South Ural State UniversityÎÁÇÎÐÍÛÅ ÑÒÀÒÜÈ ÓÄÊ 517.9 ÍÀ×ÀËÜÍÎ-ÊÎÍÅ×ÍÛÅ ÇÀÄÀ×È ÄËß ÍÅÊËÀÑÑÈ×ÅÑÊÈÕ ÌÎÄÅËÅÉ ÌÀÒÅÌÀÒÈ×ÅÑÊÎÉ

ÎÁÇÎÐÍÛÅ ÑÒÀÒÜÈ

ÓÄÊ 517.9

ÍÀ×ÀËÜÍÎ-ÊÎÍÅ×ÍÛÅ ÇÀÄÀ×ÈÄËß ÍÅÊËÀÑÑÈ×ÅÑÊÈÕ ÌÎÄÅËÅÉÌÀÒÅÌÀÒÈ×ÅÑÊÎÉ ÔÈÇÈÊÈ

Ñ.À. Çàãðåáèíà

Íåêëàññè÷åñêèìè íàçûâàþò òå ìîäåëè ìàòåìàòè÷åñêîé ôèçèêè, ÷üè ïðåäñòàâëå-

íèÿ â âèäå óðàâíåíèé èëè ñèñòåì óðàâíåíèé â ÷àñòíûõ ïðîèçâîäíûõ íå óêëàäûâà-

þòñÿ â ðàìêàõ îäíîãî èç êëàññè÷åñêèõ òèïîâ � ýëëèïòè÷åñêîãî, ïàðàáîëè÷åñêîãî èëè

ãèïåðáîëè÷åñêîãî. Ñòàòüÿ ñîäåðæèò îáçîð ðåçóëüòàòîâ àâòîðà â îáëàñòè íåêëàññè÷å-

ñêèõ ìîäåëåé ìàòåìàòè÷åñêîé ôèçèêè, äëÿ êîòîðûõ ðàññìîòðåíû íà÷àëüíî-êîíå÷íûå

çàäà÷è, îáîáùàþùèå óñëîâèÿ Êîøè è Øîóîëòåðà � Ñèäîðîâà. Àáñòðàêòíûå ðåçóëüòà-

òû ïðîèëëþñòðèðîâàíû êîíêðåòíûìè íà÷àëüíî-êîíå÷íûìè çàäà÷àìè äëÿ óðàâíåíèé è

ñèñòåì óðàâíåíèé â ÷àñòíûõ ïðîèçâîäíûõ, âîçíèêàþùèõ â ïîñëåäíåå âðåìÿ â ïðèëî-

æåíèÿõ, à èìåííî, â òåîðèè ôèëüòðàöèè, ãèäðîäèíàìèêå è ìåçîñêîïè÷åñêîé òåîðèè, è

ðàññìîòðåííûõ íà ìíîæåñòâàõ ðàçëè÷íîé ãåîìåòðè÷åñêîé ñòðóêòóðû.

Êëþ÷åâûå ñëîâà: íåêëàññè÷åñêèå ìîäåëè ìàòåìàòè÷åñêîé ôèçèêè, ìîäåëü Ïëîò-

íèêîâà, ñèñòåìà Íàâüå � Ñòîêñà, óðàâíåíèå Áàðåíáëàòòà � Æåëòîâà � Êî÷èíîé,

(ìíîãîòî÷å÷íûå) íà÷àëüíî-êîíå÷íûå çàäà÷è, îòíîñèòåëüíûé ñïåêòð.

Ââåäåíèå

Â íàñòîÿùåå âðåìÿ ðàçâèòèå ñîâðåìåííûõ âûñîêèõ òåõíîëîãèé ïðèâîäèò ê íåîáõîäè-ìîñòè èññëåäîâàíèÿ ôèçè÷åñêèõ ïðîöåññîâ, âîçíèêàþùèõ â èíæèíèðèíãå. Â ñâÿçè ñ ýòèìâîçíèêàåò íåîáõîäèìîñòü ïîñòðîåíèÿ àäåêâàòíûõ ìàòåìàòè÷åñêèõ ìîäåëåé è èõ äàëüíåéøååèçó÷åíèå. Â äàííîé ñòàòüå ïðåäïîëàãàåòñÿ ðàññìîòðåòü ñëåäóþùèå ìîäåëè.

I. Ïóñòü Ω ⊂ Rn � îãðàíè÷åííàÿ îáëàñòü ñ ãðàíèöåé ∂Ω êëàññà C∞. Â öèëèíäðå Ω×R+ðàññìîòðèì ìîäåëü Ïëîòíèêîâà [1, 2]

θt(x, t) + φt(x, t) = ∆θ(x, t) + f(x, t), ∆φ(x, t) + αφ(x, t) + βθ(x, t) + g(x, t) = 0,

∂θ

∂n(x, t) + λθ(x, t) = 0,

∂φ

∂n(x, t) + λφ(x, t) = 0, (x, t) ∈ ∂Ω× R+,

(0.1)

êîòîðàÿ ÿâëÿåòñÿ ëèíåàðèçàöèåé â íóëå ñèñòåìû óðàâíåíèé ôàçîâîãî ïîëÿ, îïèñûâàþùèõâ ðàìêàõ ìåçîñêîïè÷åñêîé òåîðèè ôàçîâûå ïåðåõîäû ïåðâîãî ðîäà. Çäåñü λ ∈ R, α, β ∈ C,ôóíêöèè f è g îòâå÷àþò âíåøíåìó âîçäåéñòâèþ íà ñèñòåìó.

II. Ïóñòü òåïåðü Ω ⊂ Rn, n ∈ {2, 3} � îãðàíè÷åííàÿ îáëàñòü ñ ãðàíèöåé ∂Ω êëàññà C∞.Ñèñòåìà óðàâíåíèé Íàâüå � Ñòîêñà

vt = ν∇2v − (v · ∇)v −∇p, ∇ · v = 0, (0.2)

ìîäåëèðóþùàÿ äèíàìèêó âÿçêîé íåñæèìàåìîé æèäêîñòè, áûëà ïîëó÷åíà â 1845 ãîäó. Çäåñüâåêòîð-ôóíêöèÿ v = (v1, v2, . . . , vn), vm = vm(x, t) ñîîòâåòñòâóåò ñêîðîñòè æèäêîñòè, ôóíê-öèÿ p = p(x, t) � äàâëåíèþ, ïàðàìåòð ν ∈ R+ õàðàêòåðèçóåò âÿçêîñòü. Çà èñòåêøåå âðåìÿ

2013, òîì 6, 2 5


óðàâíåíèÿ (0.2) èçó÷àëèñü â ðàçëè÷íûõ àñïåêòàõ, íàèáîëåå ãëóáîêèå èõ èññëåäîâàíèÿ èçëî-æåíû â [3, 4]. Îäíàêî äî ñèõ ïîð íå ðåøåí âîïðîñ î ñóùåñòâîâàíèè ñèëüíûõ ðåøåíèé çàäà÷èÊîøè � Äèðèõëå

v(x, 0) = v0(x), x ∈ Ω,

v(x, t) = 0, (x, t) ∈ ∂Ω× (0, T ) (0.3)

äëÿ óðàâíåíèé (0.2) ïðè ïðîèçâîëüíîì T ∈ R+ è n = 3.III. Óðàâíåíèå Áàðåíáëàòòà � Æåëòîâà � Êî÷èíîé

(λ−∆)ut = α∆u+ f

ìîäåëèðóåò äèíàìèêó äàâëåíèÿ æèäêîñòè, ôèëüòðóþùåéñÿ â òðåùèííîâàòî-ïîðèñòîé ñðåäå[5]. Çäåñü α è λ � âåùåñòâåííûå ïàðàìåòðû, õàðàêòåðèçóþùèå ñðåäó; ïàðàìåòð α ∈ R+, àïàðàìåòð λ ìîæåò ïðèíèìàòü è îòðèöàòåëüíûå çíà÷åíèÿ, êîòîðûå íå ïðîòèâîðå÷àò ôèçè-÷åñêîìó ñìûñëó çàäà÷è [6], ôóíêöèÿ f = f(x) èãðàåò ðîëü âíåøíåé íàãðóçêè. Êðîìå òîãî,ýòî óðàâíåíèå îïèñûâàåò òå÷åíèå æèäêîñòåé âòîðîãî ïîðÿäêà [7], ïðîöåññ òåïëîïðîâîäíî-ñòè ñ ≪äâóìÿ òåìïåðàòóðàìè≫ [8], ïðîöåññ âëàãîïåðåíîñà â ïî÷âå [9]. Íåîáõîäèìî îòìåòèòü,÷òî îäíîìåðíîå óðàâíåíèå Áàðåíáëàòòà � Æåëòîâà � Êî÷èíîé ÿâëÿåòñÿ îäíîìåðíûì àíà-ëîãîì ëèíåàðèçîâàííîé ñèñòåìû Îñêîëêîâà [10, 11], îïèñûâàþùåé äèíàìèêó âÿçêîóïðóãîéíåñæèìàåìîé æèäêîñòè Êåëüâèíà � Ôîéãòà.

Ïóñòü G = G(V;E) � êîíå÷íûé ñâÿçíûé îðèåíòèðîâàííûé ãðàô, ãäå V = {Vi} � ìíîæå-ñòâî âåðøèí, à E = {Ej} � ìíîæåñòâî äóã. Ìû ïðåäïîëàãàåì,÷òî êàæäàÿ äóãà èìååò äëèíólj > 0 è øèðèíó dj > 0. Íà ãðàôå G íàñ áóäóò èíòåðåñîâàòü çàäà÷è ñ êðàåâûìè

uj(0, t) = uk(0, t) = um(lm, t) = un(ln, t),Ej , Ek ∈ Eα(Vi), Em, Ek ∈ Eω(Vi);

(0.4)

∑Ej∈Eα(Vi)

djujx(0, t)−∑

Ek∈Eω(Vi)

dkukx(lk, t) = 0; (0.5)

óñëîâèÿìè äëÿ óðàâíåíèéλujt − ujxxt = αujxx, (0.6)

îïèñûâàþùèõ äàâëåíèå æèäêîñòè â ñëó÷àå, êîãäà ñðåäà ïðåäñòàâëÿåò ñîáîé ñâÿçíûé ïëàñò,èìåþùèé ñëîèñòóþ ñòðóêòóðó. Çäåñü ÷åðåç Eα(ω)(Vi) îáîçíà÷åíî ìíîæåñòâî äóã ñ íà÷àëîì(êîíöîì) â âåðøèíå Vi. Óñëîâèå (0.4) òðåáóåò, ÷òîáû âñå ðåøåíèÿ áûëè íåïðåðûâíûìè íàâåðøèíàõ ãðàôà; a (0.5) îçíà÷àåò, ÷òî ïîòîê ÷åðåç êàæäóþ âåðøèíó äîëæåí ðàâíÿòüñÿíóëþ � àíàëîã óñëîâèÿ Êèðõãîôôà. Åñëè, ê ïðèìåðó, ãðàô ñîñòîèò èç åäèíñòâåííîé äóãèñ äâóìÿ âåðøèíàìè, òî óñëîâèÿ (0.4) îòñóòñòâóþò, à óñëîâèÿ (0.5) ïðåâðàùàåòñÿ â óñëîâèåÍåéìàíà.

Â ïîäõîäÿùèõ ôóíêöèîíàëüíûõ ïðîñòðàíñòâàõ çàäà÷è (0.1); (0.2), (0.3); (0.4) � (0.6)ðåäóöèðóþòñÿ ê ëèíåéíîìó óðàâíåíèþ ñîáîëåâñêîãî òèïà

Lu̇ = Mu+ f. (0.7)

Âïåðâûå óðàâíåíèÿ, ñâîäÿùèåñÿ ê âèäó (0.7), ïîÿâèëèñü â ðàáîòå À. Ïóàíêàðå [12], îäíàêî èõñèñòåìàòè÷åñêîå èçó÷åíèå íà÷àëîñü ñ ðàáîòû Ñ.Ë. Ñîáîëåâà [13] (ñì. îáñòîÿòåëüíûé îáçîðâ [14]).

