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Åñëè â íà÷àëå òåïëî áûëî ðàñïðåäåëåíî íåêîòîðîé ôóíêöèåé, òî êàæäàÿ òî÷êà"êàê áû"äà¼ò âêëàä â îáùåå âëèÿíèå.E(t, 0) → +∞,ïðèt → +0E(t, x) → 0, ïðè t → +∞ (x , 0)∫E(t, x)dx = 1;E(t, x) → δ(x),ïðèt → +0δ(x)- äåëüòà ôóíêöèÿ ÄèðàêàR1Òåîðåìà 4.1: Ïóñòü â çàäà÷å Êîøè:{ut = a2 uxx , t > 0, x ∈ R1ut=0 = u0 (x)25u0 (x) ∈ C(R1 ) & |u0 (x)| 6 M0 ∀x ∈ R1 . Òîãäà ôóíêöèÿ∫ −(x−ξ)21u(t, x) = √e 4a2 t u0 (ξ)dξ, t > 0(∗)24πa t 1R1)u(t, x) ∈ C({t > 0, x ∈ R1 }) ∩ C∞ ({t > 0, x ∈ R1 })2)ÿâëÿåòñÿ êëàññè÷åñêèì ðåøåíèåì çàäà÷è Êîøè3)|u(t, x)|6 M0 , ∀t > 0, x ∈ R1Äîêàçàòåëüñòâî:u(t, x) = √Ïóñòü1I= √π∫14πa2 t 1Ry−x=η:√∫4a2 te−(x−y)24a2 tu0 (y)dy = I√2e−η u0 (x + 2a tη)dηR1√u0 (x + 2a tη) ∈ C({t > 0, x ∈ R1 , η ∈ R1 })√22|eη u0 (x + 2a tη)| 6 M0 e−ηÍåñîáñòâåííûé èíòåãðàë, çàâèñÿùèé îò ïàðàìåòðà.
Íàì áû õîòåëîñü, ÷òîáû îí ñõîäèëñÿ ðàâíîìåðíî. Íåïðåðûâíîñòü î÷åâèäíà. Ðàññìîòðèì îáëàñòüÍàQQâ âèäå ïðÿìîóãîëüíèêà|x| < A; 0 < t < Tèíòåãðàë ñõîäèòñÿ ðàâíîìåðíî.∀t > 0, ∀x ∈ R1 : u(t, x) íåïðåðûâíà, òàê êàê I - íåïðåðûâíàu(t, x) ∈ C({t > 0, x ∈ R1 )∫∫√1122|u(t, x)| 6 √e−η |u0 (x + 2a tη)|dη 6 √e−η dη = M0ππè äîîïðåäåëèì ïî íåïðåðûâíîñòè â 0:R1Ïðîâåðèì, ñóùåñòâóåò ëè ux (t, x)∫(x−y)21−ux (t, x) v√ e 4a2 t · (y − x)u0 (y)dy4a3 πR1Ïóñòü ïîäûíòåãðàëüíîå âûðàæåíèå ðàâíîλ. λ(t, x, η) ∈ C({t > 0, x ∈ R1 , η ∈ R1 })R1Íà ýòîò ðàç ðàññìîòðèì ïðÿìîóãîëüíóþ îáëàñòüôèêñèðîâàííûõt0 , T, AQ : |x| < A, 0 < t0 < t < TÄîêàæåì, ÷òî ïðè- ïîäûíòåãðàëüíîå âûðàæåíèå ñõîäèòñÿ ðàâíîìåðíî. Òåì ñàìûì ìû äîêàæåì,1÷òî îí ñõîäèòñÿ ðàâíîìåðíî ïðè t > 0, x ∈ R .Ñôîðìóëèðóåì î÷åâèäíûå óòâåðæäåíèÿ:22+ b > 2ab6 A : |y − x| > |y| − |x| > |y| − Aâ)∀|x| 6 A : |y − x| 6 |y| + |x| 6 |y| + Aà)aá)∀|x|y2|y| √y2− 2A2 =− A2|y| > A : (x − y)2 = |y − x|2 > (|y| − A)2 = |y|2 + A2 − 2 √2A > y2 + A2 −2221 2Ïðè |y| 6 A : (x − y) > − A2 1 22 2 y − A , |y| > A2(x − y) > φA (y), ãäå φA (y) = 1 − A2 , |y| 6 A2−φ(y)−(x−y)21M2Òîãäà λ 6 √ e 4a2 T (|y| + A) = Ψ(y)√ e 4a t (y − x)u0 (x) 6 3 √√ 4a3 πt t 4a πt0 t0∫∫+∞2 2Ψ(y)dy ve−β y (|y| + A)dy - ñõîäèòñÿÏðèR1−∞26Çíà÷èòñòâóåò â∀λ(x, y, t) â Q ñõîäèòñÿ àáñîëþòíî è ðàâíîìåðíî ⇒ èç ìàòåìàòè÷åñêîãî àíàëèçà ux (t, x)Q.
