分数阶抛物方程整体解的径向对称性与单调性.pdf

2023,43A(5):14091416http:/?N)?5N5/(?H1“?O?410205):T?N)?5N5.?N)?5N5,$W?9?d?K?4?n.d?,?Laplacian f?5,?/?.c:?;?N);5;N5;.MR(2010)Ka:35B50;35R10;35K60a:O175zI:A?:1003-3998(2023)05-1409-081?Xe?)?5N5ut(x,t)+()su(x,t)=f(t,|x|,u(x,t),(x,t)B1(0)R,(1.1)mC t (,),d(1.1)?)?N)(entire solution).u?N),z 13,14.z?t R,()su(x,t)=Cn,sP.V.ZRnu(x,t)u(y,t)|x y|n+2sdy=Cn,slim0ZRnB(x)u(x,t)u(y,t)|x y|n+2sdy,(1.2)0 s 1 9 P.V.L Cauchy.u C1,1loc L2s,()su?.-L2s=?u(,t)L1loc(Rn)|ZRn|u(x,t)|1+|x|n+2sdx +?,KT?Laplacian f.,?,z?x,?s 1,(4)su(x,t)u 4u(x,t).,ukf?k,Xz 15,15,z 6,17,9w(sliding methods)zvF:2022-06-07;?F:2022-03-24E-mail:78:Ig,7(12201201)Supported by the NSFC(12201201)1410n?Vol.43 A7,18,21,22.?,kf?k?(J,Xz 16,19,20.,?,?8c,u?f?%?A24.C,?2?.K?)?N5.3z 9,W?uf?Liouville n.?,z 23,q?;)(ancientsolutions)?(J.du?Laplacian?N)?(J5?uf?5,?ND?$uf,d,?#?5f?5,X:?/?,d?K9?4?n?.35(PDE)y“+,)?510!5!N512?-?5.?8?(1.1)?N)?5N5.?nXen 1.1b?u(x,t)(C1,1loc(B1(0)C(B1(0)C1(R)e?Dirichlet K?k.)ut(x,t)+()su(x,t)=f(t,|x|,u(x,t),x B1(0),t 0,x B1(0),t ,u(x,t)=0,x Bc1(0),t ,(1.3)Bc1(0)L Rnm?8.-(0,1)?2s(0,1).b?f(t,|x|,u)L(R R+R+),f u t C2sloc?,u u Lipschitz Y?9u|x|,f(t,|x|,u)4?.?,b?f(t,|x|,0)=0,fu(t,|x|,0)0,t (,),(1.4)o?)u(x,t)u?:?9u|x|4.5 1.1?#L?3u?/?.z 8?,?d?K4?n?.,?,f(t,|x|,u)?b?.2OK?(1.3)u(x,t)?5N5,I/?.OK)d?K?4?n,?n3z 9 y?.?B,3e-?n,y?.x Rn,x=(x1,x0),x0 Rn1.-T=x Rn|x1=,RL,=x Rn|x1?,x=(2 x1,x2,xn)L x u T?:.d u(x,t)?(1.3)?),u(x,t)=u(x,t)(x,t)=u(x,t)u(x,t).No.5/:?N)?5N51411w,(x,t)u T?,=(x1,x2,xn,t)=(2 x1,x2,xn,t).n 2.1(d?K9)b?L k.?d,x|2l x1,l?.b?(x,t)(C1,1loc()L2s)C1(R)3 u t k.9u x eY,(x,t)o(1)|x|?0 0 (x,t)0,(x,t)R.n 2.2(?4?n9)b?L k.,?X x1k.e(x,t)(C1,1loc()L2s)C1(R)3 u t k.9u x eY,?0 0?,(x,t)R,(2.4)o(x,t)0,(x,t)R.?,kr4?n:(x,t)0 (x,t)0,(x,t)R.n 2.1(?Hopfs n9)b?(x,t)(C1,1loc()L2s)C1(R)k.t(x,t)+()s(x,t)c(x,t)(x,t),(x,t)R,(x,t)0,(x,t)R,(x,t)=(x,t),(x,t)R.(2.5)c(x,t)ek.e3:x?(x,t)0,(x,t)R.ox1(x,t)0,(x,t)T R.5 2.1dun 1.1 k u=0 3 Bc1(0),d u L2sy3n 2.1 2.2w,.,?,3n 1.1?u(x,t)o(1)|x|?|x|,?3u?7?|x|?,?3z 9 u(x,t)o(1)|x|?|x|u?n?)3?C1.1412n?Vol.43 A3n?y?5yn 1.1.NyLXe.y-=x B1(0)|x1 1 C 1,y(x,t)0,(x,t)R.(3.3),ddn 2.1 y?(3.3).NXe:d f?Lipschitz Y5?c(x,t)k.;,?,d(x,t)0 3()R d=y(3.3).(S2)0=sup 0|(x,t)0,(x,t)R,.?8?y0=0.e06=0,od 0?0 0 3S?k 0?k 0,(3.4)Zk:=(x,t)k R|k(x,t)0,E?SkfS?,k Sk 0.?3f?xk1,yk1(1,k),zk Rn1,tk R?k(xk1,zk,tk)0.(3.6)b?xk1 x0,yk1 y0,x0,y0 1,0.(3.7)-kk(x,t)=ukk(x,t)uk(x,t),uk(x,t)=u(x1,x0+zk,t+tk),x=(x1,x0)Rn,t R.o,kk(xk1,0,0)0uk(yk1,0,0)0.(3.8)du uk(x,t)k.,z 11 u?)?K5?O,?f?