Öåëüþ íàøåãî èññëåäîâàíèÿ ÿâëÿåòñÿ ðàçðåøèìîñòü äëÿ óðàâíåíèÿ (0.7) òàê íàçûâàå-ìîé íà÷àëüíî-êîíå÷íîé çàäà÷è

Pj(u(τj)− uj) = 0, j = 0, n, (0.8)

6 Âåñòíèê ÞÓðÃÓ. Ñåðèÿ ≪Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå è ïðîãðàììèðîâàíèå≫


−∞ ≤ a < τ0 < τ1 < τ2 < . . . < τn < b ≤ +∞, Pj � îòíîñèòåëüíî ñïåêòðàëüíûå ïðîåêòîðû(êîòîðûå áóäóò îïðåäåëåíû ïîçäíåå). Çàìåòèì, ÷òî åñëè n = 1, òî (0.8) ïðåâðàòèòñÿ â áîëååïðîñòóþ çàäà÷ó

Pin(u(0)− u0) = 0, Pfin(u(τ)− uτ ) = 0. (0.9)

Èñòîðèÿ çàäà÷è (0.7), (0.9) íà÷èíàåòñÿ ñ îäíîé ñòîðîíû â [15], ãäå îíà íàçâàíà çàäà÷åéÂåðèãèíà, à ñ äðóãîé ñòîðîíû è íåçàâèñèìî � â [16], ãäå îíà íàçâàíà çàäà÷åé ñîïðÿæåíèÿ.Îäíàêî â îáîèõ ñëó÷àÿõ âìåñòî îòíîñèòåëüíî ñïåêòðàëüíûõ ïðîåêòîðîâ Pin è Pfin ðàññìàò-ðèâàþòñÿ ñïåêòðàëüíûå ïðîåêòîðû îïåðàòîðà L, ïðè÷åì L âäîáàâîê ïðåäïîëàãàåòñÿ ñàìîñî-ïðÿæåííûì. Íàø ïîäõîä îñíîâàí íà êîíöåïöèè îòíîñèòåëüíîãî ñïåêòðà, ïðåäëîæåííîé Ã.À.Ñâèðèäþêîì [17], è ðàçâèòîé åãî ó÷åíèêàìè [18�20], â ÷àñòíîñòè, Â.Å. Ôåäîðîâûì. Êðîìåòîãî, ìåòîäû, ïðåäëîæåííûå Ã.À. Ñâèðèäþêîì, ñòàëè ôóíäàìåíòîì àëãîðèòìîâ ÷èñëåííîãîðåøåíèÿ óðàâíåíèé ëåîíòüåâñêîãî òèïà (ò.å. êîíå÷íîìåðíûõ óðàâíåíèé ñîáîëåâñêîãî òèïà),êîòîðûå â ñâîþ î÷åðåäü ñûãðàëè âàæíóþ ðîëü â ÷èñëåííûõ èññëåäîâàíèÿõ ýêîíîìè÷åñêèõ[21�23] è òåõíè÷åñêèõ ìîäåëåé [24, 25].

Ïåðâûå ðåçóëüòàòû â ýòîì íàïðàâëåíèè èçëîæåíû â [26], ãäå ðàññìîòðåí ÷àñòíûé ñëó÷àéçàäà÷è (0.9) ïðè÷åì ñ áîëåå æåñòêèìè ÷åì çäåñü óñëîâèÿìè íà L-ñïåêòð îïåðàòîðà M . Â [27]ðàññìîòðåíà çàäà÷à (0.9), íî äëÿ òåõ æå óñëîâèé íà L-ñïåêòð îïåðàòîðàM , ÷òî è â [26], îäíà-êî â ýòîì ñëó÷àå îòìå÷åíà âîçìîæíîñòü áîëüøåãî ïðîèçâîëà â îòíîñèòåëüíî ñïåêòðàëüíûõóñëîâèÿõ.

Íåîáõîäèìî îòìåòèòü, ÷òî â íàñòîÿùåå âðåìÿ íà÷àëüíî-êîíå÷íûå çàäà÷è äëÿ íåêëàññè-÷åñêèõ óðàâíåíèé ìàòåìàòè÷åñêîé ôèçèêè àêòèâíî èçó÷àþòñÿ, â òîì ÷èñëå è íà ìíîæåñòâàõðàçëè÷íîé ãåîìåòðè÷åñêîé ñòðóêòóðû [28�30]. Çàìåòèì åùå, ÷òî åñëè σLfin(M) = ∅, òî çàäà÷à(0.9) ïðåâðàùàåòñÿ â çàäà÷ó Øîóîëòåðà � Ñèäîðîâà [31] P (u(0) − u0) = 0 è ïîýòîìó ñ÷è-òàåòñÿ åñòåñòâåííûì îáîáùåíèåì ïîñëåäíåé [32], êîòîðàÿ, â ñâîþ î÷åðåäü, îáîáùàåò çàäà÷óÊîøè.

Ñòàòüÿ êðîìå ââîäíîé ÷àñòè è ñïèñêà ëèòåðàòóðû ñîäåðæèò ñåìü ïàðàãðàôîâ. Ïåðâûéïàðàãðàô ïîñâÿùåí èññëåäîâàíèþ çàäà÷è (0.7), (0.9), ïðè÷åì îïåðàòîð M ñèëüíî (L, p)-ðàäèàëåí [33]. Â êà÷åñòâå êîíêðåòíîé èíòåðïðåòàöèè âî âòîðîì ïàðàãðàôå ñòàòüè ïîêàçû-âàåòñÿ îäíîçíà÷íàÿ ðàçðåøèìîñòü íà÷àëüíî-êîíå÷íîé çàäà÷è (0.9) äëÿ ñèñòåìû óðàâíåíèé(0.1). ×àñòíûé ñëó÷àé ýòèõ ðåçóëüòàòîâ áûë ïîëó÷åí â [34]. Â-òðåòüåì ïàðàãðàôå ðàññìàò-ðèâàåòñÿ çàäà÷à (0.7), (0.9) â ñëó÷àå (L, p)-ñåêòîðèàëüíîñòè îïåðàòîðà M [35]. Äàííûå àá-ñòðàêòíûå ðåçóëüòàòû ïðîèëëþñòðèðîâàíû êîíêðåòíûì ïðèìåðîì, ïðèâåäåííûì â ÷åòâåð-òîì ïàðàãðàôå. Çäåñü ïðèâåäåíà òåîðåìà îá îäíîçíà÷íîé ðàçðåøèìîñòè íà÷àëüíî-êîíå÷íîéçàäà÷è äëÿ ñèñòåìû Íàâüå � Ñòîêñà (0.2), (0.3) [36]. Â ïÿòîì ïàðàãðàôå ïðèâîäèòñÿ îáîáùåí-íàÿ òåîðåìà î ðàñùåïëåíèè, êîòîðàÿ èñïîëüçóåòñÿ â øåñòîì ïàðàãðàôå ïðè äîêàçàòåëüñòâåîäíîçíà÷íîé ðàçðåøèìîñòè ìíîãîòî÷å÷íîé íà÷àëüíî-êîíå÷íîé çàäà÷è (0.8) äëÿ óðàâíåíèÿñîáîëåâñêîãî òèïà ñ (L, p)-îãðàíè÷åííûì îïåðàòîðîìM (0.7) [37]. Â ïîñëåäíåì, ñåäüìîì, ïà-ðàãðàôå ðàññìàòðèâàåòñÿ íà÷àëüíî-êîíå÷íàÿ çàäà÷à äëÿ óðàâíåíèé Áàðåíáëàòòà � Æåëòî-âà � Êî÷èíîé íà êîíå÷íîì ñâÿçíîì îðèåíòèðîâàííîì ãðàôå (çàäà÷à (0.4) � (0.6)), ðåçóëüòàòûêîòîðîé îïóáëèêîâàíû â [38].

Íàêîíåö çàìåòèì, ÷òî âñå ðàññìîòðåíèÿ ïðîâîäÿòñÿ â âåùåñòâåííûõ áàíàõîâûõ ïðî-ñòðàíñòâàõ, îäíàêî ïðè ðàññìîòðåíèè ≪ñïåêòðàëüíûõ âîïðîñîâ≫ ââîäèòñÿ èõ åñòåñòâåííàÿêîìïëåêñèôèêàöèÿ. Âñå êîíòóðû îðèåíòèðîâàííû äâèæåíèåì ïðîòèâ ÷àñîâîé ñòðåëêè èîãðàíè÷èâàþò îáëàñòè, ëåæàùèå ñëåâà ïðè òàêîì äâèæåíèè.

2013, òîì 6, 2 7


1. Îòíîñèòåëüíî ñèëüíî p-ðàäèàëüíûé îïåðàòîð

Ïóñòü U è F � áàíàõîâû ïðîñòðàíñòâà, îïåðàòîðû L ∈ L(U;F) (ò.å. ëèíååí è íåïðåðûâåí)è M ∈ Cl(U;F) (ò.å. ëèíååí, çàìêíóò è ïëîòíî îïðåäåëåí). Îáîçíà÷èì ÷åðåç

ρL(M) = {µ ∈ C : (µL−M)−1 ∈ L(F;U)}

L-ðåçîëüâåíòíîå ìíîæåñòâî îïåðàòîðà M , σL(M) = C \ ρL(M) � L-ñïåêòð îïåðàòîðà M ,RLµ(M) = (µL − M)−1L � ïðàâóþ L-ðåçîëüâåíòó îïåðàòîðà M , à ÷åðåç LLµ(M) = L(µL −M)−1 � ëåâóþ.

Ïóñòü îïåðàòîð M ñèëüíî (L, p)-ðàäèàëåí (òåðìèíîëîãèÿ è ðåçóëüòàòû ñì. ãë. 3 [17]).Èçâåñòíî, ÷òî ïðè óñëîâèè ñèëüíîé (L, p)-ðàäèàëüíîñòè ñóùåñòâóåò åäèíèöà ðàçðåøàþùåéïîëóãðóïïû îäíîðîäíîãî óðàâíåíèÿ (0.2), êîòîðàÿ ÿâëÿåòñÿ ïðîåêòîðîì, ðàñùåïëÿþùèìïðîñòðàíñòâî U

P = U0 = s- limt→0+

U t, U t = s- limk→∞

(k(p+ 1)

tRLk(p+1)

t

(M)

)k(p+1).

Àíàëîãè÷íî ìîæíî ïîñòðîèòü ïðîåêòîð äëÿ ïðîñòðàíñòâà F

Q = F 0 = s- limt→0+

F t, F t = s- limk→∞

(k(p+ 1)

tLLk(p+1)

t

(M)

)k(p+1).

Ââåäåì â ðàññìîòðåíèå ÿäðà kerU · = U0, kerF · = F0 è îáðàçû imU · = U1, imF · = F1 ýòèõïîëóãðóïï. Â ñèëó ñèëüíîé (L, p)-ðàäèàëüíîñòè îïåðàòîðà M [39]

U0 ⊕ U1 = U (F0 ⊕ F1 = F). (A1)

Îáîçíà÷èì ÷åðåç Lk (Mk) ñóæåíèå îïåðàòîðà L (M) íà Uk (domM ∩ Uk), k = 0, 1. È åñëè

îïåðàòîð M ñèëüíî (L, p)-ðàäèàëåí, p ∈ {0} ∪ N, òî Lk ∈ L(Uk;Fk), Mk ∈ Cl(Uk;Fk), k =0, 1, ïðè÷åì ñóùåñòâóåò îïåðàòîð M−10 ∈ L(F0;U0). Â ñëó÷àå ñèëüíîé (L, p)-ðàäèàëüíîñòèîïåðàòîðà M , p ∈ {0} ∪ N âûïîëíÿåòñÿ åùå îäíî óñëîâèå [40] �

ñóùåñòâóåò îïåðàòîð L−11 ∈ L(F1;U1). (A2)

Íàêîíåö, ââåäåì åùå îäíî âàæíîå óñëîâèå �

L− ñïåêòð σL(M) îïåðàòîðà M ïðåäñòàâèì â âèäåσL(M) = σLfin(M) ∪ σLin(M), ïðè÷åì σLfin(M)ñîäåðæèòñÿ â îãðàíè÷åííîé îáëàñòè D ⊂ Cñ êóñî÷íî ãëàäêîé ãðàíèöåé γ, ãäå γ ∩ σL(M) = ∅.

(A3)

Ïîñòðîèì îòíîñèòåëüíî ñïåêòðàëüíûé ïðîåêòîð

Pfin =1

2πi

∫γRLµ(M)dµ,

ïðè ýòîì â ñëó÷àå ñèëüíîé (L, p)-ðàäèàëüíîñòè îïåðàòîðà M ñïðàâà âûïîëíÿåòñÿ PfinP =PPfin = Pfin. Çíà÷èò, â äàííîì ñëó÷àå ñóùåñòâóåò ïðîåêòîð Pin = P − Pfin. ÏîëîæèìU1in(fin) = imPin(fin), F

1in(fin) = imQin(fin), è ÷åðåç Lin(fin) (Min(fin)) îáîçíà÷èì ñóæåíèå

îïåðàòîðà L (M) íà ïîäïðîñòðàíñòâà U1in(fin) ñîîòâåòñòâåííî.

Èòàê, ïóñòü âûïîëíåíû óñëîâèÿ (A1) � (A3), ôèêñèðóåì τ ∈ R+, u0, uτ ∈ U.



Îïðåäåëåíèå 1.2. Âåêòîð-ôóíêöèþ u ∈ C([0, τ ];U) ∩ C1((0, τ);U), óäîâëåòâîðÿþùóþóðàâíåíèþ (0.7), íàçîâåì ðåøåíèåì íà÷àëüíî-êîíå÷íîé çàäà÷è (0.7), (0.9), åñëè îíà óäîâëå-òâîðÿåò óðàâíåíèþ (0.7), è lim

t→0+Pin(u(t)− u(0)) = 0, lim

t→τ−Pfin(u(t)− u(τ)) = 0.