Áåðÿ â êà÷åñòâå Q âñåâîçìîæíûå äîïóñòèìûå ïðèìîóãîëüíèêè, èìååì:ñóùå-òî÷êåux (t, x) ∈ C({t > 0, x ∈ R1 })Àáñîëþòíî àíàëîãè÷íî äîêàçûâàåòñÿ, ÷òîu(t, x) ÿâëÿåòñÿ áåñêîíå÷íî äèôôåðåíöèðóåìîé ôóíêöèåéíà äàííîé îáëàñòè.Ïðîâåðèì, ÿâëÿåòñÿ ëè u(t, x) ðåøåíèåì óðàâíåíèÿ òåïëîïðîâîäíîñòè.−(x−y)21e 4a2 tE(t, x − y) = √4πa2 t1 (x − y) −(x−y)Ex (t, x − y) = − √ · 3 3/2 · e 4a2 t2 π 2a t[]2(x − y)2 −(x−y)−114a2 t+eExx (t, x − y) = √4a5 t5/22 π 2a3 t3/2[]2(x − y)2 −(x−y)1−124atEt (t, x − y) = √+e2a3 t3/24a3 t3/2∫2 πut − a2 uxx = [Et (t, x − y) − a2 Exx (t, x − y)]u0 (y)dy = 0|{z}R1≡0∀t > 0, x ∈ R12Òåïåðü ïðîâåðèì íà÷àëüíûåäàííûå:∫ut=01= u(0, x) = √πe−η2R1√1u0 (x + 2a tη)dη = √ u0 (x)π| {z }0∫e−η2dη= u0 (x)R1Äîêàçàòåëüñòâî çàâåðøåíî.Çàäà÷à Êîøè äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè â{ut = a2 ∆x u, t > 0, x = (x1 , . .
. , xn ) ∈ Rnut=0 = u0 (x), x ∈ RnÑíà÷àëà ðàññìîòðèì ÷àñòíûé ñëó÷àé:Rn(∗)u0 (x) = φ1 (x1 ), . . . , φn (xn ),φk (xk ) ∈ C(R1 );|φk (xk )| 6 M0{21ukt (t, xk ) = a uxk xk (t, xk ), t > 0, xk ∈ Ruk t=0 = φk (xk ), xk ∈ R1∫ −(x −y )2kk1ku (t, xk ) = √e 4a2 t φk (yk )dyk4πa2 t 1RÄîêàæåì, ÷òî1)u(t, x)=n∏=k = 1, nuk (t, xk )k=1u (t, xk ) ∈ C({t > 0, x ∈ R1 }) ∩ C∞ ({t > 0, x ∈ R1 })k=12)ut (t, x)u(t, x) =n∏ñåðèÿ çàäà÷ Êîøè,kn(∏-î÷åâèäíînn [nn [])] ∏∏∑∏jjut (t, x j )·a2 ux j x j (t, x j )·uk (t, xk ) =uk (t, xk ) =uk (t, xk ) =tk=1j=1j=1k=1,k,jnn( n)∑∑∂2 ∏ k∂22= a2u(t,x)=a[u(x, t)] = a2 ∆x u(t, x)k22∂x∂xj k=1jj=1j=1nn∏∏u(0, x) =uk (0, xk ) =φl (xk )k=1,k, jk=1k=1Äîêàçàòåëüñòâî çàâåðøåíî.Òî åñòü â ñëó÷àå ðàçäåëåíèÿ ïåðåìåííûõ,ìû èìååì:(1u(t, x) = √4πa2 tx, y ∈ Rn)n (R1R1−(x1 −y1 )2 +...+(xn −yn )24a2 te(1φ1 (y1 ) .
