Strong global attractors for non-damping weak dissipative abstract evolution equations

Transcript

1 17 3 Journal of East China Normal University Natural Science No. Mar. 17 : ,, 737 :,, V θ V θ L µr + ; V θ. : ; ; : O175.9 : A DOI: /j.issn Strong global attractors for non-amping weak issipative abstract evolution equations ZHANG Yu-bao, WANG Xuan College of Mathematics an Statistics, Northwest Normal University, Lanzhou 737, China Abstract: In this paper, by using the theory of semigroup, contractive function an the metho of efining functionals, the existence of the global attractors for nonamping weak issipative abstract evolution equations with strong solutions in the space V θ V θ L µr + ; V θ was obtaine when the nonlinear term satisfies the weaker issipative conition. Key wors: abstract evolution equation; contractive function; global attractor R 3, αu t t u tt t + ka θ ut + k sa θ ut ss + gut = f, x, t R +, ux, t =, x, t R,.1 ux, t = u x, t, x, t, : : ; 145RJZA11 :,,,. blcx7@gmail.com. :,,,,. wangxuan@nwnu.eu.cn.

2 , : 9. θ >, k, k > s R +, k s. ux, t,,, u = ux,, x..1 [1-], [3].,.,,.,,,., V θ V θ L µr + ; V θ. u tt t + αu t t + ka θ ut + k sa θ ut ss + gut = f, x, t R +,.. A =, θ = 1,, [4-5],. [6]., k s, [7], g., gu = sin u, Sine-Goron [8]. A =, θ =, [9], u.,. [1], María Anguiano. [11],. [1],,..1, [13],, : C >, g C 1 R lim inf y y gss ygy C y y, liminf gss y y. [13],, g 1.5,.,,.,, u t + σu ;,,, Soblev, C.1..1 [14] Banach X {St} t C, ε > B X, tb > X 1, {PStx x B, t tb}, I PStx < ε, t tb, x B,

3 1 17 P : X X 1.,, V θ V θ L µr + ; V θ.3. : 1,, ;, V θ V θ L µr + ; V θ,,. 1 H = L, Au, v = bu, v, u, v H, bu, v H,. A H, DA H. {λ j } j N, {ω j } j N A, {ω j } j N H, Aω j = λ j ω j, < λ 1 λ λ j, λ j j., A A θ, DA θ H. DA = H, DA θ = { V θ, DA θ = Vθ, } DAθ = V θ. V θ = u H : λ θ j u, ω j <. : u, v θ = j=1 λ θ ju, ω j v, ω j, u θ = u, u θ, u, v V θ. j=1 Hilbert, A r DA s DA s r, s, r R. A, A θ. θ 1 θ, DA θ1 DA θ, V θ H H. :V θ V θ H, V θ V θ H = H Vθ,, H, Vθ H, V θ., H, V θ V θ : u, v = u vx, u = u x, u, v H; u, v θ = u, v θ = A θ u A θ vx, u θ = A θ u A θ vx, u θ = A θ u x, u, v V θ ; A θ u x, u, v V θ. L µr + ; V θ R + V θ Hilbert, : ϕ, ψ µ,θ = µs A θ ϕa θ ψxs, ϕ µ,θ = Hilbert H = V θ V θ L µr + ; V θ, : µs A θ ϕ xs. z H = u, u t, η t H = 1 u θ + u t θ + η t µ,θ.

4 , : 11 Poincaré, λ > λ v λ v θ v θ, v V θ. 1.1 η t s = η t x, s = ux, t ux, t s, s R +. µs = k s k = 1,.1 : u tt t + A θ ut + µsa θ η t ss + gut = f, ηts t = ηss t + u t t. - : ux, t =, η t x, s =, x, t, ux, = u x, u t x, = u t x, x, η x, s = u x ux, s, x, s R µs : h 1 µs C 1 R + L 1 R +, µs µ s, s R + ; h µss = k > ; h 3 δ >, µ s + δµs, s R + ; h 4 s >, M >, µ s L, s µ s + Mµs, s s. g :g C R, R, : g y C1 + y, y R, 1.4 gy lim inf > λ. 1.5 y + y, l >, g : 1.1 [15] g y < l. 1.6 I = [, T ], µ h 1 h 3, T >, η t s CI; L µ R+ ; V θ, δ >, η t s, ηss t µ,θ δ ηt s µ,θ. 1.1 [7,16] X Banach, B X. X X φ, B B, {x n } n=1 B,, {x n}, lim n CB B B. lim φx n, x m =, m