Èìååò ìåñòî ñëåäóþùàÿÒåîðåìà 1.1. (Òåîðåìà î ðàñùåïëåíèè) [17]Ïóñòü îïåðàòîð M ñèëüíî (L, p)-ðàäèàëåí. Òîãäà(i) îïåðàòîðû Lin(fin) ∈ L(U1in(fin);F

1in(fin)), ïðè÷åì ñóùåñòâóþò îïåðàòîðû L

−1in(fin) ∈

L(F1in(fin);U1in(fin));

(ii) îïåðàòîðû Min(fin) ∈ L(U1in(fin);F1in(fin)).

Òåîðåìà 1.2. (i) cåìåéñòâî {U tfin : t ∈ R}, U tfin = U t∣∣∣∣U1fin∈L(U

1fin)

ÿâëÿåòñÿ îäíîïàðà-

ìåòðè÷åñêîé àíàëèòè÷åñêîé ðàçðåøàþùåé ãðóïïîé îäíîðîäíîãî óðàâíåíèÿ (0.2) àíàëèòè-

÷åñêè ïðîäîëæèìîé âî âñþ êîìïëåêñíóþ ïëîñêîñòü, ïðè÷åì Pfin = U0fin [17, ãë. 3];

(ii) ñåìåéñòâî {U tin : t > 0}, U tin = U t∣∣∣∣U1in∈L(U1in)

ÿâëÿåòñÿ îäíîïàðàìåòðè÷åñêîé ðàçðå-

øàþùåé ñèëüíî íåïðåðûâíîé ïîëóãðóïïîé îäíîðîäíîãî óðàâíåíèÿ (0.2), ïðè÷åì Pin = U0in =

s- limt→0+

U tin [17, ãë. 3];

(iii) ñåìåéñòâî {Rtfin : t ∈ R},

Rtfin =1

2πi

∫Γfin

(µLfin −Mfin)−1eµtdµ, t ∈ R,

ýêñïîíåíöèàëüíî îãðàíè÷åíî è àíàëèòè÷åñêè ïðîäîëæèìî âî âñþ êîìïëåêñíóþ ïëîñêîñòü

[20, ãë. 2];(iv) ñåìåéñòâî {Rtin ∈ L(F1in;U1in) : t ≥ 0}

Rtin = s- limk→∞

((Lin−

t

k(p+ 1)Min

)−1Lin

)k(p+1)(Lin−

t

k(p+ 1)Min

)−1, R0in = s- lim

t→0+Rtin

ýêñïîíåíöèàëüíî îãðàíè÷åíî è ñèëüíî íåïðåðûâíî [20, ãë. 2].(v) U sfin(in)R

tfin(in) = R

s+tfin(in), R

0in(fin) = L

−1in(fin)Qin(fin) [33].

Ïîäåéñòâóåì íà óðàâíåíèå (0.7) ïîñëåäîâàòåëüíî ïðîåêòîðàìè I−Q è Qin(fin) è ñâåäåìåãî ê ýêâèâàëåíòíîé ñèñòåìå èç òðåõ íåçàâèñèìûõ óðàâíåíèé

Hu̇0 = u0 +M−10 f0, (1.1)

u̇1in = Sinu1in + L

−1in f

1in, (1.2)

u̇1fin = Sfinu1fin + L

−1finf

1fin, (1.3)

ãäå H = M−10 L0 ∈ L(U0), íèëüïîòåíòåí ñòåïåíè p ∈ {0} ∪ N; Sin(fin) = L−1in(fin)Min(fin) ∈

L(U1in(fin)), ïðè÷åì ñïåêòð σ(Sin(fin)) = σLin(fin)(M); f

0 = (I − Q)f , f1in(fin) = Qin(fin)f ,u0 = (I− P )u, u1in(fin) = Pin(fin)u.

Òåîðåìà 1.3. [33] Ïóñòü îïåðàòîð M ñèëüíî (L, p)-ðàäèàëåí, è ÷àñòü ñïåêòðà σLfin(M)îãðàíè÷åíà. Òîãäà äëÿ ëþáûõ âåêòîðîâ u0, uτ ∈ U è ëþáîé âåêòîð-ôóíêöèè f : [0, τ ] →F, òàêîé, ÷òî f0 ∈ Cp+1((0, τ);F0), f1in ∈ C([0, τ ];F1in), f1fin ∈ C([0, τ ];F1fin), ñóùåñòâóåò

2013, òîì 6, 2 9


åäèíñòâåííîå ðåøåíèå u ∈ C([0, τ ];U)∩C1((0, τ);U) çàäà÷è (0.7), (0.9), êîòîðîå ê òîìó æåèìååò âèä

u(t) = −p∑

q=0

HqM−10dq

dtqf0(t) + U tinu0 +

∫ t0Rt−sin f

1in(s)ds+ U

t−τfin uτ −

∫ τt

Rt−sfinf1fin(s)ds. (1.4)

2. Ëèíåàðèçîâàííàÿ ñèñòåìà óðàâíåíèé ôàçîâîãî ïîëÿ

Ïóñòü Ω ⊂ Rn � îãðàíè÷åííàÿ îáëàñòü ñ ãðàíèöåé ∂Ω êëàññà C∞. Â öèëèíäðå Ω × R+ðàññìîòðèì ìîäåëü Ïëîòíèêîâà

θt(x, t) + φt(x, t) = ∆θ(x, t) + f(x, t),

∆φ(x, t) + αφ(x, t) + βθ(x, t) + g(x, t) = 0,(2.1)

∂θ

∂n(x, t) + λθ(x, t) = 0,

∂φ

∂n(x, t) + λφ(x, t) = 0, (x, t) ∈ ∂Ω× R+. (2.2)

Çäåñü èñêîìûìè ôóíêöèÿìè ÿâëÿþòñÿ θ(x, t), φ(x, t).Ðåäóöèðóåì çàäà÷ó (2.1), (2.2) ê óðàâíåíèþ

Lu̇ = Mu+ f. (2.3)

Äëÿ ýòîãî, ñíà÷àëà ñäåëàåì çàìåíû

θ(x, t) + φ(x, t) = u(x, t), φ(x, t) = v(x, t).

Òîãäà ñèñòåìà (2.2) ïðèìåò âèä

ut(x, t) = ∆u(x, t)−∆v(x, t) + f(x, t),

∆v(x, t) + (α− β)v(x, t) + βu(x, t) + g(x, t) = 0,

∂u

∂n(x, t) + λu(x, t) = 0,

∂v

∂n(x, t) + λv(x, t) = 0, (x, t) ∈ ∂Ω× R+.

(2.4)

Ïóñòü F = U = (L2(Ω))2. Ïîñòðîèì îïåpàòîpû

L =

(I OO O

)∈ L(U;F), M =

(∆ −∆βI (α− β)I +∆

)∈ Cl(U;F),

ïðè÷åì kerL = {0} × L2(Ω), à

domM = {(u, v)∈(H2(Ω))2 :(

∂

∂n+ λ

)u(x)=

(∂

∂n+ λ

)v(x) =0, x∈∂Ω}.

Ïóñòü Aw = ∆w, domA = {w ∈ H2(Ω) : ∂w∂n

(x) + λw(x) = 0, x ∈ ∂Ω}, òîãäàA ∈ Cl(L2(Ω)). ×åðåç {φk : k ∈ N} îáîçíà÷èì îðòîíîðìèðîâàííûå â ñìûñëå ñêàëÿðíîãîïðîèçâåäåíèÿ ⟨·, ·⟩ â L2(Ω) ñîáñòâåííûå ôóíêöèè îïåðàòîðà A, çàíóìåðîâàííûå ïî íåâîçðàñ-òàíèþ ñîáñòâåííûõ çíà÷åíèé {λk : k ∈ N} ñ ó÷åòîì èõ êðàòíîñòè. Òîãäà L-ñïåêòð îïåðàòîðàM èìååò âèä

σL(M) =

{µk =

(α+ λk)λkα+ λk − β

, k ∈ N \ {l : λl = β − α}}.



Ïîíÿòíî, ÷òî äëÿ òàêîãî ìíîæåñòâà ìîæíî ïîäîáðàòü êîíòóð γ ∈ C, êîòîðûé áû óäîâëåòâî-ðÿë óñëîâèþ (A3).

Ëåììà 2.1. [20] Ïóñòü β − α /∈ σ(A). Òîãäà îïåðàòîð M ñèëüíî (L, 0)-ðàäèàëåí.Ïîñòðîèì ïðîåêòîðû

Pin =

∑

Reµk∈σLin(M)⟨·, φk⟩φk O∑

Reµk∈σLin(M)

β⟨·,φk⟩φkβ−α−λk O

, Pfin =

∑Reµk∈σLfin(M)

⟨·, φk⟩φk O∑Reµk∈σLfin(M)

β⟨·,φk⟩φkβ−α−λk O

è áóäåì èñêàòü ðåøåíèå íà÷àëüíî-êîíå÷íîé çàäà÷è

Pin(u(0)− u0) = 0, Pfin(u(τ)− uτ ) = 0 (2.5)

äëÿ ñèñòåìû (2.4). Ïðîñòîòû ðàäè îãðàíè÷èìñÿ ñëó÷àåì f ≡ const è g ≡ const.Òåîðåìà 2.1. Ïóñòü β − α, −α, 0 /∈ σ(A), è ñóùåñòâóþò òàêèå λk, ÷òî Reµk ∈

σLfin(M). Òîãäà ïðè ëþáûõ u0 ∈ domA, uτ , f, g ∈ L2(Ω) ñóùåñòâóåò åäèíñòâåííîå ðåøåíèåu ∈ C1([0, τ ];U) çàäà÷è (2.5) äëÿ ñèñòåìû (2.4), êîòîðîå ê òîìó æå èìååò âèä

(u(t)v(t)

)=

0∞∑k=1

⟨g,φk⟩φkβ−α−λk

+

∑Reµk∈σLin(M)

exp((α+λk)λkα+λk−β t

)⟨u0, φk⟩φk

∑Reµk∈σLin(M)

β exp(

(α+λk)λkα+λk−β

t)

β−α−λk ⟨u0, φk⟩φk

+

+

∑

Reµk∈σLfin(M)exp

((α+λk)λkα+λk−β (t− τ)

)⟨uτ , φk⟩φk

∑Reµk∈σLfin(M)

β exp(

(α+λk)λkα+λk−β

(t−τ))

β−α−λk ⟨uτ , φk⟩φk

+

+

∑

Reµk∈σLin(M)

[1− exp

((α+λk)λkα+λk−β t

)](β−α−λk)⟨f,φk⟩−λk⟨g,φk⟩

λk(α+λk)φk∑

Reµk∈σLin(M)

[1− exp

((α+λk)λkα+λk−β t

)]β(β−α−λk)⟨f,φk⟩+⟨g,φk⟩

λk(α+λk)(β−α−λk) φk

−

−

∑

Reµk∈σLfin(M)

[1− exp


)](β−α−λk)⟨f,φk⟩−λk⟨g,φk⟩

λk(α+λk)φk∑

Reµk∈σLfin(M)

[1− exp


)]β(β−α−λk)⟨f,φk⟩+⟨g,φk⟩

λk(α+λk)(β−α−λk) φk

.

