A NEUTRAL ERS ON OF THE D FFUS E HUTCHINSON EQUATION ON? THE ADM SS BLY STABLE MAN FOLDS AND ASYMPTOTIC BEACH OR. HUTCH NSON EQUAT ON THE ADM SS BLY STABLE MAN FOLDS ... study asymptotic behavior of a neutral version of the diffusive Hutchinson equation.ABSTRACT. We investigate a system of distributed delay differentialdifference equations describ ing an epidemic model of susceptible, ...
A NEUTRAL ERS ON OF THE D FFUS E HUTCH NSON EQUAT ON THE ADM SS BLY STABLE MAN FOLDS AND ASYMPTOT C BEHA OR Xuan Quang Bu acult of Mathemat cs and atural Sc ences Ema l quangbx dhhp edu Ngày nh n bài: 28/3/2022 Ngày PB ánh giá: 13/4/2022 Ngày t ng: 27/4/2022 ABSTRACT Consider the neutral evolution equation d dt Fut = B (t )u (t ) + Φ (t , ut ), under the conditions that the family of linear partial differential operators U (t , s ) t s B(t ) t generates the evolutionary process having an exponential dichotomy on the half-line and the nonlinear delay operator Φ (t , ) satis es the -Lipschitz condition, where belongs to some admissible space on the half- line The existence of admissibly stable manifolds of E -class is obtained in Nguyen - Trinh Taiwan J Math 23 (2019), 897-923 This short paper discusses the application of the theoretical result to study asymptotic behavior of a neutral version of the diffusive Hutchinson equation M T PHIÊN B N TRUNG T NH C A PH NG TRÌNH HUTCHINSON CĨ KHU CH T N A T P N NH CH P NH N C V D NG I U TI M C N ABSTRACT Xét ph ng tr nh ti n hố trung tính hàm ri ng n tính B(t ) t d Fut = B (t )u (t ) + Φ (t , ut ), v i i u ki n h toán t dt sinh tr nh ti n hóa U (t , s ) t s o có nh phân m tốn t phi n Φ (t , ) th a m n i u ki n -Lipschitz, ó thu c vào m t kh ng gian ch p nh n c tr n n a ng th ng S t n t i c a a t p n nh ch p nh n c E -l p nh n c c ng tr nh Nguyen - Trinh Taiwan J Math 23 (2019), 897-923 Bài báo th o lu n v áp d ng c a k t qu l thuy t ó nghi n c u dáng i u ti m c n c a m t phi n b n trung tính c a ph ng tr nh Hutchinson có khu ch tán TR NG I H C H I PH NG d Fut = B(t )u (t ) + F(t , ut ), dt u0 = := C( -r,0 , ) B(t ) : ( D( B), D(B ) ) x D(B) F: t >0 F: D(B) X + -r,0 ut ( ) := u(t + ) Lp L p ,q d ( x + * x) - Ax - * x = f , dt t w(t , x) - kw(t - , x) - Dw(t , x) = w(1 - w(t - , x)) ( B(t ))t F(t, ) - F(t, ) (t ) - , () T P CH KHOA H C, S 52, tháng n m 2022 r>0 -r,0 w : -r, ) := C( -r,0 , ) = sup - r ,0 ( ) wt -r,0 (U (t , s))t s wt ( ) = w(t + ) U (t, t ) = I (t, s) U (t, r )U (r, s) = U (t, s) U (t, s) x x U (t , s) x x (U (t , s))t Ke (t -s) x s P(t ),t U (t, s) P(s) = P(t )U (t, s) U (t , s)| : ker P(s ) ker P(t ) U (s, t )| := (U (t , s )| ) -1 Ne- U (t, s) x U ( s, t )| x Ne (t -s ) x - (t -s ) x x P(s) x ker P(t ) (U (t , s))t H := supt t P(t ) < P(t ) ( P(t ))t P(t ) P(t ) : t P(t ) ( ) = U (t - , t ) P(t ) (0), for all P(t )2 = P(t ) F : 0, ) TR X F(t,0) (t ) F(t, ) - F(t , ) NG -r, P(t ) t (t ) + I H C H I PH NG - t + , s ( B(t ))t (U (t , s))t s D(B) Y F: D(B) Y 0 h (t ) := e - t- F(t, ) F: + () E , for all t 0, F(t, ) (t )(1+ ) C ( -r , ), D( B)) F (t , v, ) = - B(t ) Fvt + B(t )v(t ) + F (t , ) F(t, 0, 0) (t ), F (t , u , ) - F (t , v, ) , K Y ut - vt + (t ) - , u,v C( -r, ), D(B)) T P CH KHOA H C, S 52, tháng n m 2022 E (U (t , s))t s u : -r, ) t Fut = U (t , s) Fus + U (t, )F( , u, u )d , for all t > s s - class + = P(t ) ker P(t ) P(t ) y t : P (t ) = (t, + y t ( )) t + := ker P (t ), t P(t ) + ker P(t ) : t + + y t ( ) : (t , + y t ( )) , P(t ) ; P(t ) s - r, ) us = s, ) (t )ut us ( ) := Fust s-r s, us