Nonexistence of Global Solutions to Generalized Boussinesq Equation with Arbitrary Energy

Transcription

1 Nonexistence of Global Solutions to Generalized Boussinesq Equation with Arbitrary Energy M. Dimova, N. Kolkovska, N. Kutev Institute of Mathematics and Informatics - Bulgarian Academy of Sciences ICRAPAM 015, June 3 6, 015, Istanbul, Turkey Supported by Bulgarian Science Fund, Grant DFNI I - 0/9 M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 1/15

2 Outline 1 Introduction Finite time blow up by means of the potential well method 3 Finite time blow up for arbitrary high positive energy Modification of the concavity method of Levine Construction of the initial data Numerical experiments M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up /15

3 Boussinesq equation with linear restoring force (BE lrf ) β u tt u xx β 1 u ttxx + u xxxx + mu + f(u) xx = 0, u(0, x) = u 0 (x), u t (0, x) = u 1 (x), (t, x) R + R β 1 0, β > 0, m 0; m = 0 generalized double dispersion equation u 0 H 1 (R), ( ) 1/ u 0 L (R) u 1 H 1 (R), ( ) 1/ u 1 L (R) Nonlinearities l s f(u) = a k u pk 1 u b j u q j 1 u, a 1 > 0, a k 0, b j 0 k=1 j=1 1 < q s < q s 1 < < q 1 < p 1 < p < < p l f(u) = au 5 ± bu 3, a > 0, b > 0 in the theory of atomic chains and shape memory alloys M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 3/15

4 Modeling A.M. Samsonov, Nonlinear waves in elastic waveguides, Springer, 1994 A.D. Mishkis, P.M. Belotserkovskiy, ZAMM Z. Angew. Math. Mech. 79 (1999) transverse deflections of an elastic rod on an elastic foundation A.V. Porubov, Amplification of nonlinear strain waives in solids, World Scientific, 003 Numerical Study C.I. Christov, T.T. Marinov, R.S. Marinova, Math. Comp. Simulation 80 (009) M.A. Cristou, AIP Conf. Proc (011) Theoretical Investigations N. Kutev, N. Kolkovska, M. Dimova, AIP Conf. Proc. 169 (014) N. Kutev, N. Kolkovska, M. Dimova, Pliska Studia Mathematica Bulgarica 3 (014) M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 4/15

5 ) Notations: u, v = β (( ) 1/ u, ( ) 1/ v + β 1(u, v) u, v H 1, ( ) 1/ u, ( ) 1/ v L Conservation of full energy E(t) = E(0) t [0, T m ), E(t) = 1 ( ) u t, u t + u H + m ( ) 1/ u 1 L R u 0 f(y) dy dx E(0) 0 0 < E(0) d E(0) > d M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 5/15

6 ) Notations: u, v = β (( ) 1/ u, ( ) 1/ v + β 1(u, v) u, v H 1, ( ) 1/ u, ( ) 1/ v L Conservation of full energy E(t) = E(0) t [0, T m ), E(t) = 1 ( ) u t, u t + u H + m ( ) 1/ u 1 L R u 0 f(y) dy dx E(0) 0 0 < E(0) d E(0) > d Non-positive energy M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 5/15

7 ) Notations: u, v = β (( ) 1/ u, ( ) 1/ v + β 1(u, v) u, v H 1, ( ) 1/ u, ( ) 1/ v L Conservation of full energy E(t) = E(0) t [0, T m ), E(t) = 1 ( ) u t, u t + u H + m ( ) 1/ u 1 L R u 0 f(y) dy dx E(0) 0 0 < E(0) d E(0) > d Non-positive energy Subcritical and critical energy Potential Well Method M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 5/15

8 ) Notations: u, v = β (( ) 1/ u, ( ) 1/ v + β 1(u, v) u, v H 1, ( ) 1/ u, ( ) 1/ v L Conservation of full energy E(t) = E(0) t [0, T m ), E(t) = 1 ( ) u t, u t + u H + m ( ) 1/ u 1 L R u 0 f(y) dy dx E(0) 0 0 < E(0) d E(0) > d Non-positive energy Subcritical and critical energy Potential Well Method Supercritical energy M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 5/15

9 Potential well method (0 < E(0) < d) Payne LE, Sattinger DH, Israel Journal of Mathematics, 1975 wave equation Wang S, Chen G, Nonlinear Anal 006 m = 0, uf(u) ( + ε) u 0 f(s) ds, u R, ε > 0 Liu Y, Xu R, J Math Anal Appl 008 m = 0, f(u) = ± u p, ± u p 1 u Xu R, Math Meth Appl Sci 011 m = 0, β 1 = 0, combined power-type nonlinearity Polat N, Ertas A, Math Anal Appl 009 m = 0, damped Boussinesq equation Potential energy functional J(u): J(u) = 1 u H + m ( ) 1/ u 1 L R u Nehari functional I(u): I(u) = J (u)u = I(u) = u H + m ( ) 1/ u 1 L Depth of the potential well (critical energy constant) d: 0 f(y) dydx R u d = inf u N J(u), N = {u H1 : I(u) = 0, u H 1 0} 0 f(y) dydx M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 6/15

10 Potential well method (0 < E(0) < d) Theorem Suppose u 0 H 1, ( ) 1/ u 0 L, u 1 L, and ( ) 1/ u 1 L. Assume that 0 < E(0) < d. If I(0) > 0 or u 0 H 1 = 0 then the weak solution of BE lrf is globally defined for t [0, ); If I(0) < 0 then the weak solution of BE lrf blows up for a finite time. Lemma (Sign preserving property of I(u(t))) If 0 < E(0) < d then the sign of the Nehari functional I(u(t)) is invariant under the flow of BE lrf. Kalantarov Ladyzhenskaya 1978 (Levine 1974, δ = 0, µ = 0) Ψ (t)ψ(t) γψ (t) δψ(t)ψ (t) µψ (t), γ > 1, δ 0, µ 0 Ψ(t) = u(t), u(t) M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 7/15

