Vanishing of solutions of diffusion equation with convection and absorption

We study the vanishing of solutions of the Cauchy problem for the equation $$ u_t = sum_{i,j=1}^N a_{ij}(u^m)_{x_ix_j} + sum_{i=1}^N b_i(u^n)_{x_i} - cu^p, quad (x,t)in S = mathbb{R}^Nimes(0,+infty). $$ Obtained results depend on relations of parameters of the problem and growth of initial data at infinity.