具有多项式记忆的非经典反应扩散方程全局吸引子的存在性.pdf

Pure Mathematics n,2023,13(8),2330-2344Published Online August 2023 in Hans.https:/www.hanspub.org/journal/pmhttps:/doi.org/10.12677/pm.2023.138241kP?;A*?f?35?|?“O?=vF2023c7?3FF2023c8?4FuF2023c8?11F?kP?;A*?f?35 k|Faedo-GarlerkinU?O?)?35?Xy?k.8?35?f?35cP;A*?fExistence of Global Attractorsof Non-Classical Reaction-DiffusionEquation with PolynomialMemoryHan YangCollege of Mathematics and Statistics,Northwest Normal University,Lanzhou GansuReceived:Jul.3rd,2023;accepted:Aug.4th,2023;published:Aug.11th,2023:?|.kP?;A*?f?35J.n,2023,13(8):2330-2344.DOI:10.12677/pm.2023.138241?|AbstractIn this paper,we consider existence of global attractors for non-classical reaction-diffusion equation with polynomial memory.We first obtain existence and uniquenessof the solution by using Faedo-Garlerkin method and energy estimation,then proveexistence of bounded absorbing set,finally we get existence of the global attractors.KeywordsPolynomial Memory,Non-Classical Reaction-Diffusion Equation,Global AttractorCopyright c?2023 by author(s)and Hans Publishers Inc.This work is licensed under the Creative Commons Attribution International License(CC BY 4.0).http:/creativecommons.org/licenses/by/4.0/1.0?XeK:tu 4tu 4u R0(t s)|u(s)|u(s)ds+|u(t)|u(t)=0,x ,t 0,u(x,t)=0,x ,t 0,u(x,t)=u0(x),x ,t 6 0,(1.1)R3k1w.?k.m,0 6 6 0,0(s)6 0,s R+,(1.2)Z0(s)ds=1,(1.3)m1(s)6 0(s)6 m2(s),s R+,(1.4)m1,m2 0.;A*3nk?uL,y3N?.1980c,Aifantis3 1J?e,u,aA?(XpN)DOI:10.12677/pm.2023.1382412331n?|?.,e;?A*ut 4u+f(u)=g(x),?vkA*K?,?N*Lx0?5,?5,?.?,L?,?*L?.,=e?;A*ut 4ut 4u+f(u)=g(x),T?$u6N,?N99D?n?+,24?.a,c?eJ.Sl?u3 5?35 f v?.O?g H1();*?f?35;3 6,Chepyzhov?kPP?*3 f v?.O,g L2()?f?35.Jaime?.?X,L?Lyapunov y?PP,)P-?.Cc5,kN?;A*?C1,?3 8g?kPP?;A*.3Ty?5v?.O?f?35?K5.|?.P5.Z?)?35P5.?3 10?:k5P?;A*?f?35,O?P?.2.?”5,P H=V0=L2(),V=V1=H10(),QT=(0,T),k kqLm Lq()?,AO/,?q=2,Pk k2=k k.-Vs=D(As2),A=4,SO(u,v)Vs=(As2u,As2v),kukVs=kAsuk.H0 V0OL H and V?m.B,?y?C L?,?3z1$1U.n 2.1 11(Aubin-Dubinskii-Lions n)?X Y Z Banach m,X b Y.