5 [7-8,16-17] {St} t Banach X,, B, ε >, T = TB, ε φ T, CB, STx STy ε + φ T x, y, x, y B. {St} t X, {y n } n=1 X {t n }, n, {St n y n } n=1 X. 1. [14] X Banach {St} t X. {St} t X, {St} t : 1S = I ; StSs = St + s, t, s ; 3 t n t, x n x, St n x n Stx. 1.3 [14] X, Y Banach, X, Y, X Y Y X, i : X Y i : Y X. {St} t X Y,, Y. X B, {St} t SB, SB = {x B Stx B, t }. 1.4 [8,14] X Banach, {St} t X. {St} t 1 {St} t B X; {St} t..1, I = [, T ], T >, f V θ. z H, zt = ut, u t t, η t s I, zt 1., ut L I; V θ, u t t L I; V θ L I; V θ, η t s L I; L µr + ; V θ, ηts t + ηss t L I; L µr + ; V θ L I; L µr + ; V θ.,, Galerkin, ,..1, h 1 h , g CV θ ; V θ, f V θ, T > z H, zt = ut, u t t, η t s, zt L [, T ]; H..1, H St : z zt, t R +, {St} t, {St} t H.., N 1 >

6 , : 13, ν 1 >, Et = 1 ut θ + u tt θ + ηt s µ,θ, Ft = u t t µsa θ η t ssx, Gut = ut gssx, J t = N 1 Et + N 1 A θ Gut + Ft + ν 1 A θ utu t tx + C, C >, J t., N 1, ν 1 C 1 >, C >, 1 C 1 Et J t C 1 Et + C..1.1 zt = ut, u t t, η t s f V θ, g CV θ ; V θ , h 1 h 4, Q >, B H, t = t B H, zt H = 1 ut θ + u tt θ + ηt s µ,θ Q, t t. A θ u t t 1. H, 1.1, t Et + Aθ Gut + δ ηt s µ,θ f θ u t t θ k 4N 1 u t t θ + N 1 k f θ.. Ft t Ft = u tt t µsa θ η t ssx u t t µsa θ ηt t ssx..3.3, 1. u tt t µsa θ η t ssx = A θ ut µsa θ η t ssx x + µsa θ η ss t + gut µsa θ η t ssx f µsa θ η t ssx >, Young A θ ut µsa θ η t x ssx + µsa θ η ss t 1 ut θ + k η t s µ,θ..5 1

7 , 1.1 gut µsa θ η t ssx k 1 l ut + g µs A θ η t s s 1 1 ut θ + + k l λ η t s µ,θ + k 1 g..6.4, 1.1, Cauchy Young f µsa θ η t ssx k 1 f η t s µ,θ 1 ηt s µ,θ + k f θ λ..7.3, h h u t t µsa θ ηt t ssx = u t t µsa θ u t t η t sssx = k u t t s θ u t t µ sa θ η t ss + µ sa θ η t ss x s k u t t s θ + µ s u t t µ 1 s µ 1 s A θ η t s s + M µs A θ η t s s x s k u t t µ s 1 θ + µs s + Mk 1 u t t k u tt θ + 1 λ k s µs A θ η t s s µ s µs s 1 + Mk 1 η t s µ,θ..8 t Ft 1 ut θ k u tt θ + C 4 η t s µ,θ + k λ f θ + k g,.9 C 4 = k k + k l 1 λ + 1 s µ s 1 1 λ k µs s. + Mk 1 [15], gut, A θ ut λ γ ut θ 1 γ ut θ λ,.1 γ >., 1..1 A θ utu t tx u t t θ t γ ut θ λ + λ k γ ηt s µ,θ + 1 γ f θ J t N 1δ t + C 4 + ν 1λ k γ + k 4 + ν 1 η t s µ,θ + 1 γν 1 ut θ λ u t t θ + N 1 k f θ + k λ f θ + k g + ν 1 γ f θ. x 1 x

