The paper discusses the temporal behavior of the spatially spherosymmetric solution for a nonlinear parabolic equation, which is a sense related with that of the spatially spherosymmetric solution for the 3-dimentional commpressible Burgers' equation.Article信州大学理学部紀要 30(1): 1-5(1995)departmental bulletin pape

ITAYA,  Nobutoshi

MUKAI,  Kentaro

English

Shinshu University Institutional Repository

                               J, FAC. SCI., SHINSHU UNIVERSITY, Vol. 30, No, 1, pp. 1-5 1995   A Memoir on the Spatially Spherosymmetric Solution             for a Nonlinear Parabolic Equation               Nobutoshi ITAYA and Kentaro MUKAI                    Department of Mathematical Sciences,                    Faculty of Science, Shinshu University                         (Received July, 31, 1995)Abstract : The paper discusses the temporal behavior of the spatially spherosymmetricsolution for a nonlinear parabolic equation, which is a sense related with that of thespatially spherosymmetric solution for the 3-dimentional commpressible Burgers'equation. ･   As for the notation used below, it is conventional, H2'a(si), H2'"･i'g(nT), etc.,denoting HOIder spaces. ･                         '                          tt1 Introduction   The partial differential equation to be considered is:        8t V(x, t) - ip(x, t)-2ptA.V(x, t) + if(x, t)2 (lxlf{gl(>O),tl}iO) (1.l)                      (x = (xi,'''-xn), An = tt/i oOx2?. ),             lth(ar, o) -: le,( c) (E H2+a(n), fl - {xl lxlKl},o< ev<1), (1.2)       V(x, t)lixi-t =O(t2iO), (accompanied by compatibility conditions), (1.3)                'where pt is a positive constant and di (x, t) is defined by                      di (x, t) == exp(,(itif(x, r) dr]. (1.4)   Now, without proof, we give two theorems.Theorem l (Temporally local existence). For some TG(O, oo), there exists aunique solution lth(Ju,t) for (1.1)-(1.2)-(1.3) belonging to H2'a'i'S(nT)(SIT==()×[O,T]).[VViz note tha4 of IOb(x);}ie, then if(x, t)}iO.]Theorem 2 (Spatial spherosymmetry). if if(x, t)EH2"cr'i'S(siT) (O<T<co)satiSfies (1.1)-(1.2)-(1.3) with sbb(x)- uto(lxl), then if(x, t) hczs a form th(Ixl,t),tGraduate Course (Doctor), Tokyo Metropolitan University2 Nobutoshi lTAyA and Kentaro MuKAIwhere ifo(r) ciePends only on r and V(z t) dbes only on r and t. Moreovez if (z t)sakesies                       '      -5}u if(z t)= diN(r, t)-2pt[oO.22 diN(z t) + "il oO. ¢AV(z t)]÷ di--(z t)2 (1.s)                (OK r f{ l,O E{; t E{ T), ( th.r ,.,,i lrr(O,t)),                 V(r,e) =:= dio(r), dir(O,t) :- if(l,t)-O, (1.6)(IQgether zvith compatibility conditions). (q5(r,t)=exp[f6 1th(r,T)dr].) [in thediscussion of the 3-dimentional compTessible Bblrgers' equation, tize case n=5 matters.]2 Blow-up result   In this section, we show that the solution tr(x, t) for (1.1)-(1.2)-(1.3) with                                                                    '                       th(x,O) =- fio(lxl) (l}r O, th O) (2.1)                                                               'blows up in a finite time under a certain condition on ifo(r).  , First, we define 1{'i)(r) (n == 1,2,･･･) by                         f("'(r) -= C.reZe-i(B.r), (2-2)                (Cn(>O), const. such that ,CtlC'i>(r)dr =1),where k-i(s) is the Bessel function of order -ll--1 for each n and Bnl is the firstzero-point of Zij-i (s) (s>O). For example, f(i)(r) :::: Cicos-l7-z f(2)(r) ::= C27clb(B2r)(lb(B21) := O), etc. Next, let ¢(x, t) belong to ll2'a'i'g(nT) for some TE (O,oo) andsatisfy (1.1)-(1.2)-(2.1). Then, by theorem 2, if(x, t) is expressed as if(x,t) == if(lxl,t). Moreover, if(r,t) (r=lxl) satisfies (1.5)-(1.6). Multiplying both sides of(1.5) by ip (r, t)2 e J(n)(r) and integrating them in r over [O,l], we have easily,X` ipA-2-tllt- ifN (r, t)f`"'(r) de =- - "B. ,C ` ifh- (r, t)f('!'