3. Îòíîñèòåëüíî p-ñåêòîðèàëüíûé îïåðàòîð

Ïóñòü U è F � áàíàõîâû ïðîñòðàíñòâà, îïåðàòîðû L ∈ L(U;F) (ò.å. ëèíååí è íåïðåðû-âåí) è M ∈ Cl(U;F) (ò.å. ëèíååí, çàìêíóò è ïëîòíî îïðåäåëåí), ïðè÷åì îïåðàòîð M (L, p)-ñåêòîðèàëåí, p ∈ {0} ∪ N (òåðìèíîëîãèÿ è ðåçóëüòàòû ñì. ãë. 3 [17]). Ðàññìîòðèì ëèíåéíîåîäíîðîäíîå óðàâíåíèå ñîáîëåâñêîãî òèïà

Lu̇ = Mu. (3.1)

Òîãäà ñóùåñòâóþò âûðîæäåííûå àíàëèòè÷åñêèå ïîëóãðóïïû îïåðàòîðîâ

U t =1

2πi

∫ΓRLµ(M)e

µtdµ è F t =1

2πi

∫ΓLLµ(M)e

µtdµ,

2013, òîì 6, 2 11


îïðåäåëåííûå íà ïðîñòðàíñòâàõ U è F ñîîòâåòñòâåííî. Ââåäåì â ðàññìîòðåíèå ÿäðà kerU · =U0, kerF · = F0 è îáðàçû imU · = U1, imF · = F1 ýòèõ ïîëóãðóïï. Íåòðóäíî ïîêàçàòü, ÷òîU0 ⊕ U1 = U0 ⊕ U1 = U0 ⊕ U1, F0 ⊕ F1 = F0 ⊕ F1 = F0 ⊕ F1. Íàì ïîòðåáóåòñÿ áîëåå ñèëüíîåóòâåðæäåíèå

U0 ⊕ U1 = U (F0 ⊕ F1 = F), (B1)

êîòîðîå èìååò ìåñòî ëèáî â ñëó÷àå ñèëüíîé (L, p)-ñåêòîðèàëüíîñòè îïåðàòîðà M ñïðàâà(ñëåâà), p ∈ {0}∪N, ëèáî ðåôëåêñèâíîñòè ïðîñòðàíñòâà U (F) [40]. Îáîçíà÷èì ÷åðåç Lk (Mk)ñóæåíèå îïåðàòîðà L (M) íà Uk (domM ∩ Uk), k = 0, 1. È åñëè îïåðàòîð M ñèëüíî (L, p)-ñåêòîðèàëåí ñïðàâà è ñëåâà, p ∈ {0}∪N, òî Lk ∈ L(Uk;Fk), Mk ∈ Cl(Uk;Fk), k = 0, 1, ïðè÷åìñóùåñòâóåò îïåðàòîð M−10 ∈ L(F0;U0), à òàêæå ïðîåêòîð P = s− limt→0+U

t (Q = s− limt→0+

F t),

ðàñùåïëÿþùèé ïðîñòðàíñòâî U (F) ñîãëàñíî (B1), ïðè÷åì U1 = imP (F1 = imQ). Ââåäåìåùå îäíî óñëîâèå �

ñóùåñòâóåò îïåðàòîð L−11 ∈ L(F1;U1), (B2)

êîòîðîå èìååò ìåñòî â ñëó÷àå ñèëüíîé (L, p)-ñåêòîðèàëüíîñòè îïåðàòîðà M , p ∈ {0} ∪ N.(Êñòàòè, â [39] ïîêàçàíî, ÷òî (B1) âìåñòå ñ óñëîâèåì (L, p)-ñåêòîðèàëüíîñòè îïåðàòîðà M ,p ∈ {0} ∪ N, äàåò ñèëüíóþ (L, p)-ñåêòîðèàëüíîñòü îïåðàòîðà M ñïðàâà (ñëåâà), p ∈ {0} ∪ N,à åñëè ê íèì äîáàâèòü óñëîâèå (B2), òî ïîëó÷èì ñèëüíóþ (L, p)-ñåêòîðèàëüíîñòü îïåðàòîðàM , p ∈ {0} ∪ N).

Íàêîíåö, ââåäåì åùå îäíî âàæíîå óñëîâèå �

L− ñïåêòð σL(M) îïåðàòîðà M ïðåäñòàâèì â âèäåσL(M) = σLfin(M) ∪ σLin(M), ïðè÷åì σLfin(M)ñîäåðæèòñÿ â îãðàíè÷åííîé îáëàñòè D ⊂ Cñ êóñî÷íî ãëàäêîé ãðàíèöåé γ, ïðè÷åì γ ∩ σL(M) = ∅.

(B3)

Ïîñòðîèì îòíîñèòåëüíî ñïåêòðàëüíûé ïðîåêòîð [35]

Pfin =1

2πi

∫γRLµ(M)dµ,

ïðè÷åì îêàçûâàåòñÿ, ÷òî ïðè óñëîâèè ñèëüíîé (L, p)-ñåêòîðèàëüíîñòè îïåðàòîðà M ñïðà-âà PfinP = PPfin = Pfin. Çíà÷èò, â äàííîì ñëó÷àå ñóùåñòâóåò ïðîåêòîð Pin = P − Pfin.Èòàê, ïóñòü âûïîëíåíû óñëîâèÿ (B1) � (B3), ôèêñèðóåì τ ∈ R+, u0, uτ ∈ U, è äëÿ óðàâ-íåíèÿ (3.1) ðàññìîòðèì íà÷àëüíî-êîíå÷íóþ çàäà÷ó: íàéòè âåêòîð-ôóíêöèþ v ∈ ∞(R+;U),óäîâëåòâîðÿþùóþ óðàâíåíèþ (3.1) è óñëîâèÿì

Pin(u(0)− u0) = 0, Pfin(u(τ)− uτ ) = 0. (3.2)

Òîãäà ñóùåñòâóåò àíàëèòè÷åñêàÿ ãðóïïà {U tfin : t ∈ R}, ãäå

U t =1

2πi

∫ΓRLµ(M)e

µtdµ, U tfin =1

2πi

∫γRLµ(M)e

µtdµ

òàêèå, ÷òî s − limt→0+

U t = P è U0fin = Pfin. Ïîñòðîèì àíàëèòè÷åñêóþ ïîëóãðóïïó {U tin : t ∈

R+}, ãäå U tin = U t − U tfin. Î÷åâèäíî, s − limt→0+Utin = Pin. Ïîëîæèì imPin(fin) = U

1in(fin);

î÷åâèäíî, U1 = U1in ⊕ U1fin. ÑïðàâåäëèâàÒåîðåìà 3.1. Ïóñòü îïåðàòîð M (L, p)-ñåêòîðèàëåí, p ∈ {0}∪N, è âûïîëíåíû óñëîâèÿ

(B1) � (B3). Òîãäà ïðè ëþáûõ τ ∈ R+, u0, uτ ∈ U ñóùåñòâóåò åäèíñòâåííîå ðåøåíèå çàäà÷è(3.1), (3.2), êîòîðîå ê òîìó æå èìååò âèä u(t) = U tinu0 + U

t−τfin uτ .



Òîò ôàêò, ÷òî âåêòîð-ôóíêöèÿ u(t) = U tinu0 + Ut−τfin uτ óäîâëåòâîðÿåò óðàâíåíèþ (3.1),

ïðîâåðÿåòñÿ íåïîñðåäñòâåííî. Âûïîëíåíèå óñëîâèÿ (3.2) ñëåäóåò èç ñîîòíîøåíèé PinUtfin =

O è PfinU tin = O, à òàêæå U tin = PinU tin = U tinPin è U tfin = PfinU tfin = U tfinPfin ïðè âñåõt ∈ R+ â ñëó÷àå U tin è ïðè âñåõ t ∈ R â ñëó÷àå U tfin. Åäèíñòâåííîñòü ðåøåíèÿ âûòåêàåò èçýêâèâàëåíòíîñòè óðàâíåíèÿ (3.1) ñèñòåìå óðàâíåíèé

u0 = 0, u̇1in = Sinu1in, u̇

1fin = Sfinu

1fin,

ãäå Sin = L−1in Min ∈ Cl(U1in;F1in) � ñåêòîðèàëüíûé îïåðàòîð, Sfin = L

−1finMfin ∈ L(U

1fin;F

1fin);

ïîäïðîñòðàíñòâà F1in è F1fin ñòðîÿòñÿ àíàëîãè÷íî ïðîñòðàíñòâàì U

1in è U

1fin, òîëüêî âìåñòî

ïîëóãðóïïû {U t : t ∈ R+} è ãðóïïû {U tfin : t ∈ R} íàäî âçÿòü ïîëóãðóïïó {F t : t ∈ R+} èãðóïïó {F tfin : t ∈ R}, ãäå, ñîîòâåòñòâåííî,

F t =1

2πi

∫ΓLLµ(M)e

µtdµ, F tfin =1

2πi

∫γLLµ(M)e

µtdµ;

îïåðàòîðû Lin(fin) (Min(fin)) åñòü ñóæåíèå îïåðàòîðîâ L1 (M1) íà U1in(fin) (domM ∩U

1in(fin)).

4. Óðàâíåíèå Íàâüå � Ñòîêñà

Ïóñòü Ω ⊂ Rn, n = N\{1}, � îãðàíè÷åííàÿ îáëàñòü ñ ãðàíèöåé ∂Ω êëàññà C∞. Â öèëèíäðåΩ× R+ ðàññìîòðèì çàäà÷ó Äèðèõëå

v(x, t) = 0, (x, t) ∈ ∂Ω× R+

äëÿ ñèñòåìû óðàâíåíèé Íàâüå � Ñòîêñà

vt = ν∇2v −∇p, ∇ · v = 0. (4.1)

Ïðåæäå ÷åì ðåäóöèðîâàòü ñèñòåìó (4.1) ê óðàâíåíèþ (3.1), ïðåäñòàâèì åå â âèäå

vt = ν∇2v − p, ∇(∇ · v) = 0. (4.2)

Ñèñòåìà (4.2) ïîëó÷åíà èç (4.1) ïîñëå çàìåíû ∇p → p [41].Äëÿ ðåäóêöèè óðàâíåíèé (4.2) ê óðàâíåíèþ (3.1) íàì ïîòðåáóþòñÿ ôóíêöèîíàëüíûå

ïðîñòðàíñòâà èç [42]. Ïóñòü H2σ è H2π (Hσ è Hπ) � ïîäïðîñòðàíñòâà ñîëåíîèäàëüíûõ è ïîòåí-öèàëüíûõ âåêòîð-ôóíêöèé ïðîñòðàíñòâà H2 = (W 22 (Ω)∩

◦W 12(Ω))

n (L2 = (L2(Ω))n). Ôîðìó-ëîé A = diag {∇2, . . . ,∇2} çàäàåòñÿ ëèíåéíûé íåïðåðûâíûé îïåðàòîð ñ äèñêðåòíûì êîíå÷-íîêðàòíûì îòðèöàòåëüíûì ñïåêòðîì σ(A), ñãóùàþùèìñÿ ëèøü íà −∞. Îáîçíà÷èì ÷åðåçAσ(π) ñóæåíèå îïåðàòîðà A íà H2σ(π).

Ëåììà 4.1. (òåîðåìà Ñîëîííèêîâà � Âîðîâè÷à � Þäîâè÷à). Îïåðàòîð Aσ(π) ∈L(H2σ(π),Hσ(π)), ïðè÷åì σ(Aσ(π)) = σ(A) è A = AσΣ+AπΠ.

Çäåñü ÷åðåç Π ∈ L(H2,H2π) îáîçíà÷åí ïðîåêòîð âäîëü H2σ, Σ = I−Π.Ëåììà 4.2 (òåîðåìà Êàïèòàíñêîãî � Ïèëåöêàñà). Ôîðìóëîé B : u → ∇(∇ · u) çàäàåòñÿ

îïåðàòîð B ∈ L(H2,Hπ), ïðè÷åì kerB = H2σ.Ïîëîæèì U = F ≡ Hσ × Hπ × Hp, Hp = Hπ. Âåêòîð u ∈ U èìååò âèä u = (uσ, uπ, up).

Ôîðìóëàìè

L =

I O OO I OO O O

, M = νAσ O OO νAπ −I

O B O

2013, òîì 6, 2 13


çàäàþòñÿ îïåðàòîðû L ∈ L(U;F), imL = Hσ×Hπ×{0}, kerL = {0}×{0}×Hp è M ∈ Cl(U;F),dom M = H2σ ×H2π ×Hp. Èòàê, ðåäóêöèÿ óðàâíåíèé (4.2) ê óðàâíåíèþ (3.1) çàêîí÷åíà.

Ëåììà 4.3. [43]. Ïðè ëþáûõ ν ∈ R+ îïåðàòîð M ñèëüíî (L, 1)-ñåêòîðèàëåí.Ïîñòðîèì ïîäïðîñòðàíñòâà U0 = F0 = {0} × Hπ × Hp, U1 = F1 = Hσ × {0} × {0}.

Âûïîëíåíèå óñëîâèé (B1) è (B2) î÷åâèäíî, ïðè÷åì

M−10 =

O O OO O OO O O

,ãäå Bπ � ñóæåíèå îïåðàòîðà B íà H2π (èç ëåììû 2 âûòåêàåò, ÷òî Bπ : H2π → Hπ � òîïëèíåéíûéèçîìîðôèçì). Íåòðóäíî òàêæå ïðîâåðèòü, ÷òî

M−10 L0 =

O O −IO O OO O O

−íèëüïîòåíòíûé îïåðàòîð ñòåïåíè 1.

Ñïåêòð σ(A) = {λk}, ãäå λk ∈ R− � ñîáñòâåííûå çíà÷åíèÿ, çàíóìåðîâàííûå ïî íåâîçðàñ-òàíèþ ñ ó÷åòîì èõ êðàòíîñòè, òîãäà σL(M) = {ν−1λk}. Ïîíÿòíî, ÷òî äëÿ òàêîãî ìíîæåñòâàìîæíî ïîäîáðàòü êîíòóð γ ∈ C, êîòîðûé áû óäîâëåòâîðÿë óñëîâèþ (B3).

Òåïåðü ïîñòðîèì

U tin(fin) =

∑

ν−1λk∈σLin(fin)(M)

eνλkt⟨·, φk⟩φk O O

O O OO O O

.Òîãäà â ñèëó òåîðåìû 3.1 è ëåììû 4.3 ñïðàâåäëèâà ñëåäóþùàÿÒåîðåìà 4.1. [36] Ïðè ëþáûõ ν ∈ R+, u0, uτ ∈ U ñóùåñòâóåò åäèíñòâåííîå ðåøåíèå

çàäà÷è (3.2) äëÿ ñèñòåìû óðàâíåíèé (4.2), ïðè÷åì ýòî ðåøåíèå u = u(t) èìååò âèä

uσ(t) = Ut−τ1 uτσ + U

t2u0σ, uπ ≡ 0, up ≡ 0.

