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 Trudy Mat. Inst. Steklova, 2003, Volume 243, Pages 66–86 (Mi tm421)

On the Global Solvability of Semilinear Parabolic Systems with Mixed Right-Hand Side

K. O. Besov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: For a system $u_t-\mathcal L_1u\ge b_1(t,x)u^Pv^Q$, $v_t-\mathcal L_2v\ge b_2(t,x)u^Rv^S$, the nonexistence of nontrivial global nonnegative weak solutions in $\mathbb R^{N+1}_+$ is proved under the most general conditions imposed on the nonnegative parameters $P$, $Q$, $R$, and $S$ and on the behavior of the positive functions $b_1$ and $b_2$, as well as for the initial data that sufficiently slowly decrease at infinity. The second-order linear differential operators $\mathcal L_1$ and $\mathcal L_2$ in the above system are of the form $\mathcal L_k=\mathrm {div}[A_k(t,x)\nabla u]$, $k=1,2$, where $A_k$ are measurable matrices such that the corresponding quadratic forms $(A_1\cdot,\cdot )$ and $(A_2\cdot,\cdot )$ are positive semidefinite for all $t$ and $x$. An important feature of such systems with mixed right-hand sides (as compared with the diagonal systems that have been investigated much better) is that the critical exponents essentially depend on whether or not these quadratic forms are equivalent.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2003, 243, 59–79

Bibliographic databases:
UDC: 517.9

Citation: K. O. Besov, “On the Global Solvability of Semilinear Parabolic Systems with Mixed Right-Hand Side”, Function spaces, approximations, and differential equations, Collected papers. Dedicated to the 70th birthday of Oleg Vladimirovich Besov, corresponding member of RAS, Trudy Mat. Inst. Steklova, 243, Nauka, MAIK «Nauka/Inteperiodika», M., 2003, 66–86; Proc. Steklov Inst. Math., 243 (2003), 59–79

Citation in format AMSBIB
\Bibitem{Bes03} \by K.~O.~Besov \paper On the Global Solvability of Semilinear Parabolic Systems with Mixed Right-Hand Side \inbook Function spaces, approximations, and differential equations \bookinfo Collected papers. Dedicated to the 70th birthday of Oleg Vladimirovich Besov, corresponding member of RAS \serial Trudy Mat. Inst. Steklova \yr 2003 \vol 243 \pages 66--86 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm421} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2049463} \zmath{https://zbmath.org/?q=an:1074.35045} \transl \jour Proc. Steklov Inst. Math. \yr 2003 \vol 243 \pages 59--79 

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This publication is cited in the following articles:
1. K. O. Besov, “Critical Exponents for Nondiagonal Quasilinear Parabolic Systems”, Proc. Steklov Inst. Math., 248 (2005), 41–46
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