(EIP uk)-k ,Kuk(x,t)u(x,t),()suk(x,t)()s u(x,t),kk(x,t)0(x,t):=u0(x,t)u(x,t)0,(x,t)0 R,(3.9)0(x0,0,0)0.(3.10),?,u(x,t)v ut(x,t)+()s u(x,t)=f(t,|x|,u(x,t),x B1(0),t ,u(x,t)0,x B1(0),t ,u(x,t)=0,x Bc1(0),t 0.(3.12)d(3.11)(3.12)k u(x,t)0,(x,t)B1(0)R.(3.13),e(3.13),o3:(x,t)B1(0)R?u(x,t)=0=infB1(0)R u(x,t),l?ut(x,t)+()s u(x,t)0,(x0,t)Rn1 R.(3.14)?,k 0(x,t)0,(x,t)0 R.(3.15)K,e(3.15),o3:(x,t)0 R?0(x,t)=inf0R 0(x,t)=0.?0t(x,t)=0()s 0(x,t)0t(x,t)+()s 0(x,t)=f(t,|x|,u0(x,t)f(t,|x|,u(x,t)=0.g,l?(3.15).L(3.10)(3.15),du x0 1,0,?0(x0,0,0)=0 x0=0.Llimkxk1=x0=0.(3.16),?,du u 0,(x,t)(B1(0)c R,L(3.13)?:3:x B1(0)?0(x,t)0 t R,9 0(x,t)v 0t(x,t)+()s 0(x,t)c0(x,t)0(x,t),(x,t)0 R,0(x,t)0,(x,t)0 R,0(x0,t)=0(x,t),(x,t)0 R.(3.17)c0(x,t)=f(t,|x|,u0(x,t)f(t,|x|,u(x,t)u0(x,t)u(x,t)?Hopfs n 2.1,d 0 x1(x,t)0.=3 0?,ux1(x,0,0)ek.ukx1(x1,0,0)kaq?5,=3?0,?k,kukx1(x1,zk,tk)=ukx1(x1,0,0)0,x1 0,0+.No.5/:?N)?5N51415(3.8)k(xk1,zk,tk)0 g.d,(ii).(i)(ii),?0=0.du(x1,x2,xn,t)u(x1,x2,xn,t),?:t R,u(x,t)u?:.(S3)(S1)(S2)?,1 0,(x,t)R.(3.18),bX3:(x0,t0)R?(x0,t0)=0=infB1(0)R(x,t),ot(x0,t0)+()s(x0,t0)0,f(t0,|x0|,u(x0,t0)f(t0,|x0|,u(x0,t0)0.g.d(3.18).?,Hopfs n 2.1,?x1(x,t)0,(x,t)R,1 0,(x,t)R,1 0.x1-?,ux1(x,t)0,(x,t)R,0 1,d,u=u(|x|,t)vu|x|(x,t)0,(x,t)B1(0)R.l?u X|x|4?.y.z1 Berestycki H,Nirenberg L.On the method of moving planes and the sliding method.Bol Soc Brasil Mat(NS),1991,22(1):1372 Chen W,Hu Y.Monotonicity of positive solutions for nonlocal problems in unbounded domains.J FunctAnal,2021,281(9):1091873 Chen W,Li C.Classification of solutions of some nonlinear elliptic equations.Duke Math J,1991,63(3):6156224 Chen W,Li C,Li Y.A direct method of moving planes for the fractional Laplacian.Adv Math,2017,308:4044375 Chen W,Li Y,Ma P.The Fractional Laplacian.Hackensack:World Scientific,20206 Chen W,Li Y,Zhang R.A direct method of moving spheres on fractional order equations.J Funct Anal,2017,272(10):413141577 Chen W,Wu L.A maximum principle on unbounded domains and a Liouville theorem for fractionalp-harmonic functions.arXiv:1905.099861416n?Vol.43 A8 Chen W,Wang P,Niu Y,Hu Y.Asymptotic method of moving planes for fractional parabolic equations.Adv Math,2021,377:1074639 Chen W,Wu L.Liouville theorems for fractional parabolic equations.Adv Nonlinear Stud,2021,21(4):93995810 Du L,Tang Y.The steady incompressible jet flows issuing from a finitely long nozzle.Journal of DifferentialEquations,2021,280:35537411 Fern andez-Real X,Ros-Oton X.Regularity theory for general stable operators:Parabolic equations.JFunct