5. Îáîáùåííàÿ òåîðåìà î ðàñùåïëåíèè

Ïóñòü U è F � áàíàõîâû ïðîñòðàíñòâà, îïåðàòîðû L ∈ L(U;F) (ò.å. ëèíååí è íåïðåðûâåí)è M ∈ Cl(U;F) (ò.å. ëèíååí, çàìêíóò è ïëîòíî îïðåäåëåí). Ïóñòü îòíîñèòåëüíûé ñïåêòðîïåðàòîðà M σL(M) = σL0 (M) ∪ σL1 (M), ïðè÷åì

σL1 (M) ̸= ∅, ñóùåñòâóåò çàìêíóòûé êîíòóðΓ ⊂ C, îãðàíè÷èâàþùèé îáëàñòü D ⊃ σL1 (M),òàêîé, ÷òî D ∩ σL0 (M) = ∅.

(5.1)Ïîñòðîèì èíòåãðàëû òèïà Ô. Ðèññà (ïîíèìàåìûå â ñìûñëå Ðèìàíà)

P =1

2πi

∫Γ

RLµ(M)dµ, Q =1

2πi

∫Γ

LLµ(M)dµ, (5.2)

ãäå RLµ(M) (LLµ(M)) � ïðàâàÿ (ëåâàÿ) L-ðåçîëüâåíòû îïåðàòîðà M .



Ëåììà 5.1. Ïóñòü σL(M) = σL0 (M)∪σL1 (M), ïðè÷åì âûïîëíåíî (5.1). Òîãäà îïåðàòîðûP : U → U è Q : F → F � ïðîåêòîðû.

Ïîëîæèì U0(F0) = kerP (kerQ), U1(F1) = imP (imQ) è ÷åðåç Lk (Mk) îáîçíà÷èì ñóæåíèåîïåðàòîðà L (M) íà Uk (domM ∩ Uk), k = 0, 1.

Òåîðåìà 5.1. Ïóñòü âûïîëíåíû óñëîâèÿ ëåììû 5.1. Òîãäà

(i) Lk ∈ L(Uk;Fk), k = 0, 1;(ii) M0 ∈ Cl(U0;F0), M1 ∈ L(U1;F1);(iii) ñóùåñòâóþò îïåðàòîðû L−11 ∈ L(F1;U1) è M

−10 ∈ L(F0;U0).

Êàê èçâåñòíî, îáà ýòèõ óòâåðæäåíèÿ ïåðâûì ñôîðìóëèðîâàë è äîêàçàë Ã.À. Ñâèðèäþê,ïðàâäà, ïðè áîëåå îãðàíè÷èòåëüíîì óñëîâèè, à èìåííî:

σL(M) ̸= ∅, ñóùåñòâóåò çàìêíóòûé êîíòóðΓ ⊂ C, îãðàíè÷èâàþùèé îáëàñòü D ⊃ σL(M).

}(5.3)

Îäíàêî âíèìàòåëüíûé àíàëèç åãî äîêàçàòåëüñòâ (ñì. [17], ëåììà 4.1.1 è òåîðåìà 4.1.1) ïî-êàçûâàåò, ÷òî îíè ãîäÿòñÿ è â íàøåì ñëó÷àå.

Ïóñòü σL(M) =n∪

j=0

σLj (M), n ∈ N, ïðè÷åì σLj (M) ̸= ∅,

ñóùåñòâóåò çàìêíóòûé êîíòóð Γj ⊂ C,

îãðàíè÷èâàþùèé îáëàñòü Dj ⊃ σLj (M), òàêîé, ÷òî

Dj ∩ σL0 (M) = ∅ è Dk ∩Dl = ∅ ïðè âñåõ j, k, l = 1, n, k ̸= l.

(5.4)

Àíàëîãè÷íî (5.2) ïîñòðîèì èíòåãðàëû

Pj =1

2πi

∫Γj

RLµ(M)dµ, Qj =1

2πi

∫Γj

LLµ(M)dµ, j = 1, n. (5.5)

Ëåììà 5.3. [37] Ïóñòü âûïîëíåíû óñëîâèÿ (5.3), (5.4). Òîãäà îïåðàòîðû(i) Pj : U → U è Qj : F → F � ïðîåêòîðû, j = 1, n;(ii) PkPl = O, QkQl = O, k, l = 1, n, k ̸= l.

(iii) P0 = P −n∑

j=1

Pj è Q0 = Q−n∑

j=1

Qj � ïðîåêòîðû.

(Çàìåòèì, ÷òî çäåñü ðàäè ýêîíîìèè ìåñòà ïðîåêòîðû Pj è Qj , j = 1, n, èç (5.5), àïðîåêòîðû P è Q èç (5.2), íî ñ çàìåíîé óñëîâèÿ (5.1) íà óñëîâèå (5.3)).

Ïîëîæèì U0(F0) = kerP (kerQ), U1j (F1j ) = imPj(imQj), j = 0, n, è ÷åðåç L0 (M0) îáîçíà-

÷èì ñóæåíèå îïåðàòîðà L (M) íà U0 (domM ∩ U0), à ÷åðåç L1j (M1j) îáîçíà÷èì ñóæåíèåîïåðàòîðà L (M) íà U1j (domM ∩ U1j ), j = 0, n.

Òåîðåìà 5.1. (Îáîáùåííàÿ òåîðåìà î ðàñùåïëåíèè) [37] Ïóñòü âûïîëíåíû óñëîâèÿ(1.3), (1.4). Òîãäà

(i) L0 ∈ L(U0;F0), L1j ∈ L(U1j ;F1j ), j = 0, n;(ii) M0 ∈ Cl(U0;F0), M1j ∈ L(U1j ;F1j ), j = 0, n;(iii) ñóùåñòâóþò îïåðàòîðû L−11j ∈ L(F1j ;U1j ), j = 0, n, è M

−10 ∈ L(F0;U0).

6. Îòíîñèòåëüíî p-îãðàíè÷åííûå îïåðàòîðû

Ïóñòü U è F � áàíàõîâû ïðîñòðàíñòâà, îïåðàòîðû L ∈ L(U;F) (ò.å. ëèíååí è íåïðåðû-âåí) è M ∈ Cl(U;F) (ò.å. ëèíååí, çàìêíóò è ïëîòíî îïðåäåëåí), ïðè÷åì îïåðàòîð M (L, p)-îãðàíè÷åí, p ∈ {0} ∪ N (òåðìèíîëîãèÿ è ðåçóëüòàòû ñì. ãë. 5 [17]). Ðàññìîòðèì ëèíåéíîå

2013, òîì 6, 2 15


óðàâíåíèå ñîáîëåâñêîãî òèïà

Lu̇ = Mu. (6.1)

Ðåøåíèåì u = u(t) óðàâíåíèÿ (6.1) íàçîâåì âåêòîð-ôóíêöèþ u ∈ ∞(R;U), óäîâëåòâîðÿþùóþýòîìó óðàâíåíèþ.

Îïðåäåëåíèå 6.1. [17] Îòîáðàæåíèå U . ∈ C∞(R;L(U)) íàçîâåì ãðóïïîé ðàçðåøàþùèõîïåðàòîðîâ óðàâíåíèÿ (2.1), åñëè

(i) U tU s = U t+s ïðè âñåõ s, t ∈ R;(ii) ïðè âñåõ v ∈ U âåêòîð-ôóíêöèÿ u = U tv åñòü ðåøåíèå óðàâíåíèÿ (2.1).Â äàëüíåéøåì, ñëåäóÿ òðàäèöèè, áóäåì îòîæäåñòâëÿòü ãðóïïó ðàçðåøàþùèõ îïåðàòî-

ðîâ óðàâíåíèÿ (6.1) ñ åå ãðàôèêîì {U t : t ∈ R}, è â äàëüíåéøåì íàçûâàòü ïðîñòî ãðóïïîéóðàâíåíèÿ (6.1). Ãðóïïó {U t : t ∈ R} óðàâíåíèÿ (6.1) áóäåì íàçûâàòü àíàëèòè÷åñêîé, åñëèîíà àíàëèòè÷åñêè ïðîäîëæèìà âî âñþ êîìïëåêñíóþ ïëîñêîñòü ñ ñîõðàíåíèåì ñâîéñòâà (i).

Òåîðåìà 6.1.Ïóñòü âûïîëíåíû óñëîâèÿ (5.3), (5.4). Òîãäà ñóùåñòâóþò àíàëèòè÷åñêèå

ãðóïïû óðàâíåíèÿ (6.1)

U t =1

2πi

∫Γ

RLµ(M)eµtdµ, U tj =

1

2πi

∫Γj

RLµ(M)eµtdµ, j = 1, n,

ïðè÷åì

(i) U tU sj = UsjU

t = U t+sj ïðè âñåõ s, t ∈ R, j = 1, n;(ii) U tkU

sl = U

sl U

tk = O ïðè âñåõ s, t ∈ R, k, l = 1, n, k ̸= l.

(iii) U t0 = Ut −

n∑j=1

U tj � àíàëèòè÷åñêàÿ ãðóïïà óðàâíåíèÿ (6.1).

Äàëåå âîçüìåì âåêòîð-ôóíêöèþ f ∈ C∞((a, b);F) è ðàññìîòðèì ëèíåéíîå íåîäíîðîäîåóðàâíåíèå ñîáîëåâñêîãî òèïà

Lu̇ = Mu+ f. (6.2)

Âåêòîð-ôóíêöèþ u ∈ C∞((a, b);U), óäîâëåòâîðÿþùóþ óðàâíåíèþ (6.2), íàçîâåì ðåøåíèåìóðàâíåíèÿ (6.2). Ðåøåíèå u = u(t), t ∈ (a, b) óðàâíåíèÿ (6.2), óäîâëåòâîðÿþùåå óñëîâèÿì

Pj(u(τj)− uj) = 0, j = 0, n, (6.3)

íàçîâåì ðåøåíèåì ìíîãîòî÷å÷íîé íà÷àëüíî-êîíå÷íîé çàäà÷è äëÿ óðàâíåíèÿ (6.2).Òåîðåìà 6.1. [37] Ïóñòü îïåðàòîð M (L, p)-îãðàíè÷åí, ïðè÷åì âûïîëíåíî óñëîâèå (5.4).

Òîãäà äëÿ ëþáûõ f ∈ C∞((a, b);F)), uj ∈ U, j = 0, n ñóùåñòâóåò åäèíñòâåííîå ðåøåíèåçàäà÷è (6.2), (6.3), êîòîðîå ê òîìó æå èìååò âèä

u(t) = −p∑

q=0

(M−10 L0)qM−10 (I−Q)f

(q)(t) +n∑

j=0

Ut−τjj uj +

n∑j=0

∫ tτj

U t−sj L−11j Qjf(s)ds.

7. Óðàâíåíèå Áàðåíáëàòòà � Æåëòîâà � Êî÷èíîé íà ãðàôå

Ïóñòü G = G(V;E) � êîíå÷íûé ñâÿçíûé îðèåíòèðîâàííûé ãðàô, ãäå V = {Vi} � ìíîæå-ñòâî âåðøèí, à E = {Ej} � ìíîæåñòâî äóã. Ìû ïðåäïîëàãàåì,÷òî êàæäàÿ äóãà èìååò äëèíólj > 0 è øèðèíó dj > 0. Íà ãðàôå G íàñ áóäóò èíòåðåñîâàòü çàäà÷è ñ êðàåâûìè

uj(0, t) = uk(0, t) = um(lm, t) = un(ln, t),Ej , Ek ∈ Eα(Vi), Em, Ek ∈ Eω(Vi);

(7.1)



∑Ej∈Eα(Vi)

djujx(0, t)−∑

Ek∈Eω(Vi)

dkukx(lk, t) = 0; (7.2)

óñëîâèÿìè äëÿ óðàâíåíèéλujt − ujxxt = αujxx. (7.3)

Ââåäåì ìíîæåñòâî

L2(G) = {g = (g1, g2, . . . , gj , . . .) : gj ∈ L2(0, lj)},

êîòîðîå ñòàíîâèòñÿ ãèëüáåðòîâûì ïðîñòðàíñòâîì ñî ñêàëÿðíûì ïðîèçâåäåíèåì

⟨g, h⟩ =∑Ej∈E

dj

lj∫0

gj(x)hj(x) dx.

×åðåç U îáîçíà÷èì ìíîæåñòâî

U = {u = (u1, u2, . . . , uj , . . .) : uj ∈ W 12 (0, lj), è âûïîëíåíî (7.2)}.

