Consider t he Cauchy pr oblem fo r the evolutio n p-Laplacian equation
In terms of the uniform estimates to the r egulized solutions of  the problem above, w e prove
that ？ , where the Holder exponent with r espect to this great than ？2


LI Hui-lai

ZHAO Jun-ning

李辉来

赵俊宁

Xiamen University Institutional Repository

Nor theast. M ath. J.16( 1) ( 2000) , 96—98　　Received May 12, 1999.Regularity of Solutions for the Evolutionp-Laplacian EquationsLI Hui-lai (李辉来)( D ep artment of M athematics , J ilin University , Changchun, 130023)ZHAO Jun-ning (赵俊宁)( Dep ar tment of M athematics, X iamen University , X iamen, 361005)Abstract　Consider t he Cauchy problem fo r the evolution p -Laplacian equationut= div( u p - 2 u) , ( x , t) ∈ QT = Rn × ( 0, T ) ,u( x , 0) = u0( x ) ∈ L ∞ ( RN ) , x ∈ RN .In terms o f the uniform estimat es t o the r egulized solutions o f the problem above, w e pro vethat uxj∈c , / ( 1+ )loc ( QT ) , w here the Ho lder exponent w ith r espect to t is gr eat than2.Key Words regular ity ; Ho lder estimate; p -Laplacian1991 MR Subject Classification: 35K55, 35K20CLC number: O175. 26Document code: AArticle ID: 1000-1778( 2000) 01-0096-03　In this note w e consider the follow ing Cauchy problem:ut= div( u p - 2 u) , ( x , t ) ∈ QT = R n × ( 0, T ) ,u( x , 0) = u0 ( x ) ∈ L∞ ( RN ) , x ∈ R N ,( 1)w here p > 2 is a constant . It is w ell known that ther e ex ists a solution u∈C loc( QT )∩L∞( Q T ) to ( 1) , w ith ux j∈C, / 2loc ( QT ) , j = 1, 2, ⋯, N ( see [ 1] , [ 2] ) . The pro ofs of ux j∈C, / 2loc ( QT ) are very com plicated and diff icult . In this no te w e use another appr oach topr ove the Holder cont inuity . We prove uxj∈C, / ( 1+ )loc ( QT ) , w here the Holder exponentw ith respect to t is great than2.　Definition　A f unction u( x , t ) def ined in Q T is call ed a w eak solution of ( 1) , if u∈C loc( Q T)∩Lp ( 0, T : W1, ploc ( RN ) )∩L∞( Q T ) , ∈( 0, 1) , and f or any  ( x , t )∈C1 ( Q- T ) w ith = 0 w hen x is lar ge enough, it holds that∫RN u( x , t) ( x , t ) dx +∫t0∫RN [ - u t + u p - 2 u   ] dx dt=∫RN u0 ( x ) ( x , 0) dx . ( 2)　We prove the fol low ing theo rem .　Theorem　L et u0( x )∈L∞( RN) , u0≥0. T hen there ex ist constants C> 0 and ∈( 0, 1)such that the solution of ( 1) satisf iesut ( x , t) ≤C( p - 2) t,　uxj∈ c, / ( 1+ )loc ( QT ) . ( 3)　Proof　Acoording to [ 3] , the solut ion u of ( 1) is the lim it of the solutions of thefo llow ing boundary value pr oblemsut= div ( ( u p - 2 ) u) , ( x , t ) ∈ B n × ( 0, T ) ,u( x , t) = 0, ( x , t ) ∈ B n × ( 0, T ) ,u( x , 0) = u0n( x ) , x ∈ Bn ,( 4)w here B n= { x ; x < n} , u0n∈C10 ( Bn ) satisfy ‖u0n‖L∞≤‖u0‖L∞ andlimn→∞u0n = u0　　a. e. on RN.　Let un be the so lut ions of ( 4) . Setv n( x , t) = !∀un ( x , !t) , 　!> 1,　∀= 1p - 2.Then v n sat isfiesvt= div( v p - 2 v ) , ( x , t) ∈ Bn × ( 0, T ) ,v ( x , t ) = 0, ( x , t) ∈ Bn × ( 0, T ) ,v ( x , 0) = !∀u0n( x ) , x ∈ B n.