On the nonexistence of solutions of some anisotropic elliptic and backward parabolic inequalities

We study the nonexistence of weak solutions of higher-order elliptic and parabolic inequalities of the following types: (sum {_{i = 1}^Nsumnolimits_{{e_i} leqslant {alpha _i} leqslant {m_i}} {D_{{x_i}}^{{alpha _i}}left( {{A_{{alpha _i}}}left( {x,u} right)} right)} geqslant fleft( {x,u} right),} x in {mathbb{R}^N}), and ({u_t} + sum {_{i = 1}^Nsumnolimits_{{k_i} leqslant {beta _i} leqslant {n_i}} {D_{{x_i}}^{{beta _i}}left( {{B_{{beta _i}}}left( {x,t,u} right)} right)} > gleft( {x,t,u} right),left( {x,t} right)} in {mathbb{R}^N} times {mathbb{R}_ + }), where l _i , m _i , k _i , n _i ∈ N satisfy the condition l _i , k _i > 1 for all i = 1,..., N, and A _αi (x, u), B _βi (x, t, u), f(x, u), and g(x, t, u) are some given Carathéodory functions. Under appropriate conditions on the functions A _αi , B _βi , f, and g, we prove theorems on the nonexistence of solutions of these inequalities.