Ìíîæåñòâî U ÿâëÿåòñÿ ãèëüáåðòîâûì ïðîñòðàíñòâîì ñî ñêàëÿðíûì ïðèçâåäåíèåì è íîðìîé

[u, v] =∑Ej∈E

dj

lj∫0

(ujxvjx(x) + ujvjx(x)) dx, ∥u∥2U =∑Ej∈E

dj

lj∫0

(u2jx(x) + u2j (x)) dx.

Â ñèëó òåîðåì âëîæåíèÿ Ñîáîëåâà ïðîñòðàíñòâî W 12 (0, lj) ñîñòîèò èç àáñîëþòíî íåïðåðûâ-íûõ ôóíêöèé, à, çíà÷èò, ïðîñòðàíñòâî U êîððåêòíî îïðåäåëåíî, ïëîòíî è êîìïàêòíî âëîæåíîâ L2(G). Îòîæäåñòâèì L2(G) ñî ñâîèì ñîïðÿæåííûì è ÷åðåç F îáîçíà÷èì ñîïðÿæåííîå îò-íîñèòåëüíî äâîéñòâåííîñòè ⟨ ·, · ⟩ ïðîñòðàíñòâî ê U. Î÷åâèäíî, F � áàíàõîâî ïðîñòðàíñòâî,ïðè÷åì âëîæåíèå U ↪→ F êîìïàêòíî.

Ôîðìóëîé

⟨Au, v⟩ =∑Ej∈E

dj

lj∫0

(ujx(x)vjx(x) + auj(x)vj(x)) dx,

ãäå a ∈ R+, u, v ∈ U, çàäàäèì îïåðàòîð, îïðåäåëåííûé íà ïðîñòðàíñòâå U. Ïîñêîëüêó

|⟨Au, v⟩| ≤ C1∥u∥U∥v∥U

â ñèëó íåðàâåíñòâà Êîøè � Áóíÿêîâñêîãî è

C2∥u∥2U ≤ ⟨Au, u⟩ ≤ C3∥u∥2U (7.4)

ïðè âñåõ u, v ∈ U è íåêîòîðûõ Ck ∈ R+, k = 1, 2, 3, òî ëèíåéíûé îïåðàòîð A : U → F íåïðåðû-âåí è èíúåêòèâåí. Êðîìå òîãî, èç ïåðâîé îöåíêè (7.4) âûòåêàåò ñþðúåêòèâíîñòü ñîïðÿæåí-íîãî îïåðàòîðà A∗ : F∗ → U∗. Â ñèëó ðåôëåêñèâíîñòè ïðîñòðàíñòâà U è ñàìîñîïðÿæåííîñòèîïåðàòîðà A ïîëó÷àåì, ÷òî îïåðàòîð A ∈ L(U;F) áèåêòèâåí. Îòñþäà ïî òåîðåìå Áàíàõàñëåäóåò ñóùåñòâîâàíèå îïåðàòîðà A−1 ∈ L(F;U). Ïîñêîëüêó âëîæåíèå U ↪→ F êîìïàêòíî,òî îïåðàòîð A−1 ∈ L(F) ÿâëÿåòñÿ êîìïàêòíûì. Çíà÷èò, ñïåêòð îïåðàòîðà A âåùåñòâåíåí,äèñêðåòåí, êîíå÷íîêðàòåí è ñãóùàåòñÿ òîëüêî ê +∞.

Òåïåðü ôèêñèðóåì α ∈ R+ è λ ∈ R è ïîñòðîèì îïåðàòîðû

L = (λ− a)I+A, M = α(aI−A).

2013, òîì 6, 2 17


Èç ñêàçàííîãî ñëåäóåòÒåîðåìà 7.1. (ñì. íàïð. [44]). Îïåðàòîðû L,M ∈ L(U;F), ïðè÷åì ñïåêòð σ(L) îïåðà-

òîðà L âåùåñòâåíåí, äèñêðåòåí, êîíå÷íîêðàòåí è ñãóùàåòñÿ òîëüêî ê −∞.Èç òåîðåìû 7.1 âûòåêàåò, ÷òî îïåðàòîð L�ôðåäãîëüìîâ, ïðè÷åì kerL = {0}, åñëè 0 /∈

σ(L).Ëåììà 7.1. (ñì. íàïð. [44]). Ïóñòü ïàðàìåòðû α, λ ∈ R \ {0}, òîãäà îïåðàòîð M(L, 0)-

îãðàíè÷åí.

Ïóñòü {λk} � ñîáñòâåííûå çíà÷åíèÿ îïåðàòîðà A, çàíóìåðîâàííûå ïî íåóáûâàíèþ ñó÷åòîì èõ êðàòíîñòè; à {φk} � ñîîòâåòñòâóþùèå èì îðòîíîðìèðîâàííûå â ñìûñëå L2(G)ôóíêöèè. Ïî ôîðìóëàì (5.2) ïîñòðîèì ïðîåêòîðû

P =

{I, åñëè 0 /∈ σ(L);I−

∑λk=λ−a

⟨ · , φk⟩φk, åñëè 0 ∈ σ(L); Q =

{I, åñëè 0 /∈ σ(L);I−

∑λk=λ−a

⟨ · , φk⟩φk, åñëè 0 ∈ σ(L);

(çàìåòèì, ÷òî íåñìîòðÿ íà "ïîõîæåñòü" ïðîåêòîðû îïðåäåëåíû íà ðàçíûõ ïðîñòðàíñòâàõ) èðàçðåøàþùóþ ãðóïïó

U t =

∞∑k=1

′eµkt⟨ · , φk⟩φk,

ãäå øòðèõ ó çíàêà ñóììû îçíà÷àåò îòñóòñòâèå ÷ëåíîâ ðÿäà ñ íîìåðàìè k òàêèìè, ÷òî λk =λ− a; ⟨ ·, · ⟩�ñêàëÿðíîå ïðîèçâåäåíèå â L2(G). L-ñïåêòð îïåðàòîðà M èìååò âèä

σL(M) = {µk =α(a− λk)

λ− (a+ λk), k ∈ N}.

Âûïîëíåíèå óñëîâèÿ (5.3) î÷åâèäíî, âûáåðåì σLj (M), j = 0, n, òàê, ÷òîáû âûïîëíÿëîñüóñëîâèå (5.4) (ïîíÿòíî, ÷òî ýòî ìîæíî ñäåëàòü íå îäíèì ñïîñîáîì). Ïîñòðîèì ïðîåêòîðû

Pj =∑

µk∈σLj (M)

⟨·, φk⟩φk, j = 0, n. (7.5)

Âîçüìåì −∞ ≤ a < τ0 < τ1 < τ2 < . . . < τn < b ≤ +∞, uj ∈ U, j = 0, n, f ∈ C∞((a, b);F) èðàññìîòðèì çàäà÷ó (6.2), (6.3), ãäå U � ôóíêöèîíàëüíîå áàíàõîâî ïðîñòðàíñòâî ñ êðàåâûìóñëîâèåì (7.1), îïåðàòîðû L è M èç (6.2), à ïðîåêòîðû Pj , j = 0, n, èç (6.3).

Â ñèëó ëåììû 7.1 è òåîðåìû 6.1 âûòåêàåòÒåîðåìà 7.2. Ïðè ëþáûõ λ ∈ R, α ∈ R\{0}, uj ∈ U, j = 0, n, ìíîãîòî÷å÷íàÿ íà÷àëüíî-

êîíå÷íàÿ çàäà÷à (6.3), (7.1), (7.2) äëÿ óðàâíåíèé (7.3) èìååò åäèíñòâåííîå ðåøåíèå u ∈C∞((a, b);U), êîòîðîå ê òîìó æå èìååò âèä

u(t) =n∑

j=0

∑µk∈σLj (M)

eµk(t−τj)⟨uj , φk⟩φk.

Àâòîð âûðàæàåò ñâîþ èñêðåííþþ ïðèçíàòåëüíîñòü ïðîôåññîðó Ã.À. Ñâèðèäþêó çà

ïîñòàíîâêó çàäà÷è è èíòåðåñ ê ðàáîòå, à òàê æå Ì.À. Ñàãàäååâîé çà ïëîäîòâîðíûå äèñ-

êóññèè.

Ëèòåðàòóðà

1. Ïëîòíèêîâ, Ï.È. Çàäà÷à Ñòåôàíà ñ ïîâåðõíîñòíûì íàòÿæåíèåì êàê ïðåäåë ìîäåëè ôà-çîâîãî ïîëÿ / Ï.È. Ïëîòíèêîâ, Â.Í. Ñòàðîâîéòîâ // Äèôôåðåíö. óðàâíåíèÿ. � 1993. �Ò. 29, 3. � Ñ. 395�405.



2. Ïëîòíèêîâ, Ï.È. Óðàâíåíèÿ ôàçîâîãî ïîëÿ è ãðàäèåíòíûå ïîòîêè ìàðãèíàëüíûõ ôóíê-öèé / Ï.È. Ïëîòíèêîâ, À.Â. Êëåïà÷åâà // Ñèá. ìàò. æóðí. � 2001. � Ò. 42, 3. �Ñ. 651�669.

3. Ëàäûæåíñêàÿ, Î.À. Ìàòåìàòè÷åñêèå âîïðîñû äèíàìèêè âÿçêîé íåñæèìàåìîé æèäêîñòè/ Î.À. Ëàäûæåíñêàÿ. � Ì.: Ôèçìàòãèç, 1961.

4. Òåìàì, Ð. Óðàâíåíèÿ Íàâüå � Ñòîêñà. Òåîðèÿ è ÷èñëåííûé àíàëèç / Ð. Òåìàì. � Ì.:Ìèð, 1981.

5. Áàðåíáëàòò, Ã.È. Îá îñíîâíûõ ïðåäñòàâëåíèÿõ òåîðèè ôèëüòðàöèè â òðåùèíîâàòûõ ñðå-äàõ / Ã.È. Áàðåíáëàòò, Þ.Ï. Æåëòîâ, È.Í. Êî÷èíà // Ïðèêë. ìàòåìàòèêà è ìåõàíèêà.� 1960. � Ò. 24, 5. � Ñ. 58�73.

6. Ðóòêàñ, À.Ã. Çàäà÷à Êîøè äëÿ óðàâíåíèÿ A(x) +Bx(t) = f(t) / À.Ã. Ðóòêàñ // Äèôôå-ðåíö. óðàâíåíèÿ. � 1975. � Ò. 11, 11. � Ñ. 1996�2010.

7. Ting, T.W. Certain Non-Steady Flows of Second-Order Fluids / T.W. Ting // Arch. Rat.Mech. Anal. � 1963. � V. 14, no. 1. � P. 28�57.

8. Chen, P.J. On a Theory of Heat Conduction Involving Two Temperatures / P.J. Chen,M.E. Gurtin // Z. Angew. Math. Phys. � 1968. � V. 19. � P. 614�627.

9. Hallaire, M. On a Theory of Moisture-Transfer / M. Hallaire // Inst. Rech. Agronom. � 1964.� No. 3. � P. 60�72.

10. Îñêîëêîâ, À.Ï. Íåëîêàëüíûå ïðîáëåìû äëÿ îäíîãî êëàññà íåëèíåéíûõ îïåðàòîðíûõóðàâíåíèé, âîçíèêàþùèõ â òåîðèè óðàâíåíèé òèïà Ñ.Ë. Ñîáîëåâà / Îñêîëêîâ À.Ï. //Çàï. íàó÷. ñåì. ËÎÌÈ. � 1991. � Ò. 198. � Ñ. 31�48.

11. Ñâèðèäþê, Ã. À. Ôàçîâîå ïðîñòðàíñòâî çàäà÷è Êîøè � Äèðèõëå äëÿ îäíîãî íåêëàññè÷å-ñêîãî óðàâíåíèÿ / Ã.À. Ñâèðèäþê, À.Â. Àíêóäèíîâ // Äèôôåðåíö. óðàâíåíèÿ. � 2003. �Ò. 39, 11. � Ñ. 1556�1561.

12. Poincaré, H. Sur l'equilibre d'une mass �uide animée d'un mouvement de rotation /H. Poincaré // Acta Math. � 1885. � V. 7. � P. 259�380.

13. Ñîáîëåâ, Ñ.Ë. Îá îäíîé íîâîé çàäà÷å ìàòåìàòè÷åñêîé ôèçèêè / Ñ.Ë. Ñîáîëåâ // Èçâ.ÀÍ ÑÑÑÐ, ñåðèÿ ≪Ìàòåìàòèêà≫. � 1954. � Ò. 18, âûï. 1. � Ñ. 3�50.

14. Demidenko, G.V. Partial Di�erential Equations and Systems not Solvable with Respect tothe Highest � Order Deriative / G.V. Demidenko, S.V. Uspenskii. � N.-Y.; Basel; Hong Kong:Marcel Dekker, Inc., 2003.

15. Ïàíêîâ, À.À. Íåëèíåéíûå ýâîëþöèîííûå óðàâíåíèÿ ñ íåîáðàòèìûì îïåðàòîðíûì êîýô-ôèöèåíòîì ïðè ïðîèçâîäíîé / À.À. Ïàíêîâ, Ò.Å. Ïàíêîâà // Äîêë. Àêàä. íàóê Óêðàèíû.� 1993. � 9. � Ñ. 18�20.

