Geodesic
Decay of the solution of the first mixed problem for a high-order parabolic equation with minor terms
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 4, pp. 113-132 
L. M. Kozhevnikova; F. Kh. Mukminov. Decay of the solution of the first mixed problem for a high-order parabolic equation with minor terms. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 4, pp. 113-132. http://geodesic.mathdoc.fr/item/FPM_2006_12_4_a7/
@article{FPM_2006_12_4_a7,
     author = {L. M. Kozhevnikova and F. Kh. Mukminov},
     title = {Decay of the solution of the first mixed problem for a~high-order parabolic equation with minor terms},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {113--132},
     year = {2006},
     volume = {12},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2006_12_4_a7/}
}
TY  - JOUR
AU  - L. M. Kozhevnikova
AU  - F. Kh. Mukminov
TI  - Decay of the solution of the first mixed problem for a high-order parabolic equation with minor terms
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2006
SP  - 113
EP  - 132
VL  - 12
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/FPM_2006_12_4_a7/
LA  - ru
ID  - FPM_2006_12_4_a7
ER  - 
%0 Journal Article
%A L. M. Kozhevnikova
%A F. Kh. Mukminov
%T Decay of the solution of the first mixed problem for a high-order parabolic equation with minor terms
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2006
%P 113-132
%V 12
%N 4
%U http://geodesic.mathdoc.fr/item/FPM_2006_12_4_a7/
%G ru
%F FPM_2006_12_4_a7

Voir la notice de l'article provenant de la source Math-Net.Ru

In a cylindric domain $D=(0,\infty)\times\Omega$, where $\Omega\subset \mathbb{R}_{n+1}$ is an unbounded domain, the first mixed problem for a high-order parabolic equation \begin{gather*} u_t+(-1)^kD_x^k(a(x,\mathbf{y})D_x^ku)+\sum\limits_{i=l}^m\sum\limits_{|\alpha|=|\beta|=i}(-1)^i D_\mathbf{y}^{\alpha}(b_{\alpha\beta}(x,\mathbf{y})D_{\mathbf{y}}^{\beta}u)=0, \\ l\leq m,\quad k,l,m\in \mathbb{N}, \end{gather*} is considered. The boundary values are homogeneous and the initial value is a finite function. In terms of new geometrical characteristic of domain, the upper estimate of $L_2$-norm $\|u(t)\|$ of the solution to the problem is established. In particular, in domains $\{(x,\mathbf y)\in\mathbb{R}_{n+1}\mid x>0,\ |y_1|$, $0$, under the assumption that the upper an lower symbols of the operator $L$ are separated from zero, this estimate takes the form $$ \|u(t)\|\leq M\exp(-\varkappa_2t^{b})\|\varphi\|,\quad b=\frac{k-la}{k-la+2lak}. $$ This estimate is determined by minor terms of the equation. The sharpness of the estimate for the wide class of unbounded domains is proved in the case $k=l=m=1$.

[1] 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”, Mat. sb., 195:3 (2004), 115–142 | MR | Zbl

[2] Guschin A. K., “Stabilizatsiya reshenii kraevykh zadach dlya parabolicheskikh uravnenii vtorogo poryadka”, Mat. sb., 101:4 (1976), 459–499 | MR | Zbl

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

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

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

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

[7] Mukminov F. Kh., Stabilizatsiya reshenii pervoi smeshannoi zadachi dlya sistemy uravnenii Nave–Stoksa, Avtoref. dis. dokt. fiz.-mat. nauk, M., 1994 | Zbl

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

[9] Khisamutdinova N. A., Stabilizatsiya resheniya smeshannoi zadachi dlya dvumernoi sistemy uravnenii Nave–Stoksa v neogranichennoi oblasti, Dis. kand. fiz.-mat. nauk, 2004

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