, 89] For f(u) = u p and subcritical case the only steady states of the rescaled equation are {0, κ, κ} and for u 0 0 it holds w(y,s) κ = ( 1 p 1 )1/(p 1), as s. [Eberly, Troy 87, Troy 87, Budd, Qi 89, Lepin 88] In the supercritical case (and p < p JL ) there is at least a countable family {ϕ j } j of steady states satisfying { for f(u) = e u ϕ j y : 2 ϕ j +e ϕ j 1 = 0, y > 0, ϕ j (0) = α j, ϕ j (0) = 0, { for f(u) = u u p 1 ϕ j y 2 : ϕ j 1 p 1 ϕ j +ϕ j ϕ j p 1 = 0, y > 0, ϕ j (0) = β j, ϕ j (0) = 0, Blow-up profiles of solutions for the exponential reaction-diffusion equation p.5

15 Convergence ofw [Giga, Kohn, 85, 87, 89, Matos 99, Matano, Merle 04] for f(u) = u u p 1 gives that w(y,s) always converges to some stationary solution ϕ of the rescaled equation. For f(u) = e u the function w(y,s) converges to some stationary solution ϕ at least when BU type I [Matos 01, Matano, Merle 04, Fila, P. 08] Also convergence to a nonconstant ϕ occurs. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.6

16 Convergence ofw [Giga, Kohn, 85, 87, 89, Matos 99, Matano, Merle 04] for f(u) = u u p 1 gives that w(y,s) always converges to some stationary solution ϕ of the rescaled equation. For f(u) = e u the function w(y,s) converges to some stationary solution ϕ at least when BU type I [Matos 01, Matano, Merle 04, Fila, P. 08] Also convergence to a nonconstant ϕ occurs. Assume that for some stationary solution ϕ. w(y,s) ϕ(y) Blow-up profiles of solutions for the exponential reaction-diffusion equation p.6

17 Convergence ofw [Giga, Kohn, 85, 87, 89, Matos 99, Matano, Merle 04] for f(u) = u u p 1 gives that w(y,s) always converges to some stationary solution ϕ of the rescaled equation. For f(u) = e u the function w(y,s) converges to some stationary solution ϕ at least when BU type I [Matos 01, Matano, Merle 04, Fila, P. 08] Also convergence to a nonconstant ϕ occurs. Assume that for some stationary solution ϕ. w(y,s) ϕ(y) What is the behavior of u(x,t) as x 0? Blow-up profiles of solutions for the exponential reaction-diffusion equation p.6

18 Constant local profile [Herrero, Velazquez 92-93] for f(u) = u p and subcritical n = 1, u 0 0 and Ω = R n. If w(y,s) κ as s, then lim x 0 x 2/(p 1) log x 1/(p 1) u(x,t) = C, or lim x 0 x m/(p 1) u(x,t) = C, for m 4. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.7

19 Constant local profile [Herrero, Velazquez 92-93] for f(u) = u p and subcritical n = 1, u 0 0 and Ω = R n. If w(y,s) κ as s, then lim x 0 x 2/(p 1) log x 1/(p 1) u(x,t) = C, or lim x 0 x m/(p 1) u(x,t) = C, for m 4. [Velazquez 92] for Ω = R n and n 1 and type I BU. [Matos 01] for a Ω a ball and radially symmetric u. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.7

20 Constant local profile [Herrero, Velazquez 92-93] for f(u) = u p and subcritical n = 1, u 0 0 and Ω = R n. If w(y,s) κ as s, then lim x 0 x 2/(p 1) log x 1/(p 1) u(x,t) = C, or lim x 0 x m/(p 1) u(x,t) = C, for m 4. [Velazquez 92] for Ω = R n and n 1 and type I BU. [Matos 01] for a Ω a ball and radially symmetric u. [Fila, P. 08] If f(u) = e u and u radially symmeric and w(y,s) 0 as s, then lim x 0 (u(x,t)+2log x log log x ) = C, or lim x 0 (u(x,t)+mlog x ) = C, for some m 4. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.7

21 Nonconstant local profile [Matano, Merle 08] Let f(u) = u u p 1 and p > p s and u radially symmetric and BU type I. Then if and only if w(y,s) ϕ(y) constant, as s a = lim L 1 x 2/(p 1) u(x,t) < x 0 ( ) and a 0,±1. Above L p 1 = 2 p 1 n 2 2 p 1. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.8

22 Theorem 1 Let f(u) = e u and BU type I and assume that w(y,s) ϕ(y) constant, as s. Then a = lim (u(x,t)+2log x log(2(n 2))) <, x 0 and a = lim y (ϕ(y)+2log y log(2(n 2))) 0. [P., Final time blow-up profile for some superlinear reaction-diffusion equations, in prep.] Blow-up profiles of solutions for the exponential reaction-diffusion equation p.9

23 Comments The proof of [Matano, Merle 08] uses energy estimates and suitable supersolutions to obtain apriori bounds for solutions. Using those techniques is more difficult for the exponential. Our proof uses only semigroup regularization estimates and variation of constants formula. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.10

24 Comments The proof of [Matano, Merle 08] uses energy estimates and suitable supersolutions to obtain apriori bounds for solutions. Using those techniques is more difficult for the exponential. Our proof uses only semigroup regularization estimates and variation of constants formula. Also type II BU is covered by [Matano, Merle 08]. They prove: type II w(y,s) ϕ (y) (the singular stationary solution) profile u(x,t) with a = ±1. The problem with f(u) = e u is already in proving that type II BU implies w(y,s) ϕ (y), since we don t have the apriori estimates. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.10

25 Comments 2 In this result itself we do not need to assume radial symmetry. Proof works also for f(u) = u u p 1. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.11

26 Comments 2 In this result itself we do not need to assume radial symmetry. Proof works also for f(u) = u u p 1. Are there nonradial stationary solutions ϕ? Are there nonradial solutions w that converge to a radial ϕ? Blow-up profiles of solutions for the exponential reaction-diffusion equation p.11

27 Corollary Applying [Fila, P. 08] and a result in [Vazquez 99] this gives the following. If n [3,9] and u is a radially symmetric minimal L 1 -solution on (0,T ) that blows up at t = T < T, then u is regular in (T,T +ǫ) for some ǫ > 0 and log(t T)+u(x,t) <, for t (T,T +ǫ) and where lim (log(t T)+u( t Ty,t)) = ψ(y), t T { ψ + y 2 ψ +eψ +1 = 0, y > 0 ψ(0) = β, ψ(0) = 0. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.12

28 Comments 3 Immediate regularization is proved in [Fila,Matano,Polacik 05] for f(u) = e u and radially decreasing and minimal L 1 -solutions. In [Matano, Merle 08] they prove immediate regularization for f(u) = u u p 1 and also for nonminimal continuatiations. Blow-up profiles of solutions for the exponential reaction-diffusion equation p.13