Summary: We study the initial value problem for the nonlinear dissipative equations \[ u_t+(-\Delta)^{\rho/2}u+a\cdot\nabla \bigl( |u|^{\sigma+1}=0,\;(t,x) \in\mathbb{R}^+\times\mathbb{R}^n\quad u(0,x)=u_0 (x),\;x\in\mathbb{R}^n, \] where \(a\in\mathbb{R}^n\), \(\rho>1\), \(\sigma>0\). We prove that if \(a=(-1,0,\dots,0)\) and \[ \int_{\mathbb{R}^n} u_0(x)=0,\quad \int_{\mathbb{R}^n}x_1u_0(x)\,dx>0,\quad \int_{\mathbb{R}^n}x_ju_0(x)\,dx=0\text{ for }j\neq 1, \] then in the case \(\sigma=(\rho-1)/(n+1)\) solutions exist globally and decay as \[ \bigl\| u(t)\bigr\|_{L^\infty}\leq C(1+t)^{-(\rho-1)/\sigma \rho}\bigl(\log(2+t)\bigr)^{-1/\sigma}. \] And in the case \(0<\sigma<(\rho-1)/(n+1)\), \(\sigma\) is close to \((\rho-1)/(n+ 1)\), solutions have estimates \[ \bigl\| u(t)\bigr\|_{L^\infty} \leq C(1+t)^{-((\rho-1)/\sigma\rho)}. \]