Geodesic
Stabilization of a solution of the first mixed problem for a quasi-elliptic evolution equation
Sbornik. Mathematics, Tome 196 (2005) no. 7, pp. 999-1032 
L. M. Kozhevnikova. Stabilization of a solution of the first mixed problem for a quasi-elliptic evolution equation. Sbornik. Mathematics, Tome 196 (2005) no. 7, pp. 999-1032. http://geodesic.mathdoc.fr/item/SM_2005_196_7_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In a cylindrical domain $D=(0,\infty)\times\Omega$, where $\Omega$ is an unbounded subdomain of $\mathbb R_{n+1}$, one considers the first mixed problem for a higher order equation \begin{gather*} u_t+Lu=0, \\ Lu\equiv\sum_{i=q}^k(-1)^iD_x^i(a_i(x,{\mathbf y})D_x^iu)+ \sum_{i=l}^m\,\sum_{|\alpha|=|\beta|=i}(-1)^i D_{\mathbf y}^\alpha(b_{\alpha\beta}(x,{\mathbf y})D_{\mathbf y}^\beta u), \\ q\leqslant k,\quad l\leqslant m,\quad q,k,l,m\in\mathbb N,\quad x\in\mathbb R,\quad \mathbf y\in\mathbb R_n, \end{gather*} with homogeneous boundary conditions and compactly supported initial function. A new method of obtaining an upper estimate of the $L_2$-norm $\|u(t)\|$ of the solution of this problem is put forward, which works in a broad class of domains and equations. In particular, in domains $\{(x,{\mathbf y})\in\mathbb R_{n+1}:|y_1|, $0, for the operator $L$ with symbol satisfying a certain condition this estimate takes the following form: $$ \|u(t)\|\leqslant M\exp(-\kappa_2t^b)\|\varphi\|,\qquad b=\frac{q-{la}}{q-{la}+2laq}\,. $$ The estimate is shown to be sharp in a broad class of unbounded domains for $q=k=l=m=1$, that is, for second-order parabolic equations.

[1] Kondratev V. A., Eidelman S. D., “O svoistvakh reshenii lineinykh evolyutsionnykh sistem s ellipticheskoi prostranstvennoi chastyu”, Matem. sb., 81(123) (1970), 398–429 | MR | Zbl

[2] Guschin A. K., “Ob otsenkakh reshenii kraevykh zadach dlya parabolicheskogo uravneniya vtorogo poryadka”, Trudy MIAN, 126, 1973, 5–45 | Zbl

[3] Guschin A. K., “Stabilizatsiya reshenii vtoroi kraevoi zadachi dlya parabolicheskogo uravneniya vtorogo poryadka”, Matem. sb., 101 (143) (1976), 459–499 | Zbl

[4] Ushakov V. I., “Stabilizatsiya reshenii tretei smeshannoi zadachi dlya parabolicheskogo uravneniya v netsilindricheskoi oblasti”, Matem. cb., 111(153) (1980), 95–115 | MR | Zbl

[5] Lezhnev A. V., “O povedenii pri bolshikh znacheniyakh vremeni neotritsatelnykh reshenii vtoroi smeshannoi zadachi dlya parabolicheskogo uravneniya”, Matem. sb., 129:2 (1986), 186–200 | MR

[6] Mukminov F. Kh., Stabilizatsiya reshenii pervoi smeshannoi zadachi dlya sistemy uravnenii Nave–Stoksa, Dis. ... dokt. fiz.-matem. nauk, MIRAN, M., 1994

[7] Mukminov F. Kh., “Ob ubyvanii normy resheniya smeshannoi zadachi dlya parabolicheskogo uravneniya vysokogo poryadka”, Differents. uravneniya, 23:10 (1987), 1172–1180 | MR

[8] Mukminov F. Kh., “Stabilizatsiya reshenii pervoi smeshannoi zadachi dlya parabolicheskogo uravneniya vtorogo poryadka”, Matem. sb., 111 (153):4 (1980), 503–521 | MR | Zbl

[9] Tedeev A. F., “Stabilizatsiya reshenii pervoi smeshannoi zadachi dlya kvazilineinogo parabolicheskogo uravneniya vysokogo poryadka”, Differents. uravneniya, 25:3 (1989), 491–498 | MR | Zbl

[10] Bikkulov I. M., Mukminov F. Kh., “O stabilizatsii normy resheniya odnoi smeshannoi zadachi dlya parabolicheskikh uravnenii 4-go i 6-go poryadkov v neogranichennoi oblasti”, Matem. sb., 195:3 (2004), 115–142 | MR | Zbl

[11] Kozhevnikova L. M., Mukminov F. Kh., “Otsenki skorosti stabilizatsii pri $t\to\infty$ resheniya pervoi smeshannoi zadachi dlya kvazilineinoi sistemy parabolicheskikh uravnenii vtorogo poryadka”, Matem. sb., 191:2 (2000), 91–131 | MR | Zbl

[12] Khisamutdinova N. A., “Stabilizatsiya resheniya dvumernoi sistemy uravnenii Nave–Stoksa v neogranichennoi oblasti s neskolkimi vykhodami na beskonechnost”, Matem. sb., 194:3 (2003), 83–114 | MR | Zbl

[13] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[14] Moser J. A., “Harnack inequality for parabolic differential equations”, Comm. Pure Appl. Math., 17:1 (1964), 101–134 | DOI | MR | Zbl