. . φn (yn )· dy1 . . . dyn = √4πa2 t27)n ∫eRn|x−y|24a2 tu0 (y)dyÝòî òàê íàçûâàåìûé n-ìåðíûé èíòåãðàë Ïóàññîíà.Ïðèâåä¼ì ñëåäóþùèé ôàêò áåç äîêàçàòåëüñòâà:Ëþáàÿ íåïðåðûâíàÿ ôóíêöèÿ ìîæåò áûòü ïðåäñòàâëåíà ñëåäóþùèì îáðàçîì:∑φα1 (x1 ) . . . φαn (xn )αÒåîðåìà 4.2: Ïóñòü â n-ìåðíîé çàäà÷å Êîøè u0 (x) ∈()n ∫ |x−y|21u(t, x) = √e 4a2 t u0 (y)dy:4πa2 tRnn∏1)u(t, x) =uk (t, xk ) ∈ C({t > 0, x ∈ R1 }) ∩ C∞ ({t > 0, xC(Rn ),è|u0 (x)| 6 M0 ∀x ∈ Rn .Òîãäà ôóíêöèÿ:∈ R1 })k=12)ßâëÿåòñÿ êëàññè÷åñêèì ðåøåíèå çàäà÷è Êîøè (*)n3)|u(t, x)| 6 M0 , ∀t > 0, x ∈ RÏî ñóòè ìû óæå âñ¼ äîêàçàëè. Íî ïîêà ìû íè÷åãî íå ìîæåì ñêàçàòü ïðî åäèíñòâåííîñòü äàííîéçàäà÷è Êîøè.
È âîîáùå ãîâîðÿ îíî íå áóäåò åäèíñòâåííûì.Çàäà÷à Êîøè äëÿ íåîäíîðîäíîãî óðàâíåíèÿ òåïëîïðîâîäíîñòè.{ut = a2 ∆x u + f (t, x),ut=0 = 0, x ∈ Rnt > 0, x ∈ Rn(∗∗)αà)Dxf (t, x) ∈ ({t > 0, x ∈ Rn }) ∀α, |α| 6 2á)| f (t, x)| 6 M1∀t > 0, x ∈ Rnαâ)|Dx f (t, x)| 6 M1∀t > 0, x ∈ Rn ,∀α, |α| 6 2(τ > 0)Èñïîëüçóåì ìåòîä Äþàìåëÿ.{vt(t, x, τ) = a2 ∆x v(t, x, τ),ut=τ = f (τ, x), x ∈ Rn()n ∫ |x−y|212v(t, x, τ) = √e 4a (t−τ) f (τ, y)dy24πa (t − τ)nt > τ, x ∈ RnRÀíàëîãè÷íî ìåòîäó Äþàìåëÿ, êîòîðûé ìû ðàññìàòðèâàëè â âîëíîâîì óðàâíåíèè, òóò ìû òîæå ðàññìîòðåëè ñåðèþ çàäà÷.|v(t, x, τ)| 6 M1Äîêàæåì, ÷òî∫tu(t, x) =v(t, x, τ)dτ0Äëÿ ýòîãî èçó÷èì ñâîéñòâ ñëåäóþùåé ôóíêöèè:(W(t̃, x, τ) = √14πa2 t̃)n ∫e|x−y|24a2 t̃t̃ > 0, x ∈ Rn , τ > 0, y ∈ Rnf (τ, y)dyRn{Wt = a2 ∆x WW t̃=0 = f (τ, x)τ è îãðàíè÷åíèå íà u0√√y−xnÑäåëàåì çàìåíó:√ = η : y = x + 2a t̃η; dy = (2a t̃η) dη (ýëåìåíò îáú¼ìà)∫ 2a t̃√12W(t̃, x, η) = n/2e−η f (τ, x + 2a t̃η)dη, ïðè t̃ > 0π2 îòëè÷èÿ îò îáû÷íîé çàäà÷è òåïëîïðîâîäíîñòè: íàëè÷èåRnÄîêàæåì, ÷òî ýòîò èíòåãðàë ñõîäèòñÿ ðàâíîìåðíî.