(i)e F Lp(a,b;X),1 6 p 1 k.,tf e?.K F 3 Lp(a,b;Y);.e q 1,K F 3 C(a,b;Z);.(ii)e F L(a,b;X)?k.8,tF 3 Lr(a,b;Z),r 1 k.,K F 3 C(a,b;Y);.n 2.2 7(Mazur?)?p 0,K?u,v,kDOI:10.12677/pm.2023.1382412332n?|2p|u v|p+16?u|u|p v|v|p?6(p+1)|u v|(|u|p+|v|p).3.(yn 3.1 b?(1.2)(1.4).K?u0 V,(1.1)3?)u,vu L(0,);V);ut L2(0,);V).y k Faedo-Galerkin 125y)?35.?ei+i=0 H10()?,L2()?.L2()?Hilbert m,A L2()kl?f,Aei=iei,i N;0 0,3m 0,Tn?)aim(t).=um(t)3.e%C)um(t)k?O.taim+?aim(3.2)S,i l 1?m,?12ddt?kumk2+(1+?)kumk2+kum(t)k+2+2?+ktumk2+ktumk2+?kumk2+?kum(t)k+2+2?Z0(t s)|um(s)|um(s)ds,tum+?um?=0.(3.4)(?um)(t)=Z0(t s)?|um(s)|2um(s)|um(s)|2um(t)?2ds,(3.5)(0?um)(t)=Z00(t s)?|um(s)|2um(s)|um(s)|2um(t)?2ds,(3.6)DOI:10.12677/pm.2023.1382412333n?|12ddt?Z0(t s)kum(s)k+2+2ds?=12(0)kum(s)k+2+2+12Z00(t s)kum(s)k+2+2ds.(3.7)(3.5)(3.7),k?Z0(t s)|um(s)|um(s)ds,tum(t)?=12(0?um)(t)12ddt(?um)(t)12(0)kum(t)k+2+2+12ddt?Z0(t s)?kum(s)k2um(t)?2ds?12Z00(t s)?kum(s)k2um(t)?2ds.(3.8)A Young?,?Z0(t s)|um(s)|um(s)ds,um(t)?612Z0(t s)kum(s)k+2+2ds+12Z0(t s)?kum(s)k2um(t)?2ds,(3.9)r(3.8),(3.9)“(3.4)(1.4),?12ddt?kumk2+(1+?)kumk2+(?um)(t)+2+2kum(t)k+2+2+Z0(t s)kum(s)k+2+2ds Z0(t s)?kum(s)k2um(t)?2ds?+ktumk2+ktumk2+?kumk2+m1?2Z0(t s)kum(s)k+2+2ds+m12(?um)(t)+?kum(t)k+2+2m2+?2Z0(t s)?kum(s)k2um(t)?2ds60,(3.10)?v?,?m1?2 0.?=min?,?,m1?2,m2+?2,KkddtEm(t)+ktumk2+ktumk2+Fm(t)6 0,(3.11)3,t,kEm(t)6 ZtFm(s)ds+Em(),(3.12)DOI:10.12677/pm.2023.1382412334n?|Em(t)=?kumk2+(1+?)kumk2+(?um)(t)+2+2kum(t)k+2+2+Z0(t s)kum(s)k+2+2ds Z0(t s)?kum(s)k2um(t)?2ds,Fm(t)=kumk2+Z0(t s)kum(s)k+2+2ds+(?um)(t)+kum(t)k+2+2Z0(t s)?kum(s)k2um(t)?2ds.Young?,Poincar e?(1.3),Z0(t s)?kum(s)k2um(t)?2ds6+2Z0(t s)kum(s)k+2+2ds+2+2Z0(t s)kum(t)k+2+2ds6+2Z0(t s)kum(s)k+2+2ds+2+2|11kum(t)k+2+2.(3.13)(3.13)“Em(t),du +2+2|11,kEm(t)?kumk2+?2+22+2|11?kum(t)k+2+2+(?um)(t)+(1+?)kumk2+2+2Z0(t s)kum(t)k+2+2ds?kumk2+(1+?)kumk2C1(kumk2+kumk2)M1,(3.14)aq/,d Sobolev in,?Fm(t)kumk2+?1 2+2|11?kum(t)k+2+2+(?um)(t)+2+2Z0(t s)kum(t)k+2+2dsckumk2+?2kumk2C2(kumk2+kumk2)M2,(3.15)(3.14),(3.15)“(3.12)?C1(kumk2+kumk2)M16 Zt?C2(kumk2+kumk2)M2?ds+Em(),DOI:10.12677/pm.2023.1382412335n?