8 , : 15 1 = γν 1 4λ, N 1, ν 1, { N1 δ ε 1 = min.1 C 4 ν 1λ k γ Gronwall. N 1 δ C 4 ν 1λ k γ J t t >, k 4 ν 1 >., k 4 ν 1, γν } 1 N, C 5 = 1 + k + ν 1 f θ + k 4λ k λ γ g, J t t + ε 1 Et C 5. + ε 1 C 1 J t C 5 + ε 1C C 1. J t J e ε 1 C 1 t + C 1C 5 ε 1 + C..1,. f V θ, g CV θ ; V θ , h 1 h 4, {St} t B. B H, t = t B H, StB B..3, {St} t.. f V θ, g CV θ ; V θ , h 1 h 4, {St} t H. z n t = u n t, u nt t, ηn t s z mt = u m t, u mt t, ηm t s , z n = u n, u nt, η n z m = u m, u mt, η m. ωt = u nt u m t, ω t t = u nt t u mt t, ξ t s = ηn t s ηt m s, 1. ω tt t + A θ ωt + µsa θ ξ t ss + gu n t gu m t =,.1 ξt ts = ω tt ξs ts. N >, ν >, E ω t = 1 ωt θ + ω t t θ + ξ t s µ,θ, G ω t = t F ω t = A θ ω s gu n s gu m ssx, ω t t µsa θ ξ t ssx, J ω t = N E ω t + N G ω t + F ω t + ν A θ ωtω t tx + C ω,

9 16 17 C ω >, J ω t. N ν C 6, C 7, 1 C 6 E ω t J ω t C 6 E ω t + C 7. A θ ω t t.1 H, 1.1, F ω t, t F ωt = ω tt t t E ωt + G ω t + δ ξt s µ,θ ,.1 ω tt t µsa θ ξ t ssx = µsa θ ξ t ssx ω t t + A θ ωt µsa θ ξ t ssx µsa θ ξ ss t x + gu n t gu m t µsa θ ξt t ssx..14 µsa θ ξ t ssx , >, Cauchy Young, A θ ωt ωt θ + k µsa θ ξ t ssx + µsa θ ξ ss t x ξ t s µ,θ , C 8 = C gu n t gu m t C 1 + u n t + u m t ωt µsa θ ξ t ssx µsa θ ξ t ssx C 8 ωt θ + k 4λ ξ t s µ,θ, Q λ.

10 , : 17.14, h h 4 1.1, ω t t µsa θ ξtssx t = ω t t = k ω t t θ k ω t t θ + k ω t t θ + µsa θ ω t t ξ t s ssx s ω t t µ sa θ ξ t ss + s ω t t s µ sa θ ξ t ss x s µ s µ 1 s µ 1 s A θ ξ t s s + M µ s 1 µs s 1 + Mk k ω tt θ + 1 s µ s 1 λ k µs s Mk 1 ω t t µs A θ ξ t s s x 1 µs A θ ξ t s s x ξ t s µ,θ..18 t F ωt ωt θ k ω tt θ + C 9 ξ t s µ,θ + C 8 ωt θ,.19 C 9 = k + k + k + 1 s µ s 1 4λ λ k µs s. + Mk 1,.1 1.1, A θ ωtω t tx ω t t θ 1 t ωt θ + k ξt s µ,θ + C 8 ωt θ , J ω t t N δ + C 9 ν k C 8 + ν C 8 ωt θ, J ω t t N δ + C 9 ν k ν k 8λ N + ν 4N F ω t + ν 4 G ωt + ν 4N C 1 = C 8 + 9ν C ν3 8N. ξ t s µ,θ + ν ξ t s µ,θ + ν = ν, N > 1, ν < 1, N δ C 9 ν k ν k 8λ N ωt θ + k ν ω t t θ ωt θ + k 3ν ω t t θ A θ ωtω t tx + ν 4 C ω C 1 ωt θ + ν 4 C ω,.1 ν 4, k 3ν ν 4.

