一類非線性卷積擬拋物型積分微分方程整體解的存在性問題

1、26(510405)(100088)GalerkinGalerkinVolterra145,6utuxxuxxt+,t0 (ts)u(x,s),ux(x,s)xds=f(x,t,u,ux).(1)Sobolev-Galpern(1)8(1)=0,=0=0,=0,=0BBM-Burgers9(1)(1)7;=0,=0,=010uttuxxtuxxtt=(ux)x,>0,(2)2001532003(10271034)53051226Pochhammer-Chree11uttuxxtuxxuxxttf(u)xx=0,12>0(3)uttuxxtuxxuxxtt+f(u)xt=0,>

2、0(4)Volterra(1)(1)u|x=0=0,u|x=1=0,t>0,0x1.(5)(6)u|t=0=u0(x),(1),(5),(6),GalerkinBlowup=1.=(0,1),10,1×0,+)u(x,t)(1),(5),(6)T> 0 11(i)u(x,t)L0,T;H0()H2(),ut(x,t)L0,T;H0()H2(); (ii)(x,t)C00,T;L2()(ts)u(x,s),ux(x,s)xdsutuxxuxxt+00 f(x,t,u,ux)(x,t)dxdt=0;Tt (7)1()H2().(iii)u(x,0)=u0(x)H0(1),(5

3、),(6)2d(5)vn(x)µn>0d2vn=µnvn,dxvn(x)vn(x)H2()vn|x=0=vn|x=1=0, vn =1,n=1,2,···,L2()1vn(x)H0()11H0()H2()H0().(1),(5),(6)um(x,t)=nm(t)0,+)m n=1nm(t)vn(x),Galerkinm=1,2,···,3513umtumxxumxxt+=f(x,t,um,umx),vn(x),um(0)=u0m(x),u0m(x)u0(x), 0t(ts)um(x,s),umx(x,s)n=

4、1,2,···,m, xds,vn(x) (8)(9)H2()mPiccard C0,1×0,+)×R×RR),(s,p)C1,0(R×R),f (8),(9)(t)C0,+),f(x,t,s,p)(x,t)×0,+),f(x,t,s,p)C1,0(R×t=012,ff(x,t,u,ux)=g(x,t)+h(u)+(u)x,(10),h1)(s)C0,+);2)(s,q)C1(R×R), + (s,q) C,(s,q)R×R,(0,0)=0, (s,q) h(u),uB1 u 2+B2

5、,B1,B2C3)h(s)C(R),4)(u)C1(R);u0(x)g(x,t)15)u0(x)H0()H2();6)g(x,t)L2(0,T;L2(),T>0.1)6)um(x,t)um(x,t)(8),(9)0,+)(1),(5),(6)11)6)T>0,0t um(·,t) 22+ umx(·,t) 22+LLEi umx 2L2dE1(T),0tT.(11)m(8)nm(t),tn1m (ts)um(x,s),umx(x,s)ds,umumtumxxumxxt+x0 =g(x,t),um+h(um),um+(um)x,um. (12)(um,um)(um

6、xxt,um)= 1d 2 um 2+ u mxL2,L22dt(umxx,um)= umx 2L2,51426 t (ts)um(x,s),umx(x,s)ds,um(x,t) x0 t = (ts)um(x,s),umx(x,s)umx(x,t)dxds 0 t (ts) ds um(x,s),umx(x,s)umx(x,t) dx0 t 2 21 umx(·,s) 2dsM1(T)TM1(T) um(x,s),umx(x,s) L2+L4TM(T)10 t 2 um(·,),umx(·,) 22d+1 umx(·,t) 22,TM1(T)tT,(1

7、3)LL40M1(T)=sup (t) .0tT C|s|+|p|,(s,p)R×R.ab2a+2b.2) (s,p) um(x,t),umx(x,t) 2dx CPoincar´e13 um(x,t) 2dx+ umx(x,t) 2dx. umx(x,t) 2dx(14) w(x) 2dx1µ1 2 w(x) dx,1wH0().(14)(13)t (ts)um(x,s),umx(x,s)umx(x,t)dxds 0 t 21 um 2 2, dt+(·,t)tT.M2(T)u 2mxLx4L0(15)(12)B1 um 2L2+B2, u(u)=0(