16. Pyatkov, S.G. Operator Theory. Nonclassical Problems / S.G. Pyatkov. � Utrecht; Boston;K�oln; Tokyo: VSP, 2002.

17. Sviridyuk G.A. Linear Sobolev Type Equations and Degenerate Semigroups of Operators /G.A. Sviridyuk, V.E. Fedorov. � Utrecht; Boston; K�oln; Tokyo: VSP, 2003.

18. Çàìûøëÿåâà, À.À. Ëèíåéíûå óðàâíåíèÿ ñîáîëåâñêîãî òèïà âûñîêîãî ïîðÿäêà: ìîíîãð./ À.À. Çàìûøëÿåâà. � ×åëÿáèíñê: Èçä. öåíòð ÞÓðÃÓ, 2012.

19. Ìàíàêîâà, Í.À. Çàäà÷è îïòèìàëüíîãî óïðàâëåíèÿ äëÿ óðàâíåíèé ñîáîëåâñêîãî òèïà:ìîíîãð. / Í.À. Ìàíàêîâà. � ×åëÿáèíñê: Èçä. öåíòð ÞÓðÃÓ, 2012.

20. Ñàãàäååâà, Ì.À. Äèõîòîìèè ðåøåíèé ëèíåéíûõ óðàâíåíèé ñîáîëåâñêîãî òèïà: ìîíîãð./ Ì.À. Ñàãàäååâà. � ×åëÿáèíñê: Èçä. öåíòð ÞÓðÃÓ, 2012.

2013, òîì 6, 2 19


21. Êåëëåð, À.Â. Àëãîðèòì ðåøåíèÿ çàäà÷è Øîóîëòåðà � Ñèäîðîâà äëÿ ìîäåëåé ëåîíòüåâ-ñêîãî òèïà / À.Â. Êåëëåð // Âåñòíèê ÞÓðÃÓ. Ñåðèÿ: Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèåè ïðîãðàììèðîâàíèå. � ×åëÿáèíñê, 2011. � 4 (241), âûï. 7. � Ñ. 40�46.

22. Ñâèðèäþê, Ã.À. ×èñëåííîå ðåøåíèå ñèñòåì óðàâíåíèé ëåîíòüåâñêîãî òèïà / Ã.À. Ñâè-ðèäþê, Ñ.Â. Áðû÷åâ // Èçâ. âóçîâ. Ìàòåìàòèêà. � 2003. � 8. � Ñ. 46�52.

23. Ñâèðèäþê, Ã.À. Àëãîðèòì ðåøåíèÿ çàäà÷è Êîøè äëÿ âûðîæäåííûõ ëèíåéíûõ ñè-ñòåì îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé ñ ïîñòîÿííûìè êîýôôèöèåíòàìè /Ã.À. Ñâèðèäþê, È.Â. Áóðëà÷êî // Æóðí. âû÷èñë. ìàòåìàòèêè è ìàò. ôèçèêè. � 2003. �Ò. 43, 11. � Ñ. 1677�1683.

24. Shestakov, A.L. Optimal Measurement of Dynamically Distorted Signals / A.L. Shestakov,G.A. Sviridyuk // Âåñòíèê ÞÓðÃÓ. Ñåðèÿ: Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå è ïðîãðàì-ìèðîâàíèå. � 2011. � No. 17 (234), issue. 8. � P. 70�75.

25. Øåñòàêîâ, À.Ë. ×èñëåííîå ðåøåíèå çàäà÷è îïòèìàëüíîãî èçìåðåíèÿ / À.Ë. Øåñòàêîâ,À.Â. Êåëëåð, Å.È. Íàçàðîâà // Àâòîìàòèêà è òåëåìåõàíèêà. � 2012. � 1. � C. 107�115.

26. Ñâèðèäþê, Ã.À. Çàäà÷à Âåðèãèíà äëÿ ëèíåéíûõ óðàâíåíèé ñîáîëåâñêîãî òèïà ñ îòíîñè-òåëüíî p-ñåêòîðèàëüíûìè îïåðàòîðàìè / Ã.À. Ñâèðèäþê, Ñ.À. Çàãðåáèíà // Äèôôåðåíö.óðàâíåíèÿ. � 2002. � Ò. 38, 12. � Ñ. 1646�1652.

27. Çàãðåáèíà, Ñ.À. Î çàäà÷å Øîóîëòåðà � Ñèäîðîâà / Ñ.À. Çàãðåáèíà // Èçâ. âóçîâ. Ìà-òåìàòèêà. � 2007. � 3. � Ñ. 22�28.

28. Çàãðåáèíà, Ñ.À. Íà÷àëüíî-êîíå÷íàÿ çàäà÷à äëÿ ýâîëþöèîííûõ óðàâíåíèé ñîáîëåâñêîãîòèïà íà ãðàôå / Ñ.À. Çàãðåáèíà, Í.Ï. Ñîëîâüåâà // Âåñòíèê ÞÓðÃÓ. Ñåðèÿ: Ìàòåìà-òè÷åñêîå ìîäåëèðîâàíèå è ïðîãðàììèðîâàíèå. � ×åëÿáèíñê, 2008. � 15 (115), âûï. 1.� Ñ. 23�26.

29. Ìàíàêîâà, Í.À. Îá îäíîé çàäà÷å îïòèìàëüíîãî óïðàâëåíèÿ ñ ôóíêöèîíàëîì êà÷åñòâàîáùåãî âèäà / Í.À. Ìàíàêîâà, À.Ã. Äûëüêîâ // Âåñòí. Ñàì. ãîñ. òåõí. óí-òà. Ñåð. Ôèç.-ìàò. íàóêè. � Ñàìàðà, 2011. � 4 (25). � Ñ. 18�24.

30. Çàìûøëÿåâà, À.À. Íà÷àëüíî-êîíå÷íàÿ çàäà÷à äëÿ íåîäíîðîäíîãî óðàâíåíèÿ Áóññè-íåñêà � Ëÿâà / À.À. Çàìûøëÿåâà // Âåñòíèê ÞÓðÃÓ. Ñåðèÿ: Ìàòåìàòè÷åñêîå ìîäåëè-ðîâàíèå è ïðîãðàììèðîâàíèå. � ×åëÿáèíñê, 2011. � 37 (254), âûï. 10. � Ñ. 22�29.

31. Ñèäîðîâ, Í.À. Îá îäíîì êëàññå âûðîæäåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé ñ êîíâåð-ãåíöèåé / Í.À. Ñèäîðîâ // Ìàò. çàìåòêè. � 1984. � Ò. 35, 4. � C. 569�578.

32. Ñâèðèäþê, Ã.À. Çàäà÷à Øîóîëòåðà � Ñèäîðîâà êàê ôåíîìåí óðàâíåíèé ñîáîëåâñêîãîòèïà / Ã.À. Ñâèðèäþê, Ñ.À. Çàãðåáèíà // Èçâ. Èðêóò. ãîñ. óí-òà. Ñåðèÿ ≪Ìàòåìàòèêà≫.� 2010. � Ò. 3, 1. � Ñ. 104�125.

33. Çàãðåáèíà, Ñ.À. Íà÷àëüíî-êîíå÷íàÿ çàäà÷à äëÿ óðàâíåíèé ñîáîëåâñêîãî òèïà ñ ñèëüíî(L, p)-ðàäèàëüíûì îïåðàòîðîì / Ñ.À. Çàãðåáèíà // Ìàò. çàìåòêè ßÃÓ. � ßêóòñê, 2012.� Ò. 19, âûï. 2. � Ñ. 39�48.

34. Çàãðåáèíà, Ñ.À. Îáîáùåííàÿ çàäà÷à Øîóîëòåðà � Ñèäîðîâà äëÿ óðàâíåíèé ñîáîëåâñêîãîòèïà ñ ñèëüíî (L, p)-ðàäèàëüíûì îïåðàòîðîì / Ñ.À. Çàãðåáèíà, Ì.À. Ñàãàäååâà // Âåñòí.ÌàÃÓ. Ñåðèÿ ≪Ìàòåìàòèêà≫. � Ìàãíèòîãîðñê, 2006. � Âûï. 9. � Ñ. 17�27.

35. Çàãðåáèíà, Ñ.À. Çàäà÷à Øîóîëòåðà � Ñèäîðîâà � Âåðèãèíà äëÿ ëèíåéíûõ óðàâíåíèé ñî-áîëåâñêîãî òèïà / Ñ.À. Çàãðåáèíà // Íåêëàññè÷åñêèå óðàâíåíèÿ ìàòåìàòè÷åñêîé ôèçè-êè: òð. ìåæäóíàð. êîíô. ≪Äèôôåðåíöèàëüíûå óðàâíåíèÿ, òåîðèÿ ôóíêöèé è ïðèëîæå-íèÿ≫, ïîñâÿù. 100-ëåòèþ ñî äíÿ ðîæäåíèÿ àêàä. È. Í. Âåêóà / îòâ. ðåä. À. È. Êîæàíîâ;Ðîñ. Àêàä. íàóê, Ñèá. îòä., èí-ò ìàòåìàòèêè èì. Ñ. Ë. Ñîáîëåâà. � Íîâîñèáèðñê, 2007. �Ñ. 150�157.



36. Çàãðåáèíà, Ñ.À. Íà÷àëüíî-êîíå÷íàÿ çàäà÷à äëÿ ëèíåéíîé ñèñòåìû Íàâüå � Ñòîêñà /Ñ.À. Çàãðåáèíà // Âåñòíèê ÞÓðÃÓ. Ñåðèÿ: Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå è ïðîãðàì-ìèðîâàíèå. � ×åëÿáèíñê, 2011. � 4 (221), âûï. 7. � Ñ. 35�39.

37. Çàãðåáèíà, Ñ.À. Ìíîãîòî÷å÷íàÿ íà÷àëüíî-êîíå÷íàÿ çàäà÷à äëÿ ëèíåéíîé ìîäåëè Õîôôà/ Ñ.À. Çàãðåáèíà // Âåñòíèê ÞÓðÃÓ. Ñåðèÿ: Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå è ïðî-ãðàììèðîâàíèå. � ×åëÿáèíñê, 2012. � 5 (264), âûï. 11. � Ñ. 4�12.

38. Çàãðåáèíà, Ñ.À. Îá îäíîé íîâîé çàäà÷å äëÿ óðàâíåíèé Áàðåíáëàòòà � Æåëòîâà � Êî÷è-íîé / Ñ.À. Çàãðåáèíà, À.C. Êîíêèíà // Âåñòí. ÌàÃÓ. Ñåðèÿ ≪Ìàòåìàòèêà≫. � Ìàãíè-òîãîðñê, 2012. � Âûï. 14. � Ñ. 67�77.

39. Ôåäîðîâ, Â.Å. Âûðîæäåííûå ñèëüíî íåïðåðûâíûå ïîëóãðóïïû îïåðàòîðîâ / Â.Å. Ôåäî-ðîâ // Àëãåáðà è àíàëèç. � 2000. � Ò. 12, âûï. 3. � Ñ. 173�200.

40. Ôåäîðîâ, Â.Å. Î íåêîòîðûõ ñîîòíîøåíèÿõ â òåîðèè âûðîæäåííûõ ïîëóãðóïï îïåðà-òîðîâ / Â.Å. Ôåäîðîâ // Âåñòíèê ÞÓðÃÓ. Ñåðèÿ: Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå èïðîãðàììèðîâàíèå. � ×åëÿáèíñê, 2008. � 15 (115), âûï. 1. � Ñ. 89�99.

41. Çàãðåáèíà, Ñ.À. Î ñóùåñòâîâàíèè è óñòîé÷èâîñòè ðåøåíèé óðàâíåíèé Íàâüå � Ñòîêñà /Ñ.À. Çàãðåáèíà // Âåñòí. ÌàÃÓ. Ñåðèÿ ≪Ìàòåìàòèêà≫. � Ìàãíèòîãîðñê, 2005. � Âûï. 8.� Ñ. 74�86.

42. Ñâèðèäþê, Ã.À. Îá îäíîé ìîäåëè äèíàìèêè ñëàáîñæèìàåìîé âÿçêîóïðóãîé æèäêîñòè /Ã.À. Ñâèðèäþê // Èçâ. âóçîâ. Ìàòåìàòèêà. � 1994. � 1. � C. 62�70.

43. Ñâèðèäþê, Ã.À. Îá îòíîñèòåëüíî ñèëüíîé p-ñåêòîðèàëüíîñòè ëèíåéíûõ îïåðàòîðîâ /Ã.À. Ñâèðèäþê, Ã.À. Êóçíåöîâ // Äîêë. Àêàä. íàóê. � 1999. � Ò. 365, 6. � Ñ. 736�738.

