Critical Exponents for Nondiagonal Quasilinear Parabolic Systems

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 $f_i$.