We study asymptotic behavior of solutions to multifractal Burgers-type equation $u_t+f{\left(u\right)}_x=Au$, where the operator A is a linear combination of fractional powers of the second derivative $-{\partial }^2/\partial x^2$ and f is a polynomial nonlinearity. Such equations appear in continuum mechanics as models with fractal diffusion. The results include decay rates of the $L^p$-norms, 1 ≤ p ≤ ∞, of solutions as time tends to infinity, as well as determination of two successive terms of the asymptotic expansion of solutions.