( 5)Set w = v n- un. Then by Com parison Principle w≥0 and∫Bnw ( x , t)  ( x , t) dx -∫t0∫Bnw tdx d#+∫t0∫Bn( v n p- 2 v n - un p - 2 un )   dx d#=∫Bn( !∀- 1) u0n ( x ) ( x , 0) dx , ( 6)fo r any  ∈C1( B n×( 0, T ) ) , w ith  = 0 near B n. In ( 6) by an approx imate pr ocess, wecan take = ( w - k) + ,　　k = (!∀- 1)‖u0‖L∞.Then w e get∫Bn( w - k - ∃) 2+ dx + 2∫t0∫Bn∩{ w> k}( v np - 2v n- unp- 2un) ( v n - un ) dx d#= 0.This implies w≤k a. e. on B n×( 0, T ) . Thus0≤ !∀un( x , !t ) - un ( x , t ) ≤ ( !∀- 1)‖u0‖L∞. ( 7)97NO. 1 L I H . L . & ZH A O J . N . EVOLU TION p -L APL ACIAN EQU ATIONS　Divide ( 7) by !- 1 and let !→1+ w e get∀un( x , t) + tunt( x , t) ≤ ∀‖u0‖L∞.This inequality impliesut ( x , t ) ≤Ct.　No tice that for f ix ed t∈( 0, T ) , u( x , t) is a solut ion o f the elliptic equationdiv( up- 2u) = u t( x , t) , 　x ∈ RN.By [ 4] , ther e ex ist constants ∈( 0, 1) , C> 0 depending only on u t L∞, u L∞ such thatu( x 1, t ) - u( x 2 , t) ≤ C x 1 - x 2 . ( 8)We now prove that u is Holder cont inuous in t. For conver nience, we assum e that u isa sm ooth solut ion; other wise, by the uniqueness of so lut ion w e can consider theregularized problem. T ake the x j -der iv ative in ( 1) to obtainux jt= ( div( u p- 2 u) ) xj. ( 9)Let x 0∈RN, 0< t1≤t2 , %t = t2 - t1 , B ( % t ) = B ( % t) &( x 0) . Integr at ing ( 9) by parts overB( %t )×( t1, t2) , w e get∫B( % t) ( ux j ( x , t2) - ux j ( x , t 1) ) dx =∫t2t1∫B( % t) ( div( up- 2u) ) xjdx dt=∫t2t1∫B( %t) div( u p- 2 u)∋j d(dt =∫t2t1∫B( % t) ut∋j d(dt . ( 10)w here ∋= (∋1 ,∋2,⋯,∋N ) is the outw ard normal unit v ector of B ( % t) . By the mean valuetheorem , ther e ex ists x* ∈B ( %t ) such thatuxj( x * , t2 ) - uxj( x * , t 1) ≤ C( % t) 1- &. ( 11)Combining ( 8) w ith ( 11) and taking &= 11 +, w e getuxj( x 0 , t2 ) - uxj( x 0, t 1)≤ uxj( x 0 , t2 ) - uxj( x * , t 2) + uxj( x * , t2 ) - uxj( x * , t 1)+ uxj( x * , t 1) - uxj( x 0, t 1) ≤ C( %t ) / ( 1+ ) .Therefo re ux j∈C, / ( 1+ )loc ( RN) and the theorem is proved.References[ 1] Chen Yazhe, Ho lder continuit y o f the gr adient o f the so lutions of cer tain degenerate pa rabolicequations, Chinese A nn. M ath. , 8B( 3) ( 1987) , 343—356.[ 2] DiBenedetto , E. and F riedman, A . , Holder estima tes for nonlinear degenerate par abolic sy st em,J . R eine A ngew . M ath. , 357( 1985) , 1—22.[ 3] Wu Zhuoqun, Zhao Junning, Y in Jingxue and Li Huilai, Nonlinear Diffusion Equat ions, PublishingHouse of Jilin Univer sity , Changchun, 1996.[ 4] Peter T olksdor f, Regularity for a mo re general class of qua silinear ellipt ic equations, J .Diff erential Eq uations, 51( 1984) , 126—150.98 NORTHEAS T . MAT H. J. VOL. 16　

Regularity of Solutions for the Evolution p-Laplacian Equations

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Regularity of Solutions for the Evolution p-Laplacian Equations

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