(r) dr + ygt ipN2 if-- (r, t) 2f("'(r) dr, (2 . 3)            (N.B.: ddt2, f{n)(r)-(n-i) ti;(f(n)iSr))=:=-a,l(n)(r)).             'By the strength of the equality ipN t= ip-V trN , it is seen that                                                                  ',4'`ip--22t th-U(r, t)J`n'(r)de- El .C`ipA"2 ifA-(r, t)f(n'(r)cth'-2II'`ip--2 th--(r, t)2f`"'(r)dr.(2.4)                                                               'Thus it holds that    '    '  ddt rd`ip-'2 thN(r, t)f`"'(r)dr == -ptB.:,Ciifn"(r, t)f("'(r) de -i- 3rdtipN2 ifN(r, t)2f(n'(r)dr                      2}i -yB.,Ci ipN2 i}r (r, t)f(n'(r) dr   A Memoir on the Spatially Spherosynimetric Solution for a Nonlinear Parabolic Equation 3                 NN +3{f6ip2iffiiil.J)?i.'()riv)dr}2･ (2･s)           (N.B.: ip2 th1(n) = ipN(J(n))iip th (f(n))t, ¢ }) 1, f(") l}i O(t O) ) .Let y.(t) denote f6 di2Vf(n)dr (> O). Then from (2.5) follows an inequality, ･   ' ddt yn(t)2LptBnzxn(t) rmF 3yn(t)'2(211tyn(T) dz" rri-1)7', (2.6).        (N.B.: 2il ta`¢N2f(")dr = Jll di2, ut-"f(")dr = 2y., .4`ip-"(r, o)2f(")de = 1).Hereafter, we write yn(t), f(n}(r) and Bn simply y(t), f(r) and B, respectively.Defining Q(t) by                       (?(t) =- 2.4tgy(T) dr+1 (;}i 1), (2.7)we rewrite (2.6) in the following way:                                                               '                   S2 zi(, (> 2) -es(>-S2+-2- (?2 ((> == Eil Q), (2.s)                       Q(o) =- 1, o(o)-2y(e)(> o) (2.g)According to the relation -2Si/- Q = Q ddQ Q' (N.B.: Q' :=: 2y > O), from (2.8) we have aninequality                                      '- ･ i;zl> ({> }i -y+-g--2S-(v-ptB), , (2.lo)                                          t.or, what is the same,                                                                     '                                                           ''                      tt                                     '                  . ddQ Q' -;-fli 2)mv. (2.ii)                                                         '                        'Hence,                                             '     Q-e Q - (? (O) -i C> (O) - Q-g Q - 2y (O) l}i - yY[i f,,=, Q-g dQ - 2y(Q-t - 1). (2 . I2)Thus, we have                Q }i (2gy (O) -2y) Qg+2vQ > (2y(O) -2y) Qg, (2.13)      ･ Q(O) =- 1, (2.14)which leads us to the assertion that, if y(O) satisfies                    y(O) (- .CtifNo(r)J(r) dr) >y- ptB, (2.15)then Q(t)> {1-(y(ol)-,)t}, (o ,gt< y(o)1-,). '' (2･l6)4 Nobutoshi lTAyA and Kentaro MuKAIFrom the discussion made above, we have:Theorem 3. 71Vze solution if(a} t) for (1.1)-(1.2)-(1.3)-(2.1) blows mp in a finitetime under the condition (2.15).3 Global existence   Here, we give a theorem which asserts the temporally global existence of a uniquesolution for our problem (1.1)-(1.2)-(1.3)-(2.1) under an additional condition ontho(r). (In the case of tho(r) ffg O), the existence of such a solution is obvious.)Theorem 4. 