√nÇàìåòèì,÷òî f (τ, x + 2a t̃η) ∈ C({τ > 0, t̃ > 0, x ∈ R , η ∈√2| f (τ, x + 2a t̃η)| 6 M1 e−η ⇒Rn })∫Êðîìå òîãî:e−η dη2òàê êàê- ñõîäèòñÿ, òî ðàññìàòðèâàåìûé èíòåãðàë ñõîäèòñÿRnàáñîëþòíî è ðàâíîìåðíî, çíà÷èò:28W(t̃, x, τ) ∈ C({τ > 0, t̃ > 0, x ∈ Rn }) îäíîðîäíîé çàäà÷å òåïëîïðîâîäíîñòè ïðîèçâîäíûå íåëüçÿ áûëî ïðîäîëæèòü äî0.Ïîñìîòðèì,÷òî íàáëþäàåòñÿ∫ â íàøåì ñëó÷àå.√2e−η fxi (t, x + 2a t̃η)dηWx1 (t̃, x, η) vRnÎïÿòü ïîëó÷èëñÿ íåñîáñòâåííûé èíòåãðàë,çàâèñÿùèé îò ïàðàìåòðà.
Ïîäûíòåãðàëüíàÿ ôóíêöèÿíåïðåðûâíà âïëîòü äî ãèïåðïëîñêîñòè−η|e2√2fxi (t, x + 2a t̃η)| 6 Me−η ,t=0òî åñòü∫Wx1 (t̃, x, η) =√2e−η fxi (τ, x + 2a t̃η)dηRnÀíàëîãè÷íî äîêàçûâàåòñÿ, ÷òî:∫√2e−η fxi xi (τ, x + 2a t̃η)dηWx1 x1 (t̃, x, η) =RnÒî åñòüW(t̃, x, η) =1πn/2∫√2e−η f (τ, x + 2a t̃η)dηRnèçâîäíûìè 2 ïîðÿäêà ïî x âïëîòü äîÇàìåòèì, ÷òî ïðîèçâîäíóþ ïît- íåïðåðûâíà, è å¼ ìîæíî ïðîäîëæèòü âìåñòå ñ ïðî-t=0òîæå ìîæíî ïðîäîëæèòü âïëîòü äîïðàâóþ ÷àñòü ìîæíî ïðîäîëæèòü âïëîòü äîv(t, x, τ) = W(t − τ, x, τ) ⇒v(t, x, τ) íåïðåðûâíà ñàìà, è áîëåå òîãî:v, vt , vx1 , vx1 x1 ∈ C({τ > 0, t > τ, x ∈ Rn })∫tu(t, x) =v(t, x, τ)dτ áóäåò íåïðåðûâíîé,t = 0.t = 0,òàê êàêWt = a2 ∆x W ,àè áîëåå òîãî:0u, ut , ux1 , ux1 x1 ∈ C({t > 0, x ∈ Rn })Çíàê ñòðîãîãî íåðàâíåñòâî ïîêàçûâàåò, ÷òî ðåøåíèå áóäåò êëàñ-ñè÷åñêèì.Òåïåðü ïîêàæåì, ÷òî òàêàÿ ôóíêöèÿ óäîâëåòâîðÿåò ãðàíè÷íûì äàííûì:∫t∫t∫tut = v(t, x, t)+vt (t, x, τ)dτ = f (t, x)+0a2 ∆x v(t, x, τ)dτ = f (t, x)+a2 ∆x0Îïðåäåëåíèå: ÏóñòüQ- îáëàñòü âRn+1 ,v(t, x, τ)dτ = f (t, x)+a2 ∆x u(t, x)0ïåðìåííûõ(t, x1 , .