|d,?R M2C2,3 t0?ku(t0)k2+ku(t0)k26 R.(3.16)d(3.11)?ddtEm(t)+ktumk2+ktumk26 C3,(3.17)3 0,),kZ0ktumk2+ktumk2ds 6 C4,t (0,T).(3.18)?O,um3 L(0,);V H)k.;tum3 L2(0,);V H)k.d,lS?um JfS?(E,P um)?um*u 3 L(0,);V H);tum*tu 3 L2(0,);V H);4um*4u 3 L2(0,T;V0 H0);4tum*4tu 3 L2(0,T;V0 H0).q V,H,|n 2.1?,um 3 L2(0,T;H)r.=um 3 L2(QT)r,?3fS?(EP um).d um u 3 QTA?.q|um|umY,?|um|um|u|u 3 QTA?,Sobolev in V,L6(),L2(+1)()(3.18),ZT0Z?|um|um?dxdt 6 ZT0kumk2(+1)2(+1)dt 6 C,(3.19)d(3.18)Lions n,B?|um|um*|u|u 3 L2(QT).n?Z0(t s)|um|umds*Z0(t s)|u|uds 3 L2(QT).DOI:10.12677/pm.2023.1382412336n?|Vm=spane1,e2,.em)?m,?d(3.3)?,?e Vm,k(tum,e)(4tum,e)(4um,e)+?|um(t)|um(t),e?Z0(t s)|um(s)|um(s)ds,e?=0,(3.20)3 C0(0,T)t 3(0,T),?ZT0(tum,e)dt ZT0(4um,e)dt+ZT0?|um(t)|um(t),e?dtZT0?Z0(t s)|um(s)|um(s)ds,e?dt ZT0(4tum,e)dt=0,(3.21)3(3.21)-m (5(J,?ZT0(tu,e)dt ZT0(4u,e)dt+ZT0?|u(t)|u(t),e?dtZT0?Z0(t s)|u(s)|u(s)ds,e?dt ZT0(4tu,e)dt=0,(3.22)Ktu 4tu 4u+|u(t)|u(t)Z0(t s)|u(s)|u(s)ds=0.(3.23):um*u 3 V H,tum*tu 3 V H,d3 V um(0)*u(0),?3 V um(0)u0,?4?5,?u(0)=u0.ey5.b?u v (1.1)k?),-w=u v e?):wt 4wt?Z0(t s)|u(s)|u(s)ds Z0(t s)|v(s)|v(s)ds?4w+?|u(t)|u(t)|v(t)|v(t)?=0.(3.24)w 3 L2()S?12ddt?kw(t)k2+kw(t)k2?+kw(t)k2=?Z0(t s)|u(s)|u(s)ds Z0(t s)|v(s)|v(s)ds,w(t)?+?|v(t)|v(t)|u(t)|u(t),w(t)?.(3.25)DOI:10.12677/pm.2023.1382412337n?|eO?O?m?,d Mazur,H older?9 Sobolev in,2(3.16),?|v(t)|v(t)|u(t)|u(t),w(t)?6Z?|v|v|u|u?|w(t)|dx6(+1)Z|w|2?|u|+|v|?dx6(+1)kwk2+2?|u|+|v|?+26(+1)kwk2+2?kuk+2+kvk+2?6(+1)kwk2(kuk+kvk)6C(+1)kwk2,(3.26)aq/,?Z0(t s)|u(s)|u(s)ds Z0(t s)|v(s)|v(s)ds,w(t)?6Z0(t s)?|u(s)|u(s)|v(s)|v(s)?ds6C(+1)kwk2.(3.27)(3.26),(3.27)“(3.25)(Poincar e?,12ddt?kw(t)k2+kw(t)k2?6 1kwk2+?C(+1)+C(+1)?kw(t)k2,(3.28)2 Gronwall n,kw(t)k2+kw(t)k26 C?kw(0)k2+kw(0)k2?,w,?t 0,kw(t)k=0,=u(t)=v(t).5?y.d(3.16),?R M2C2,3 t0=t0(B),t t0,?ku(t0)k2+ku(t0)k26 R.-B0=tS(t)B00,B00=u0 V?ku(t0)k2+ku(t0)k26 R,K3 R0 0,?k.8 B V,B0=B(0,R0)K(1.1)?)