11 18 17 ε = ν 4N,.1 t J ωt + ε J ω t C 1 ωt θ + ν 4 C ω, e εt, t [, T ], ǫ [, T ], t > ǫ, C 11 = C 11 ǫ C 1, J ω T J ω e εt + C 11 T ǫ J ω e εt + C 11 T ωt θ t + C 1 ǫ ωt θt + ν ǫ 4 C ωe εǫ T. ωt θ t + ν ǫ 4 C ωe εǫ T φ T z n, z m = φ T z n t, z m t = C 11 T. ωt θ t + ν ǫ 4 C ωe εǫ T, φ T, z n t = u n t, u nt t, ηn t s z n = u n, u nt, η n B, V θ V θ, u n C[, T ]; V θ, lim ν ε, lim T n m..3, u n s u m s θs =,. ν ǫ lim ǫ 4 C ωe εǫ T =..3 lim n lim φ Tz n, z m =. m 1., {St} t.. 1.4,..3 f V θ, g CV θ ; V θ , h 1 h {St} t H B,, {St} t H A, H - H. [ ] [ 1 ] COLEMAN B D, NOLL W. Founations of linear viscoelasticity [J]. Reviews of Moern Physics, 1961, 33: [ ] DAFERMOS C M. Asymptotic stability in viscoelasticity [J]. Archive for Rational Mechanics an Analysis, 197, 374: [ 3 ] FABRIZIO M, MORRO A. Mathematical Problems in Linear Viscoelasticity [M]. Pennsylvania: Society for Inustrial an Applie Mathematics, 199. [ 4 ] GIORGI C, RIVERA J M, PATA V. Global attractors for a semilinear hyperbolic equations in viscoelasticity [J]. J Math Anal Appl, 1, 61: [ 5 ] PATA V, ZUCCHI A. Attractors for a ampe hyperbolic equation with linear memory [J]. Av Math Sci Appl, 1, 11:

12 , : 19 [ 6 ] MA Q Z, ZHONG C K. Exitence of strong global attractors for hyperbolic equation with linear memory [J]. Applie Mathematics an Computation, 4, 1573: [ 7 ] SUN C Y, CAO D M, DUAN J Q. Non-autonomous wave ynamics with memory-asymptotic regularity an uniform attractors [J]. Discrete & Continuous Dynamical Systems, 8, 93-4: [ 8 ] TEMAM R. Infinite Dimensional Dynamical System in Mechanics an Physics [M]. n e. New York: Spring- Verlag, [ 9 ],,. [J]., 7, 75: [1] ANGUIANO M, MARIN-RUBIO P, REAL J. Regularity results an exponential growth for pullback attractors of a non-autonomous reaction-iffusion moel with ynamical bounary conitions [J]. Nonlinear Analysis Real Worl Applications, 14, 1: [11],,,. [J]., 15, 353: [1] MA Q Z, XU L. Ranom attractors for the extensible suspension brige equation with white noise [J]. Computers & Mathematics with Applications, 15, 71: [13],,. [J]., 1, 83: [14] ZHONG C K, YANG M H, SUN C Y. The existence of global attractors for the norm-to-weak continuous semigroup an application to the nonlinear reaction-iffusion equations [J]. Journal of Differential Equations, 6, 3: [15] WANG X, YANG L, ZHONG C K. Attractors for the nonclassical iffusion equations with faing memory [J]. Journal of Mathematical Analysis & Applications, 1, 36: [16] SUN C Y, CAO D M, DUAN J Q. Non-autonomous ynamics of wave equations with nonlinear amping an critical nonlinearity [J]. Nonlinearity, 6, 1911: [17] ROBINSON J C. Infinite-imensional Dynamical Systems: An Introuction to Dissipative Parabolic PDEs an the Theory of Global Attractors [M]. Cambrige: Cambrige University Press, 1. : 7 [ 9 ] MADAN D B, CARR P P, CHANG E C. The variance gamma process an option pricing [J]. European Finance Review, 1998, : [1] BRODY D C, HUGHSTON L P, MACKIE E. General theory of geometric Lévy moels for ynamic asset pricing [J]. Proceeings of The Royal Society A, 1, 468: [11] MEERSCHAERT M M, SCHEFFLER H P. Triangular array limits for continuous time ranom walks [J]. Stochastic Processes an Their Applications, 8, 118: [1] ZHANG Y X, GU H, LIANG J R. Fokker-Planck type equations associate with suborinate processes controlle by tempere α-stable processes [J]. Journal of Statistical Physics, 13, 154: [13] GAJDA J, A. WYLOMANSKA A. Anomalous iffusion moels: Different types of suborinator istribution [J]. Acta Physica Polonica B, 1, 43: 11. [14] PODLUBNY I, Fractional Differential Equations [M]. New York: Acaemic Press, :