. . , xn ),àQ̂- ìíîæåñòâî, ïîëó÷åííîå èçQïóò¼ì äîáàâëåíèÿ ê íåìó íåêîòîðîãî êîëè÷åñòâà ãðàíè÷íûõ òî÷åê.C1,2t,x - ïîäïðîñòðàíñòâî ôóíêöèé èç C(Q), è òàêèõ,÷òî1)ut , ux1 , ux1 x1 ∈ C(Q)2) ýòè ïðîèçâîäíûå äîïóñêàþò íåïðåðûâíîå ïðîäîëæåíèå íàÀíàëîãè÷íî îïðåäåëÿþòñÿQ̂0,1C0,2t,x , Ct,xÒåîðåìà 4.3: Ïóñòü çàäà÷à Êîøè (*){ut = a2 ∆x u + f (t, x),t > 0, x ∈ Rnnu t=0 = u0 (x), x ∈ R(∗)∈ C(Rn ) & |u0 (x)| 6 M0 , ∀x ∈ Rn0,2ná) f (t, x) ∈ Ct,x ({t > 0, x ∈ R }) & | f (t, x)| 6 M1 , | fx1 (t, x)| 6 M1 , | fx1 x1 (t, x)| 6 M1 ;à)u0 (x)∀t > 0, x ∈ Rnòîãäà:(1u(t, x) = √4πa2 t1)u(t, x))∫−|x−y|2e 4a2 t∫t [(u0 (y)dy +Rn0)∫1√4πa2 tn∈ C({t > 0, x ∈ Rn }) ∩ C1,2t,x ({t > 0, x ∈ R })−|x−y|2e 4a2 t6 M0 + tM1 ,f (τ, y)dy dτRn2)u(t, x) ÿâëÿåòñÿ êëàññè÷åñêèì ðåøåíèåì çàäà÷è Êîøè3)|u(t, x)|]∀t > 0, x ∈ RnÏðèìå÷àíèå: â ïóíêòå á) íà ñàìîì äåëå ìîæåò áûòü(∗)f (t, x) ∈ C0,1t,x ,íî äîêàçàòåëüñòâî áóäåò ñëîæíåå.Ïåðâûé äâà ïóíêòà î÷åâèäíî äîêàçûâàþòñÿ ÷åðåç ïðèíöèï ñóïåðïîçèöèè è ñâåäåíèÿ, ñôîðìóëèðîâàííûå ðàíåå.
Äîêàæåì òîëüêî òðåòèé ïóíêò. Äåéñòâèòåëüíî:29∫t|u(t, x)| 6 |u1 (t, x)| + |u2 (t, x)| 6 M0 + M1dτ = M0 + tM10Åäèíñòâåíîñòü.Ñôîðìóëèðóåì ïðèíöèï ìàêñèìóìà äëÿ ïàðàáîëè÷åñêèõ óðàâíåíèé:Ω - îãðàíè÷åííàÿ îáëàñòü â Rn ; ∂Ω - å¼ ãðàíèöà, T > 0 -÷èñëîQT = (0, T) × Ω - öèëèíäðè÷åñêàÿ îáëàñòü â Rn+1ÏóñòüÏàðàáîëè÷åñêàÿ ãðàíèöà îáëàñòè:Ωτ- ïåðåñå÷åíèå öèëèíäðàΩ0- íèæíåå îñíîâàíèå,(−∞, +∞) × Ω è t = τ - ñå÷åíèåΩT - âåðõíåå îñíîâàíèå.öèëèíäðà.ΓT = Ω0 ∪ {[0, T] × ∂Ω} = ∂Qt \ΩTÏàðàáîëè÷åñêèé îïåðàòîð:∈ C1,2t,x (QT )22)Lu = ut − a ∆x u1)u(t, x)Òåîðåìà 4.4(ïðèíöèï ìàêñèìóìà):ÏóñòüÒîãäà1,2u(t, x) ∈ Ct,x(QT ) ∩ C(QT ); (0 < T < +∞); Lu(t, x) 6 0, ∀(t, x) ∈ QTmax u(t, x) äîñòèãàåòñÿ íà ïàðàáîëè÷åñêîé ãðàíèöå ΓT îáëàñòè