Geodesic
A priori estimates of strong solutions of semilinear parabolic equations
Matematičeskie zametki, Tome 64 (1998) no. 4, pp. 564-572 
G. G. Laptev. A priori estimates of strong solutions of semilinear parabolic equations. Matematičeskie zametki, Tome 64 (1998) no. 4, pp. 564-572. http://geodesic.mathdoc.fr/item/MZM_1998_64_4_a8/
@article{MZM_1998_64_4_a8,
     author = {G. G. Laptev},
     title = {A priori estimates of strong solutions of semilinear parabolic equations},
     journal = {Matemati\v{c}eskie zametki},
     pages = {564--572},
     year = {1998},
     volume = {64},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_64_4_a8/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We study an initial boundary value problem for the semilinear parabolic equation $$ \frac{\partial u}{\partial t} +\sum_{|\alpha|\le2b}a_\alpha(x,t)D^\alpha u =f(x,t,u,Du,\dots,D^{2b-1}u), $$ where the left-hand side is a linear uniformly parabolic operator of order $2b$. We prove sufficient growth conditions on the function $f$ with respect to the variables $u,Du,\dots,D^{2b-1}u$, such that the apriori estimate of the norm of the solution in the Sobolev space $W_p^{2b,1}$ is expressible in terms of the low-order norm in the Lebesgue space of integrable functions $L_{l,m}$.

[1] Pokhozhaev S. I., “O razreshimosti kvazilineinykh ellipticheskikh uravnenii proizvolnogo poryadka”, Matem. sb., 117:2 (1982), 251–265 | MR | Zbl

[2] Pokhozhaev S. I., “Ob apriornykh otsenkakh reshenii kvazilineinykh ellipticheskikh uravnenii proizvolnogo poryadka”, Differents. uravneniya, 117:1 (1983), 101–110 | MR

[3] Besov O. V., Ilin V. A., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 | Zbl

[4] Solonnikov V. A., O kraevykh zadachakh dlya lineinykh parabolicheskikh sistem differentsialnykh uravnenii obschego vida, Tr. MIAN, 83, Nauka, M., 1965 | MR

[5] Wahl von W., “Klassische Lösbarkeit im Großen für nichtlineare parabolische Systeme und das Verhalten der Lösungen für $t\to\infty$”, Nachr. Acad. Wiss. Göttingen Math.-Phys. Kl. II, 1981, no. 5, 131–177 | MR | Zbl

[6] Wahl von W., “Extention of a result of Ladyženskaja and Ural'ceva concerning second-order parabolic equations of arbitrary order”, Ann. Pol. Math., 41:1 (1983), 63–72 | MR | Zbl

[7] Besov O. V., “Integralnye predstavleniya funktsii i teoremy vlozheniya dlya oblasti s usloviem gibkogo roga”, Tr. MIAN, 170, Nauka, M., 1984, 12–30 | MR | Zbl

[8] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1976