11 Arbitrary high positive energy Theorem (Finite time blow up) Suppose u 0 H 1, ( ) 1/ u 0 L, u 1 L and ( ) 1/ u 1 L. If ( ) 1/ m 0 = min m, 1 β β1 and u 0, u 0 0, u 0, u 1 > 0, m 0 (p 1 1) (p 1 + 1) u 0, u u 0, u 1 u 0, u 0 > E(0) > 0 then the weak solution of BE lrf blows up for a finite time t <. More precisely: If If m 0 m 0 (p 1 1) (p 1 + 1) u0, u0 E(0), then t t1 = (p 1 1) u0, u0 E(0), then (p 1 + 1) t t = (p 1 1) m 0 u 0, u 0 (p 1 1) u 0, u < ; 1 u 0, u 0. (p 1 1) u0, u0 + 1 u 0,u 1 E(0) (p 1 +1) u 0,u 0 M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 8/15

12 Kutev, Kolkovska, Dimova 015 Ψ (t)ψ(t) γψ (t) αψ (t) βψ(t), γ > 1, α 0, β > 0 Straughan 1975, Korpusov 01 Ψ (t)ψ(t) γψ (t) βψ(t), γ > 1, β 0 Lemma (Kutev, Kolkovska, Dimova 015) Suppose that a twice-differentiable function Ψ(t) satisfies on t 0 the following inequality and initial conditions: Ψ (t)ψ(t) γψ (t) αψ (t) βψ(t), γ > 1, α 0, β > 0, Ψ(0) > 0, Ψ (0) > 0, β < Then Ψ(t) for t t <. (γ 1) Ψ (0) Ψ(0) + αψ(0). M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 9/15

13 Comparison with previous results, f(u) = a u p 1 u, a > 0 Cai Donghong, Ye Jianjun, Blow-up of solution to Cauchy problem for the generalized damped Boussinesq equation, WSEAS Transactions on Mathematics, 13 (014), 1 131, m > 0 1 u 0, u 1 u 0, u 0 > E(0) > 0 Ψ (t)ψ(t) γψ (t) βψ(t), γ > 1, β 0 Kutev, Kolkovska, Dimova 015 m 0 (p 1) (p + 1) u 0, u u 0, u 1 u 0, u 0 > E(0) > 0 Ψ (t)ψ(t) γψ (t) αψ (t) βψ(t), γ > 1, α 0, β > 0 M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 10/15

14 Construction of initial data Theorem For every positive constant K there exist infinitely many initial data u K 0, u K 1 such that E(uK 0, uk 1 ) = K and the existing time for the corresponding solution u K is finite. Choice of initial data u K 0 (x) = qw (x), u K 1 (x) = qw (x) + sv (x), w(x), v(x) H (R) : w H 0, v H 0, w, v = 0, w, v = 0 q, s positive constants u K 0, u K 0 = 0, u K 0, u K 1 > 0, E(u K 0, u K 1 ) = E(0) > K, m 0(p 1 1) (p 1 + 1) uk 0, u K 0 + uk 0, u K 1 u K 0, uk 0 > E(0) > 0 K is an arbitrary positive constant M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 11/15

15 Family of finite difference schemes, f(u) = a u p 1 u, a > 0 ( v n+1 i B vi n + ) vn 1 i τ Λ xx vi n + β Λ xxxx vi n mvi n ( a v n+1 (p + 1) Λxx i (p+1) v n 1 i v n+1 i B = (β + m τ )I (β 1 + στ )Λ xx + στ β Λ xxxx v n 1 i (p+1) ) = 0, v n i a discrete approximation to u at (x i, t n), τ is a time-step Λ xx v i = v i+1 vu i +v i 1 h, Λ xxxx v i = v i+ 4v i+1 + 6v i 4v i 1 + v i h 4, σ - parameter Properties: Convergence: The schemes have second order of convergence in space and time O(h + τ ). Stability: The schemes are unconditionally stable for σ > 1/4. Conservativeness: The discrete energy is conserved in time, i.e. E h (v (n) ) = E h (v (0) ), n = 1,,... M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 1/15

16 Example 1: β 1 = β = 1, m = 1, m 0 = 1, a =, p = 3 (f(u) = u 3 ), K = 10 > d u 0 (x) = qw (x), u 1 (x) = qw (x) + sv (x) w(x) = e x e (0.4x ) e (0.4x+) + 1, q = 1., s = 5.78, E h (0) > d v(x) = w (x) Figure: Profiles of the numerical solution u(x, t) at evolution times t = 0 and t=1.997; t = blow up time (τ = ). M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 13/15

17 Example : β 1 = β = 1, m = 1, m 0 = 1, a =, p = 3 (f(u) = u 3 ), K = 0 > d u 0 (x) = qw (x), w(x) = (cosh(x)) 1, u 1 (x) = qw (x) + sv (x) v(x) = sinh(x) tanh(x) q = 0.85, s = 3.64, E h (0) > d Figure: Profiles of the numerical solution u(x, t) at evolution times t = 0 and t=1.3; t = 1.33 blow up time (τ = ) M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 14/15

18 Thank you for your attention! M. Dimova, N. Kolkovska, N. Kutev - ICRAPAM 015 Generalized Boussinesq Equation Blow up 15/15