8、s)ds. 22g(x,t),umh(u g + u ,),umL2mmL2 (um)x,um=(um),umx=x(um),1=0,(12) t 2d um 22 um L2+umx(·,t)L2+ ddtxL20 t 2 u2 m2M3(T) um L2+ umx(·,t) L2+ 2d+M4(T).xL0GronwallSobolev11)6)T>0, um L×LE2(T).T351521)6)T>0,tumx 2L2×LT+ umxx 2L2×LT+0 umxx(·,) 22dE3(T),L(16)umxx L&

9、#215;LE4(T).T(8)µnnm(t),tn1m(ts)um(x,s),umx(x,s)umtumxxumxxt+0 =g(x,t),umxx+h(um),umxx+(um)x,umxx. xds,umxx (17)1d 2 umx 2(umtumxxt,umxx)=+ u 22mxxL,L2dt t (ts)um(x,s),umx(x,s)xds,umxx 0 t (ts)dsum(x,s),umx(x,s)xumxx(x,t) dx0 t um(x,s),umx(x,s) 2dxds+1 umxx(·,t) 22,TM1(T)Lx400tT.(18)2) um(

10、x,),umx(x,) x um 2um +um(x,),umx(x,)= um(x,),umx(x,) 2sxpx 2 2u 2 u m m (x,) + (x,) .C xx1t 2u(x,),u(x,) dxdmmxx0 t 2 2 2 t u um m(x,) dxd+C(x,) dxd xx200 t umxx(x,) 22d+M6(T).M5(T)L0(19)(19)(18)t (ts)um(x,s),umx(x,s)xds,umxx 0 t 2 2 1 M7(T)umxx(x,)L2d+M8(T)+umxx(·,t) L2.40(20)516(17)(g(x,t),um

11、xx) 1 g 22+1 umxx 22,(h(u 2 Lh(u 2Lm),umxx) Cm) umxx L2C1+ umxx 22(u (u LL),m)x,umxx) m) L umx L2 umxx L2C umx 2L2+ umxx 2L2).(17)ddt umx 2L2+ umxx 2tL2+ umxx(x,) 2L2dM 09(T) umx 2L2+ umxx 2tL2+ umxx(x,) 2d +M0L210(T).Gronwall(16)Sobolev(16)31)6)T>0,tumt 2L2×LT+ umxt 2L2×LT+ u) mxx(

12、3;, 20L2dE5(T),umt 2L×LTE6(T). nm(t)(8)n1m1,umt 2L2+ umxt 2L2 u+ (umx Lm) L umxt L1+ g(x,t) L2+ h(um) L2+ t(ts) (um(x,s),umx(x,s)xds L2 umt L2M11(T) umxt L2+M12(T) umt L2.umt 2L2×L+ umxt 2L2×LTTE5(T),Sobolev(21)41)6)T>0,umxxt L2×LTE7(T), umxt L2×LTE8(T).(8)µn nm(t),n

13、1mu umxxt 2L2mtL2+ umxx L2+ g(x,t) L2+ h(um) L2+ t(ts) um(x,s),umx(x,s) xds+ (u L2m) L umx L2 umxxt L226(21)1,2(22)(23)3517tt22M13(T)+M14(T) umxx L2d+M15(T) umx L2d umxxt L200M16(T) umxxt L2.umxxt(t)Sobolev1L2M17(T),0tT.(24)(23)um(x,t) 11)6)4um(x,t)(1),(5),(6)(8),(9)0,+)0,1×0,+)tum (x,t) 14T>

14、;0, um(x,t) um (x,t)u(x,t)um (x,t)u(x,t)um t(x,t)ut(x,t)u(x,t),m 1L0,T;H2()H0() 1L0,T;H2()H0() L20,T;H2()*um (x,t)u(x,t)×0,Tum x(x,t)ux(x,t)×0,T(s,p)C1(R×R),(s)C1(R),h(s)C(R) um (x,t),um x(x,t)u(x,t),ux(x,t)×0,T um (x,t)u(x,t)×0,T hum (x,t)hu(x,t)×0,T um (x,t),um x(x,t)