QT(t,x)∈QT0 < δ < T; QT−δ ; ΓT−δM = max u(t, x); m = max u(t, x)Âîçüì¼ìÏóñòü(t,x)∈ΓT−δ(t,x)∈QT−δ(îáà ñåìåéñòâà ìíîæåñòâ, ïî êîòîðûì ìû áåð¼ì ìàêñèìóì çàìêíóòû)Òîãäà óñëîâèå òåîðåìû ìîæíî òðàêòîâàòü êàêm>M< M,∃(t , x ) : u(t , x ) = M, (t , x1 ) < ïàðàáîëè÷åñêîéÏðåäïîëîæèì ïðîòèâíîå: m1 11 11òîãäàãðàíèöåut (t1 , x1 ) = 0, òàê êàê ýòî òî÷êà ëîêàëüíîãî ìàêñèìóìà.ux1 x1 (t1 , x1 ) 6 0 èç êàêèõ-òî(?!?!?!?) ñîîáðàæåíèé.Lu(t1 , x1 ) > 0 Òîãäà ïîñòðîèì íîâóþ ôóíêöèþ:n∑vβ (t, x) = u(t, x) + β|x − x1 |2 = u(t, x) + β(xk − x1k )2M−mβ=,2d2k=1ãäåd- äèàìåòðQT , β > 0Òîãäà íà ïàðàáîëè÷åñêîé ãðàíèöå äëÿ öèëèíäðà âûñîòîéM−m 2 M+mvβ (t, x) = u(t, x) + β|x − x | 6 m +d =<M22d21 11 1vβ (t , x ) = u(t , x ) + 0 = M∃(t2 , x2 ), â êîòîðîé vβ (t, x) äîñòèãàåò ìàêñèìóìà.Lvβ (t2 , x2 ) > 0Lvβ (t2 , x2 ) = Lu(t2 , x2 ) − 2na2 β > 0Lu(t2 , x2 ) > 2na2 β > 0 - ïðîòèâîðå÷èå.T−δáóäåò âûïîëíÿòüñÿ:12Ïóñòüm∗ = max u(t, x)(t,x)∈ΓTu(t, x) 6 max u(t, x) 6 max u(t, x) = m∗(t,x)∈ΓT(t,x)∈ΓT−δÒåïåðü, ÷òî êàñàåòñÿ âåðõíèõ òî÷åê:u(t, x) 6 m∗∀(t, x) ∈ QT \ΩT∀t, x ∈ QT−δ , ∀δ > 0(ïåðåøëè ê ïðåäåëó ïðèδ → 0)Äîêàçàòåëüñòâî çàâåðøåíî.Ñëåäñòâèå èç òåîðåìû 4.4: Ïóñòü∈∩ C(Q)2)ut − a ∆u = 0, ∀t, x ∈ QT1)u(t, x)2u(t, x):C1,2t,x (QT )Òîãäàmaxèmin u(t, x)äîñòèãàþòñÿ íà ïàðàáîëè÷åñêîé ãðàíèöå îáëàñòèÄîêàçàòåëüñòâî ðàçîáü¼ì íà äâå ÷àñòè:: Lu 6 0(Lu = 0 6 0) ⇒ ïî: v(t, x) = −u(t, x)v(t, x) ñïðàâåäëèâ ïðèíöèï1)maxòåîðåìå 4.4 ïîëó÷àåì òðåáóåìîå2)minÄëÿìàêñèìóìà:∃(t∗ , x∗ ) ∈30ïàðàáîëè÷åñêîé ãðàíèöåQt :QT∀t, x ∈ QT v(t, x) 6 v(t∗ , x∗ ):−u(t, x) = v(t, x) 6 v(t∗ , x∗ ) = −u(t∗ , x∗ ) :u(t, x) > u(t∗ , x∗ ) ∀t, x ∈ QTÒîëüêî èíòåãðàë Ïóàññîíà íåïðåðûâíî çàâèñèò îò âõîäíûõ äàííûõ.
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