71Pzere exists a zanique temporally global sointion di(x, t) of the Problem(1.1)-(l.2)-(1.3)-(2.1) belonging to H2'a'i"g(nT) for an arbiva71y TE (O,oo) zanderthe condition on tho(r),                   k-b;,ISO) =-llbblSP)< 6i'l", (=:= B,), (I-[O,l]). (3.1)Moreoveag it holtis that    ' futN(e,t)lg)-fth(･,t)lg-O)E{;i÷2B4k,t, (t)}ie). (3.2)Outline of the Proof Let 1th(ar,t) satisfy (1.1)-(1.2)-(l.3)-(2.1)-(3.1) in S)T forsome TE (O,oo). (N.B.: th has the form th(x, t) == if(lxl,t).) Now, we define w(x,t;a,k)(or, simply, w(x,t)) by             w(c,t;a,k) == (1 + aitrtt) lth(Jq t) -- (1 + avkrit) lth (l ul,t) (3.3)                       (a and k, positive constants).Then w(x,t) satisfies                                  '            zvt(x,t) = ¢fiV(IxLt)m2 ptAnw(x,t) + W(X't)12 ++ aak'tr"itW (X't), (3.4)                        w(x,O)=V(a7, O)- dio(lxl), (3.5)              w(x,t)li.i.t= th(jc, t)1i.i.., == V(l,t) =O(O f{{ t f{; T). (3.6)Moreover, ab(x,t) defined by                                         k                                           1ti2          ab(x,t) == w(x,t) +                                         2nsatisfies                zb,(= wt) == <3'T2A.zD +(W12++ aaXtZV pt ,trt {3'L2)., (3.7)               zD(v,O) = gNb'o(IJtrl)+ 21nb lizr12, zb(v,t)lxl==i= 2k. l2- (3.8)After a somewhat lengthy calculation concerning (3.7)-(3.8), on the basis of themaximum principle, we obtain a sufficient condition on 1 ifolP) under which IzDl S-O,) is   A Memoir on the Spatially SpherosyiTllnetric Solution for a Nonlinear Parabolic Equation 5bounded from above by known quantities and the term {･･･}A is smaller than O. As afinal result, we have our assertion.Q.E.D4 Concludingremark    The blowup-nonblowup problem of the spatially spherosymmetric solution for the3-dimentional compressible Burgers' equation              vt(x,t) == p(.pt,t) (A ÷-l}-v･div)v(x,t) - (v･v)v(x,t), (4.1)                        pt(x,t)+div(p(x,t)v(x,t))-O (4.2)is closely related to our problem (1.1)-(1.2)-(1.3)-(2.I) (n==5) in a technical sense,although the former is more complicated and more diffLcult than the latter. Ourdiscussion above will be useful in treating the former problem, whose settlementconsists in estimating p(x,t) == fi(lxl,t) in an a pn'ori way.                                  References[1] A. Friedman, Pkertial DW2irential Eauations of 1larabolic T){Pe, Prentice Hall, 1964.[2] H. Fujita, On the blowing mp of the solution of the Ckeucdy problem for ut=Au+ zai'a, J.   Fac. Sci. Univ. Tokyo, Sect. I, 13 (1966), 109-124.[3] N. Itaya, A note on the blowmp-nonblowmp P7oblem in nonlinear Pavabolic equations, Proc.   Japan. Acad., 55-A (1979), 241-244.[4] N. Itaya, On some subjects related with the blowmp Problem in nonlinear Pavabolic equations,   Lecture Notes of Num. and Appl. Anal,, 2 (Kinokuniya, 1980), 27-38.[5] N. Itaya, On a sphen'cally symmetn'c timeglobal solution for the compressible ws equation,   K6be Sh6dai `Jimmon Ronshtr, 21 (1985), 1･2, 1-10, (in Japanese).[6] N, Itaya, A note on the blowi7zg-mp P7oblem of a certain system of nonlinear Partzbolic   equations, Stud. Math. and Appl., 18 (Kinokuniya-NorthHolland, 1986), 419-429.[7] N. Itaya, A mport on the blowing-mp Problem for nonlinear Parabolic eqzscztions, Publ. 60-th   Anniv. K6be Sh6dai (1990), 771-786, (in Japanese).[8] N. Itaya A comment on the global existence of the solution for lhe nonlinear Pafabolic   equation with a drut term, J. Fac. Sci. Shinshu Univ., 29, No,1 (1993), 17-26.[ 9 ] O. A. JIaAhi)KeHcKafi, va np., ,JIzHethHhie H KBa3HrmHetiHbie YpaBHeHMH IIapa6o"Hqecicoro Tmia HayKa,    1964.[10] K. Mukai, On the blowtip-nonblowmp Problem for a cerimn nonlinear Parabolic equation of a   mixed mpe, Thesis for Master's Degree, Shinshu University, 1993, (in Japanese).

A Memoir on the Spatially Spherosymmetric Solution for a Nonlinear Parabolic Equation

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A Memoir on the Spatially Spherosymmetric Solution for a Nonlinear Parabolic Equation

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