44. Ñâèðèäþê, Ã.À. Óðàâíåíèÿ Áàðåíáëàòòà � Æåëòîâà � Êî÷èíîé íà ãðàôå / Ã.À. Ñâèðè-äþê, Â.Â. Øåìåòîâà // Âåñòíèê ÌàÃÓ. Ñåðèÿ ≪Ìàòåìàòèêà≫. � Ìàãíèòîãîðñê, 2003. �Âûï. 4. � Ñ. 129�139.

Ñîôüÿ Àëåêñàíäðîâíà Çàãðåáèíà, êàíäèäàò ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, äîöåíò, êà-ôåäðà ≪Óðàâíåíèÿ ìàòåìàòè÷åñêîé ôèçèêè≫, Þæíî-Óðàëüñêèé ãîñóäàðñòâåííûé óíèâåð-ñèòåò (ã. ×åëÿáèíñê, Ðîññèéñêàÿ Ôåäåðàöèÿ), [email protected].

Bulletin of the South Ural State University.

Series ≪Mathematical Modelling, Programming & Computer Software≫,

2013, vol. 6, no. 2, pp. 5�24.

MSC 35K70, 60H30

The Initial-Finite Problems for Nonclassical Models ofMathematical Physics

S.A. Zagrebina, South Ural State University, Chelyabinsk, Russian Federation,[email protected]

2013, òîì 6, 2 21


The models of Mathematical Physics, whose representation in the form of equations

or systems of partial di�erential equations do not �t one of the classical types such as

elliptic, parabolic or hyperbolic, are called nonclassical. The article provides an overview of

the author's results in the �eld of nonclassical models of Mathematical Physics for which

the initial-�nite problems, generalizing the Cauchy and Showalter, Sidorov conditions,

are considered. Basic method for the research is the Sviridyuk relative spectrum theory.

Abstract results are illustrated by the speci�c initial-�nite problems for the equations and

systems of equations in partial derivatives occurring in applications, namely, the theory

of �ltration, �uid dynamics and mesoscopic theory, considered on the sets of di�erent

geometrical structure.

Keywords: nonclassical models of Mathematical Physics, Plotnikov model, the Navier

� Stokes system, the Barenblatt � Zheltov � Kochina equation, the (multipoint) initial-�nite

problems, the relative spectrum.

References

1. Plotnikov P.I., Starovoitov V.N. The Stefan Problem with Surface Tension as a Limit of thePhase Field Model. Di�erential Equations, 1993, vol. 29, no. 3, pp. 395�405.

2. Plotnikov P.I., Klepacheva A.V. The Phase Field Equations and Gradient Flows of MarginalFunctions. Siberian Mathematical Journal, 2001, vol. 42, no. 3, pp.551�567.

3. Ladyzhenskaya O.A. Mathematical Problems in the Dynamics of a Viscous IncompressibleFluid. Moscow, Nauka, 1970. (in Russian)

4. Temam R. Navier�Stokes Equations. Theory and Numerical Analysis. Amsterdam, N.-Y.,Oxford, North Holland Publ. Co., 1979.

5. Barenblatt G. I., Zheltov Yu. P., Kochina I. N. Basic Concepts in the Theory of Seepage ofHomogeneous Fluids in Fissurized Rocks. J. Applied Mathematics and Mechanics (PMM),1960, vol. 24, no. 5, pp. 1286�1303.

6. Rutkas A.G. The Cauchy Problem for the Equations Ax′(t) + Bx(t) = f(t). Di�erentialEquations, 1975, vol. 11, no. 11, pp. 1996�2010. (in Russian).

7. Ting T.W. Certain Non-Steady Flows of Second-Order Fluids. Arch. Rat. Mech. Anal., 1963,vol. 14, no. 1, pp. 28�57.

8. Chen P.J., Gurtin M.E. On a Theory of Heat Conduction Involving Two Temperatures.Z. Angew. Math. Phys., 1968, vol. 19, pp. 614�627.

9. Hallaire M. On a Theory of Moisture-Transfer. Inst. Rech. Agronom., 1964, no. 3, pp. 60�72.

10. Oskolkov A.P. Nonlocal Problems for Some Class Nonlinear Operator Equations Arising inthe Theory Sobolev Type Equations. Zap. Nauchn. Sem. LOMI. Problems in the theoryof representations of algebras and groups. Part 2, Nauka, St. Petersburg, 1991, vol. 198,pp. 31�48. (in Russian).

11. Sviridyuk G.A., Ankudinov A.V. The Phase Space of the Cauchy � Dirichlet Problem for aNonclassical Equation. Di�erential Equations, 2003, vol. 39, no. 11, pp. 1639�1644.

12. Poincaré H. Sur l'equilibre d'une mass �uide animée d'un mouvement de rotation. Acta Math.,1885, vol. 7, pp. 259�380.

13. Sobolev S.L. On a New Problem of Mathematical Physics. Izv. Akad. Nauk SSSR Ser. Mat.,1954, vol. 18, issue 1, pp. 3�50. (in Russian).

14. Demidenko G.V., Uspenskii S.V. Partial Di�erential Equations and Systems not Solvablewith Respect to the Highest � Order Deriative. New York, Basel, Hong Kong, Marcel Dekker,Inc., 2003.



15. Pankov A.A., Pankova T.E. Nonlinear Evolution Equations with Irreversible OperatorCoe�cient for the Derivative. Dokl. Akad. Nauk Ukraine, 1993, no. 9, pp. 18�20.(in Russian)

16. Pyatkov S.G. Operator Theory. Nonclassical Problems. Utrecht, Boston, K�oln, Tokyo,VSP, 2002.

17. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups ofOperators. Utrecht, Boston, K�oln, Tokyo, VSP, 2003.

18. Zamyshlyaeva A.A. Linear Sobolev Type Equations of High Order. Chelyabinsk, Publ. Centerof the South Ural State University, 2012. (in Russian)

19. Manakova N.A. Optimal Control Problem for the Sobolev Type Equations. Chelyabinsk, Publ.Center of the South Ural State University, 2012. (in Russian)

20. Sagadeeva M.A. Dichotomy of Solutions of Linear Sobolev Type Equations. Chelyabinsk,Publ. Center of the South Ural State University, 2012. (in Russian)

21. Keller A.V. The Algorithm for Solution of the Showalter � Sidorov Problem for LeontiefType Models. Bulletin of the South Ural State University. Series ≪Mathematical Modelling,Programming & Computer Software≫, 2011, no. 4 (241), issue 7, pp. 40�46. (in Russian)

22. Sviridyuk G.A., Brychev S.V. Numerical Solution of Systems of Equations of Leontief Type.Russian Mathematics (Izvestiya VUZ. Matematika), 2003, vol. 47, no. 8, pp. 44�50.

23. Sviridyuk G.A., Burlachko I.V. An Algorithm for Solving the Cauchy Problem forDegenerate Linear Systems of Ordinary Di�erential Equations. Computational Mathematicsand Mathematical Physics, 2003, vol. 43, no. 11, pp. 1613�1619.

24. Shestakov A.L., Sviridyuk G.A. Optimal Measurement of Dynamically Distorted Signals.Bulletin of the South Ural State University. Series ≪Mathematical Modelling, Programming

& Computer Software≫, 2011, no. 17 (234), issue 8, pp. 70�75.

25. Shestakov A.L., Keller A.V., Nazarova E.I. Numerical Solution of the Optimal MeasurementProblem. Automation and Remote Control, 2012, vol. 73, no. 1, pp. 97�104.

26. Sviridyuk G.A., Zagrebina S.A. Verigin's Problem for Linear Equations of the SobolevType with Relatively p-Sectorial Operators. Di�erential Equations, 2002, vol. 38, no. 12,pp. 1745�1752.

27. Zagrebina S.A. On the Showalter � Sidorov Problem. Russian Mathematics (Izvestiya VUZ.Matematika), 2007, vol. 51, no. 3, pp. 19�24.

28. Zagrebina S.A., Solovyeva N.P. The Initial-Finish Problem for the Evolution of Sobolev-TypeEquations on a Graph. Bulletin of the South Ural State University. Series ≪MathematicalModelling, Programming & Computer Software≫, 2008, no. 15 (115), issue 1, pp. 23�26. (inRussian)

29. Manakova N.A., Dylkov A.G. On One Optimal Control Problem with a Penalty Functionalin General Form. Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya≪Fiziko-Matematicheskie Nauki≫, 2011, no. 4 (25), pp. 18�24.(in Russian)

30. Zamyshlyaeva A.A. The Initial-Finish Value Problem for Nonhomogenious Boussinesque �L�ove Equation. Bulletin of the South Ural State University. Series ≪Mathematical Modelling,Programming & Computer Software≫, 2011, no. 37 (254), issue 10, pp. 22�29. (in Russian)

31. Sidorov N.N. A Class of Degenerate Di�erential Equations with Convergence. MathematicalNotes, 1984, vol. 35, no. 4, pp. 300�305.

32. Sviridyuk G.A., Zagrebina S.A. The Showalter � Sidorov Problem as Phenomena of theSobolev-Type Equations. News of Irkutsk State University. Series ≪Mathematics≫, 2010,vol. 3, no. 1, pp. 51�72. (in Russian)

2013, òîì 6, 2 23


33. Zagrebina S.A. The Initial-Finish Problem for Sobolev Type Equations with Strongly (L, p)-Radial Operator. Math. notes of YSU, 2012, vol. 19, no. 2, pp. 39�48. (in Russian)

34. Zagrebina S.A., Sagadeeva M.A. The Generalized Showalter � Sidorov Problem for theSobolev type Equations with strongly (L, p)-radial operator. Vestnik Magnitogorskogogosudarstvennogo universiteta. Seria ≪Matematika≫ [Bulletin of Magnitogorsk StateUniversity. Series ≪Mathematics≫], 2006, issue 9, pp. 17�27. (in Russian)

35. Zagrebina S.A. The Showalter � Sidorov � Verigin Problem for the Linear Sobolev TypeEquations. Neklassicheskie uravneniya matematicheskoy �ziki [Nonclassical MathematicalPhysics Equations], Novosibirsk, 2007, pp. 150�157. (in Russian)

36. Zagrebina S.A. The Initial-Finish Problem for the Navier � Stokes Linear System. Bulletin ofthe South Ural State University. Series ≪Mathematical Modelling, Programming & Computer

Software≫, 2010, no. 4 (221), issue 7, pp. 35�39. (in Russian)

37. Zagrebina S.A. The Multipoint Initial-Finish Problem for Ho� Linear Model. Bulletin ofSouth Ural State University. Series ≪Mathematical Modelling, Programming & Computer

Software≫, 2012, no. 5 (264), issue 11, pp. 4�12. (in Russian)

38. Zagrebina S.A., Konkina A.S. On a New Problem for the Barenblatt � Zheltov � KochinaEquation. Vestnik Magnitogorskogo gosudarstvennogo universiteta. Seria ≪Matematika≫

[Bulletin of Magnitogorsk State University. Series ≪Mathematics≫], 2012, issue 14, pp. 67�77.(in Russian)

39. Fedorov V.E. Degenerate Strongly Continuous Semigroups of Operators. St. PetersburgMathematical Journal, 2001, vol. 12, issue 3, pp. 471�489.

40. Fedorov V.E About Some Relations in the Theory of Degenerate Operator Semigroups.Bulletin of te South Ural State University. Series ≪Mathematical Modelling, Programming

& Computer Software≫, 2008, no. 15 (115), issue 7, pp. 89�99. (in Russian)

41. Zagrebina S.A. On the Existence and Stability of Solutions Navier � Stokes Equations.Vestnik Magnitogorskogo gosudarstvennogo universiteta. Seria ≪Matematika≫ [Bulletin ofMagnitogorsk State University. Series ≪Mathematics≫], 2005, issue 8, pp. 74�86. (in Russian)

42. Sviridyuk G.A. On a Model of the Dynamics of a Weakly Compressible Viscoelastic Fluid.Russian Mathematics (Izvestiya VUZ. Matematika), 1994, vol. 38, no. 1, pp. 59�68.

43. Sviridyuk G.A., Kuznetsov G.A. Relatively Strongly p-Sectorial Linear Operators. DokladyMathematics, 1999, vol. 59, issue 2, pp. 298�300.

44. Sviridyuk G.A., Shemetova V.V. The Barenblatt � Zheltov � Kochina Equations on a Graph.Vestnik Magnitogorskogo gosudarstvennogo universiteta. Seria ≪Matematika≫ [Bulletin ofMagnitogorsk State University. Series ≪Mathematics≫], 2003, issue 4, pp. 129�139. (inRussian)

Ïîñòóïèëà â ðåäàêöèþ 12 ìàðòà 2013 ã.


Documents

South Ural State UniversityÎÁÇÎÐÍÛÅ ÑÒÀÒÜÈ ÓÄÊ 517.9 ÍÀ×ÀËÜÍÎ-ÊÎÍÅ×ÍÛÅ ÇÀÄÀ×È ÄËß ÍÅÊËÀÑÑÈ×ÅÑÊÈÕ ÌÎÄÅËÅÉ ÌÀÒÅÌÀÒÈ×ÅÑÊÎÉ