Geodesic
Critical Exponents for Nondiagonal Quasilinear Parabolic Systems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Studies on function theory and differential equations, Tome 248 (2005), pp. 46-51.

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

Using the method of nonlinear capacity (integral relations), critical exponents for the nonexistence of nontrivial nonnegative solutions are found for the Cauchy problem for systems of inequalities of the form $(u_i)_t-\operatorname {div} A_i(t,x,u_i,\nabla u_i)\ge b_i\prod _{j=1}^2 u_j^{Q_{ij}}+f_i$, where $Q_{ij}\ge 0$, $u_i=u_i(t,x)\ge 0$, $b_i=b_i(t,x)\ge 0$, and $f_i=f_i(t,x)\ge 0$, $x\in \mathbb R^N$, $t\ge 0$. Under additional assumptions concerning the functions $A_i$, a priori estimates and estimates of the lifespan of solutions are obtained in terms of the behavior of initial data and the functions $f_i$. 
@article{TM_2005_248_a4,
     author = {K. O. Besov},
     title = {Critical {Exponents} for {Nondiagonal} {Quasilinear} {Parabolic} {Systems}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {46--51},
     publisher = {mathdoc},
     volume = {248},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2005_248_a4/}
}
TY  - JOUR
AU  - K. O. Besov
TI  - Critical Exponents for Nondiagonal Quasilinear Parabolic Systems
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2005
SP  - 46
EP  - 51
VL  - 248
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2005_248_a4/
LA  - ru
ID  - TM_2005_248_a4
ER  - 
%0 Journal Article
%A K. O. Besov
%T Critical Exponents for Nondiagonal Quasilinear Parabolic Systems
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2005
%P 46-51
%V 248
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2005_248_a4/
%G ru
%F TM_2005_248_a4
K. O. Besov. Critical Exponents for Nondiagonal Quasilinear Parabolic Systems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Studies on function theory and differential equations, Tome 248 (2005), pp. 46-51. http://geodesic.mathdoc.fr/item/TM_2005_248_a4/

[1] Besov K.O., “O globalnoi razreshimosti polulineinykh parabolicheskikh sistem so smeshannoi pravoi chastyu”, Tr. MIAN, 243 (2003), 66–86 | MR | Zbl

[2] Deng K., Levine H.A., “The role of critical exponents in blow-up theorems: The sequel”, J. Math. Anal. and Appl., 243 (2000), 85–126 | DOI | MR | Zbl

[3] Escobedo M., Levine H.A., “Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations”, Arch. Rat. Mech. and Anal., 129 (1995), 47–100 | DOI | MR | Zbl

[4] Galaktionov V.A., “Ob usloviyakh nesuschestvovaniya v tselom i lokalizatsii reshenii zadachi Koshi dlya odnogo klassa nelineinykh parabolicheskikh uravnenii”, ZhVMiMF, 23:6 (1983), 1341–1354 | MR

[5] Galaktionov V.A., Kurdyumov S.P., Mikhailov A.P., Samarskii A.A., “O neogranichennykh resheniyakh zadachi Koshi dlya parabolicheskogo uravneniya $u_t=\nabla(u^\sigma\nabla u)+u^\beta$”, DAN SSSR, 252:6 (1980), 1362–1364 | MR | Zbl

[6] Mitidieri E., Pokhozhaev S.I., Apriornye otsenki i otsutstvie reshenii nelineinykh uravnenii i neravenstv v chastnykh proizvodnykh, Tr. MIAN, 234, Nauka, M., 2001 | MR

[7] Pokhozhaev S.I., Tesei A., “O kriticheskikh pokazatelyakh otsutstviya reshenii dlya sistem kvazilineinykh parabolicheskikh neravenstv”, Dif. uravneniya, 37:4 (2001), 521–528 | MR | Zbl

[8] Samarskii A.A., Galaktionov V.A., Kurdyumov S.P., Mikhailov A.P., Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii, Nauka, M., 1987 | MR