ut ) (t )ut ut ( ) = Fut - , P(t ) t s t s t for all - r ker P(t ) and t P(t ) ker P(t ) = graph( yt ) F K Y e () k1 := N (1 + H )e e (t ) = e - TR NG |t - | E r max E + h () 2K Y + E , N1 L1T1+ Nk1e r k , - k1 - Y 1- Y I H C H I PH NG + N L1 - e0 := C( -1,0 , ) := L2 (W) B : D(B) B( f ) = f + f D( B) = H (W) := { f W 2,2 0, : f (0) = f ( ) = F F: D( B), F ( f ) := f (0) - lf (-1) X F: + F(t, ) := (t) ( -1 )( ) , for all , -1 -1 B(t ) := a(t )B (T (t ))t >0 (B) = {-1 + , -4 + , n - n2 + , (B) i = T P CH KHOA H C, S 52, tháng n m 2022 (T (t )) = et ( B) = et ( (T (t )) { -1) , et ( - 4) :| |=1 = , - n2 ) et ( , , for all t > 0 { :| |1 P = P(1) Q := I - P t >0 Q TQ (t ) := T (t )Q (T (t ))t >0 TQ (t ) T (t ) |P Ne- t , TQ (-t ) = TQ (t ) ( B(t ))t for all t 0, -1 Ne - t , = (a(t ) B)t for all t (U (t , s))t U (t , s) := T (T (t ))t (U (t , s))t N = := t s a( )d s 0 U (t , s ) |P = T t s a( ) d U ( s, t )| = U (t , s ) |ker P Y |P -1 Ne - (t - s ) , t = TQ - a ( )d s Ne - (t - s ) , | l | h (t ) | b | e- t (q( - ))1/ q h (t ) Lp h1 := k 1- Y N (1 + H )e K | l | | b | (e - e - ) K | l | |b| max + , + 1/ q 1/ p 1- | l | (1 - e ) ( q) ( p) (q( - ))1/ q h2 := Nk1e r - k1 - Y N (1 + H )e 2 K | l | (1 - e - )+ | b | (e - e - ) (1- | l |)(1 - e - ) - N (1 + H )e 2K | l | (1 - e - )+ | b | (e - e - ) Lp max h1 , h2 < d Fut = B(t )u (t ) + F(t , ut ) dt F T P CH KHOA H C, S 52, tháng n m 2022 REFERENCES E Hernández, J Wu (2018) Traveling wave front for partial neutral differential equations Proceedings of the American Mathematical Society 146 (2018), 1603 1617 E Hernández, S Tro mchuk (2020) Traveling waves solutions for partial neutral differential equations Journal of Mathematical Analysis and Applications 481 (2020), 123458 N.V Minh, J Wu (2004) Invariant manifolds of partial functional differential equations Journal of Differential Equations 198 (2004), 381 421 N.V Minh, F Räbiger, and R Schnaubelt (1998) Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line Integral Equations Operator Theory 32 (1998), 332 353 T.H Nguyen (2009) Invariant manifolds of admissible classes for semi-linear evolution equations Journal of Differential Equations 246 (2009), 1820 1844 T.H Nguyen, X.-Q Bui (2016) Sectorial TR NG I H C H I PH NG operators and inertial manifolds for partial functional differential equations in admissible spaces Applicable Analysis and Discrete Mathematics 10 (2016), 262 291 T.H Nguyen, X.-Q Bui (2022) On the existence and regularity of admissibly inertial manifolds with sectorial operators Dynamical Systems, https://www.tandfonline.com/doi/abs/10.1 080/14689367.2022.2049706 Published online: 27 Mar 2022 T.H Nguyen, X.Y Trinh (2019) Admissibly stable manifolds for a class of partial neutral functional differential equations on a halfline Taiwanese Journal of Mathematics 23 (2019), no 4, 897 923 H Petzeltová, O.J Staffans (1997) Spectral decomposition and invariant manifolds for some functional partial differential equations Journal of Differential Equations 138 (1997), no 2, 301 327 10 V.D Trinh, T.H Nguyen (2018) Stable and center-stable manifolds of admissible classes for partial functional differential equations Journal of Integral Equations and Applications 30 (2018), no 4, 543 575 ... 327 10 V .D Trinh, T.H Nguyen (2018) Stable and center -stable manifolds of admissible classes for partial functional differential equations Journal of Integral Equations and Applications 30 (2018),... 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