+S(t)t3 V?k.8.=?k.8 B V,3 t0 0,?t t0,kS(t)B B0.DOI:10.12677/pm.2023.1382412338n?|y S(t)?C;5,S(t)S1(t)S2(t),S1(t)3 V;,S2(t)P.d,?k.8 B V,k(S(t)B)6(S1(t)B)+(S2(t)B)=(s2(t)B)0,t .-u=v+w,v w Oe?):vt 4vt 4v=R0(t s)|u(s)|u(s)ds|u(t)|u(t),x ,t 0,v(x,t)=0,x ,t 0,v(x,0)=v0(x),x .(3.29)wt 4wt 4w=0,x ,t 0,w(x,t)=0,x ,t 0,w(x,0)=w0(x),x .(3.30)n 3.2 Xe:ut+Aut+Au=0,x ,t 0,u(x,t)=0,x ,t 0,u(x,0)=u0(x),x .(3.31)e u0 V,u (3.31)?),K u Cb(R+;V),u vkuk2+kuk26?ku0k2+ku0k2?e1t,(3.32)1 0.y u (3.31)3 H S?12ddt?kuk2+kuk2?+kuk2=0,?,kddt?kuk2+kuk2?+1?kuk2+kuk2?6 0,Gronwall n,kvk2+kvk26?ku0k2+ku0k2?e1t.(3.33)K 3.3 9 u (3.33)?),u Cb(R;Vs+1),s R.K u Cb(R;V+1).DOI:10.12677/pm.2023.1382412339n?|y 4 A2u(3.33),-w=A2u,o(3.33)?wt+Awt+Aw=0,t R,q u Cb(R;Vs+1),?w Cb(R;Vs+1),dAX,w Cb(R;V).?u=A2w Cb(R;V+1).?e5y S1(t)3 V;.n 3.4?0 0,?tF(x,t)L(R+;V1),(3.34),?k.8 B V,3 C(B)0,?u0 V,ksupuVktF(x,t)kL(R+;V+1)6 C(B),(3.35)F(x,t)=Z0(t s)|u(s)|u(s)ds|u(t)|u(t).(3.36)y d Sobolev inH10()L6();Hs()L632s(),0 6 s 32.(3.37)?1 =2,=max,Kk0 2(1 )2,(3.38)9+2 62nn 2(1 ),(3.39)?L2nn2(1)L+2 L+2 L+2,(3.40)d?,d?n Sobolev in,?V1 H1 L2nn2(1),(3.41)duk?iY?,Kd(3.40)(3.41)V1 L+2 L+2 L+2.(3.42)DOI:10.12677/pm.2023.1382412340n?|qn1 =2n?1?O,|H older,Young?+1+2+1+2=1?(1.4),?Z00(t s)|u(s)|u(s)ds,w(t)?6 m1Z0(t s)ku(s)k+1+2dskw(t)k+26 m1+1+2Z0(t s)ku(s)k+2+2dskw(t)k+2 m11+2kw(t)k+2,(3.47)aq/,k?(0)|u(t)|u(t),w(t)?6(0)ku(t)k+1+2kw(t)k+26(0)+1+2ku(t)k+2+2kw(t)k+2+(0)1+2kw(t)k+2,(3.48)|H older?2(1)2n+12+n2(1)2n=1?(3.44),?(+1)|u(t)|ut(t),w(t)?6(+1)ku(t)kn1kut(t)kkw(t)k2nn2(1)6C(+1)ku(t)kkut(t)kkw(t)k2nn2(1),(3.49)(3.47)-(3.49)“(3.46)(3.40),(3.41)?tF(x,t),w(t)?6 C(B)kw(t)kV1,(3.50)DOI:10.12677/pm.2023.1382412341n?|C(B)=supu0B,tR+?(0)(+1)C+2ku(t)k+2+2m1C+2+C(+1)ku(t)kkut(t)k+(0)C+2m1(+1)C+2Z0(t s)ku(s)k+2+2ds?.n,C(B)k.?,?tF(x,t)3 V1?m V1,3 V1k.n 3.5 u?t,f S1(t)3 V;.y u (1.1)?),dn 3.4 u Cb(R+;V),F(x,t)Cb(R+;H),v(3.30)?),(3.30)u t,-w=vt?wt 4wt 4w=R00(t 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