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

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.