Anal,2017,272(10):4165422112 Geng S,Tang Y.Convergence rates to nonlinear diffusive waves for solutions to nonlinear hyperbolic system.Acta Mathematica Scientia,2019,39B(1):465613 Hamel F,Nadirashvili N.Entire solutions of the KPP equation.Comm Pure Appl Math,1999,52(10):1255127614 Hamel F,Nadirashvili N.Travelling fronts and entire solutions of the Fisher-KPP equation in RN.ArchRation Mech Anal,2001,157(2):9116315 Li C.Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains.Commun Partial Differ Equ,1991,16:49152616 Li C.Some Qualitative Properties of Fully Nonlinear Elliptic and Parabolic Equations.New York:NewYork University,198917 Li Y,Zhu M.Uniqueness theorems through the method of moving spheres.Duke Math J,1995,80(2):38341718 Liu Z,Chen W.Maximum principles and monotonicity of solutions for fractional p-equations in unboundeddomains.arXiv:1905.0649319 Pol a cik P.Estimates of solutions and asymptotic symmetry for parabolic equations on unbounded domains.Arch Ration Mech Anal,2007,183:599120 Pol a cik P.Symmetry properties of positive solutions of parabolic equations on RN,I:Asymptotic symmetryfor the Cauchy problem.Commun Partial Differ Equ,2005,30:1567159321 W?,.De Giorgi.5?)?N5.I:,2022,52(1):122Wu L Y,Chen W X.Monotonicity of solutions for fractional equations with De Giorgi type nonlinearities.Sci Sin Math,2022,52(1):12222 Wu L Y,Chen W X.The sliding methods for the fractional p-Laplacian.Adv Math,2020,361:Article ID10693323 Wu L Y,Chen W X.Ancient solutions to nonlocal parabolic equations.Advances in Mathematics,2022,408:10860724 Zhang B,L u Z.Symmetry and nonexistence of solutions for a fully nonlinear nonlocal system.Pac J Math,2019,299:237255The Radial Symmetry and Monotonicity of Entire Solutions forFractional Parabolic EquationsTang Yanjuan(School of Mathematics and Statistics,Hunan First Normal University,Changsha 410205)Abstract:This paper mainly develops the radial symmetry and monotonicity of entire solutions forfractional parabolic equations.To obtain the symmetry and monotonicity of entire solutions,the narrowregion principle and maximum principle for antisymmetric functions in 9 are needed.Furthermore,to circumvent the difficulty from nonlocality for the fractional Laplacian,a fractional parabolic versionof the method of moving planes will be adopted.Key words:Fractional parabolic equations;Entire solutions;Symmetry;Monotonicity;The methodof moving planes.MR(2010)Subject Classification:35B50;35R10;35K60