15、,um (x,t),hum (x,t)m QT=×0,T0,T(8)Td(t)C00,T,t=00 t (ts)um(x,s),umx(x,s)xds,d(t)vn(x)dtumtumxxumxxt+ g(x,t),d(t)vn(x)dt+TT 0T h(um),d(t)vn(x)dt(25)0+(25)0 (um)x,d(t)vn(x)dt.m=m ,0T um tum xxum xxt,d(t)vn(x)dt0T utuxxuxxt,d(t)vn(x)dt.Lebesgue0T h(um ),d(t)vn(x)dt(um )x,d(t)vn(x)dt= 0T hu(x,t),d(

16、t)vn(x)dt,TT00 (um ),d(t)vn(x)dt5180T0T26(u),d(t)vn(x)dt=tT(u)x,d(t)vn(x)dt,0T(ts)um (x,s),um x(x,s)xds,d(t)vn(x)dtt=Tt0(ts)um (x,s),um x(x,s)ds,d(t)vn(x)dt(ts)u(x,s),u(x,s)ds,d(t)vn(x)dtt =0T(ts)u(x,s),u(x,s)xds,d(t)vn(x)dt.d(t)C00,T,=00Tutuxxuxxt+t(ts)u(x,s),u(x,s)xds,d(t)vn(x)dt0TTg(x,t),d(t)vn(x

17、)dt+hu(x,t),d(t)vn(x)dt+T(u)x,d(t)vn(x)dt.1),2)(1),(5),(6)vn(x)L2()(9)m=m 1d(t)C00,Tu(x,t)u(x,t)3(1),(5),(6)(8),(9)514G(z1,z2,···,zh)z1,z2,···,zhzi(x,t)L0,T;Hk(),i=1,2,···,h,k(k1)kDxG(z1,z2,···,zh) 22C(M,k,h)h zi 2k,Li=1M=max1ih(x,t)×0

18、,Tmaxzi(x,t) .61)6)7)(s,p)lk1,R×Rkl(0,p)|=0;k8)h(s)(,+)lk1h(l)(0)=0;k1h(k1)(s)(,+)9)(s)(l)(0)=0;k(k)(s)lk1k1l(),Dxg(x,t)L20,T;L2(),1lk1;10)g(x,t)H0k()Hk+1(),11)u0(x)H03519T>0,k Dxum 2L2×LT+k+1 Dxum 2L2×LT +0tk+1 Dxum 2L2dE9(T).(26)(26)k=2(8)µ2nnm(t),n1mt umtumxxumxxt+(ts)um(x,

19、s),umx(x,s)xds,umxxxx0 =g(x,t),umxxxx+h(um),umxxxx+(um)x,umxxxx.(27)(umtumxxumxxt,umxxxx)=1d 22 umxx 2L2+ umxxx L2)+ umxxx L2,2dt(13)t (ts)um(x,s),umx(x,s)xds,umxxxx 0 t 2 = (ts)u(x,s),u(x,s)u(x,t)dxds mmxmxxx20x t 2 3 (ts) ds um(x,s),umx(x,s) um(x,t) dxx0x t 2 1 22TM1(T)u(x,),u(x,) dxd+ umxxx 2mmxL

20、2,0(28)2um3um2(um,umx)(um,umx)(um,umx)=+x2sx2px3u 222um2umm(um,umx)+(um,umx)+2sxspxx 2u 22m+2(um,umx).px27)1,2,2 2u 2 3u m m +M(T) 2(um,umx) M18(T)+M19(T) . 20xx2x3(28)t (ts)um(x,s),umx(x,s)xds,umxxxx 0 t t 12M21(T)+M22(T) umxxx L2d+M23(T)|umxx|4dxd+ umxxx 2L2,40052026Nirenberg-Gagliard151 Dju LpC(p

21、,q,r,m,j,) Dmu Lr u Lq,1q,r<+,0jm,j<1;=+(mj).q=2,r=2,p=4,m=1,j=0,u=2u,|umxx|4dxC() umxxx21C2() umxx L2 umxx 3L2 umxxx L2+4 6L2.2,t(ts) um(x,s),umx(x,s)0xds,umxxxx)tM124(T)+M25(T)0 umxxx 2L2d+4 umxxx 2L2.(27)d tdt umxx 2L2+ umxxx 2L2+ umxxx 2L2dM 0 t27(T)+M28(T) umxx 2L2+ umxxx 2L2+ umxxx 2 L2d

