Uniqueness for singular backward parabolic inequalities

Abstract

We prove that the only solution of the partial differential inequality \[ ( u t + ∑ i = 1 n [ u x i x i + ( k i x i ) u x i ] ) 2 ⩽ c [ u 2 + | u x | 2 ] {\left ( {{u_t} + \sum \limits _{i = 1}^n {\left [ {{u_{{x_i}}}_{{x_i}} + \left ( {\frac {{{k_i}}}{{{x_i}}}} \right ){u_{{x_i}}}} \right ]} } \right )^2} \leqslant c\left [ {{u^2} + |{u_x}{|^2}} \right ] \] on a bounded region with homogeneous initial and boundary conditions is the trivial one.