22、.Gronwallk=2(26)k1(26)kSobolevum L×Lk1T, umx L×LT,···, Dxum L×LTmk(8)µnnm(t),n1mt(uumtumxxmxxt+(ts)um(x,s),umx(x,s)2kxds,(1)kDxum)= g(x,t),(1)kD2xku 0m+h(um),(1)kD2xkum+(um)x,(1)kD2xkum.5,umtumxxumxxt,(1)kD2xkum=1d Dkxumt 2L2+ Dkx+1um 2L2+ Dkx+1um 2L2 d t2,t(ts) ux

23、,s)m(x,s),umx(k2k xds,(1)Dxum0 tTM k12(T)0 xm(x,s),umx(x,s) 2dxd+1 Dkx+1uum 2L24tM29(T)+M30(T)k+110 Dxum 2L2d+4 Dkx+1um 2L2,(29)35212kk1k+1 g(x,t),(1)kDxum= Dxg(x,t)·Dxumdxk1 21k+1 Dxg(x,t) L2+ Dxum 2L2,4 k1 2kk+1 h(um),(1)kDxum=Dxh(um),Dxum k1 21k+1 Dxh(um) L2+ Dxum 2L241k+1k1M31(T) Dxum 2um 2

24、L2+ DxL2,4 k 2kk+1 (um)x,(1)kDxum = Dx(um),Dxum k 21k+1 Dx(um) L2+ Dxum 2L241k+1k Dxum 2M32(T) Dxum 2+2LL2,4(29)Gronwall6(26)k7kk+1 Dxumt 2umt 2L2×L+ DxL2×LE10(T).TT(30)µknnm(t)(8)(30).n1m1()Hk+1(),Hk+1(),ut(x,t)L0,T;H026(1),(5),(6) 1u(x,t)L0,T;H0()k2.431)6)(1),(5),(6)u,v(1),(5),(6).t

25、w=uv wtwxxwxxt+(ts)u(x,s),ux(x,s)v(x,s),vx(x,s)xds0 =h(u)h(v)+(u)(v)x,w|x=0=w|x=1=0,(31)(32)w|t=0=0.(31)w(x,t),1 t d 222 wx L2d w L2+ wx L2+dt0 t 222M33(T) w L2+ wx L2+ wx L2d.GronwallT>0,w(t) 22+ wx(t) 22+LL 0t wx 2L2d=0,0tT.52226w(x,t)0×0,+41)6)u0(x)v0>0u(x,t)v(x,t) 1<.H×L(1),(5

26、),(6)u,vT>0,1L0,T;H0()(1),(5),(6)u|t=0=u0v|t=0=>0, u0v0 H1<,T5Blowuputuxxuxxt+t(ts)u(x,s)xxds=f(u),(33)(5),(6)blowup5u0(x),f,a)(t)0,t0,+);b)(s)C2(R)y>0,Lipschitzc)f(s)g(s),sR,g(s)+s01d)=0u0(x)sinxdx>0.(s)(y)(0);g(s)2s(33),(5),(6) u(·,t) p+,1p+.L()blowup,TtT6u0(x),f,a)(t)0,t0,+;b)

27、(s)C2(R)y>0,Lipschitzc)f(s)g(s),sR,g(s)+s01d)=0u0(x)sinxdx>0.(s)(y)(0);g(s)2s(33),(5),(6) u(·,t) p+,1p+.L()blowup,TtT56y(t)=(33)Jensen 1u(x,t)sinxdx.sinx,xy(t)+y(t)+y(t)+222t(ts)(0)y(s)ds1 222y(t)+y(t)+y(t)+(ts)(0)u(x,s)sinxdxds200t=y(t)+2y(t)+2y(t)+2(ts)ds(u)xxsinxdx201 1=f(u)sinxdxg(u)s

28、inxdxg(y).2200t3523t 21y(t)(ts)y(s)(0)ds(g(y)2y),221+01+1y(0)=(34)y(t)>0,0u0(x)sinxdx=>0.(34)t>0.t1>0,0,t1)y(t)>0t0,t1) y(s)(0)>0,s(0,t),t0,t1).(34)y(t)>0,t(0,t1),y(t)0,t1t1>0,y(t1)>0. t>0,y(t)>0.t>0gy(t)2y(t)>0, s(0,t),y(s)(0)>0.(34)y(t)0,Ty(t1)=0,y(0)=>0, gy(t)2y(t)y(t)(3