In this paper, we consider the Cauchy problem for a generalized parabolic-elliptic Keller-Segel equation with a fractional dissipation and advection by a weakly mixing (see Definition 2.4). Here the attractive kernel has strong singularity, namely, the derivative appears in the nonlinear term by singular integral. Without advection, the solution of equation blows up in finite time. Under a suitable mixing condition on the advection, we show the global existence of classical solution with large initial data in the case of the derivative of dissipative term is higher than that of nonlinear term. Since the attractive kernel is strong singularity, the weakly mixing has destabilizing effect in addition to the enhanced dissipation effect, which makes the problem more complicated and difficult. In this paper, we establish the L ∞ -criterion and obtain the global L ∞ estimate of the solution through some new ideas and techniques. Combined with [48], we discuss all cases of generalized Keller-Segel system with mixing effect, which was proposed by Kiselev, Xu (see [36]) and Hopf, Rodrigo (see [31]). Based on more precise estimate of solution and the resolvent estimate of semigroup operator, we introduce a new method to study the enhanced dissipation effect of mixing in generalized parabolic-elliptic Keller-Segel equation with a fractional dissipation. And the RAGE theorem is no longer needed in our analysis.

We are concerned with oscillation of the first order neutral delay differential equation[x(t)−px(t−τ)]′+qx(t−σ)=0with constant coefficients, and we obtain some necessary and sufficient conditions of oscillation for all the solutions in respective cases0<p<1andp>1.

This paper is concerned with the Cauchy problem for a fractal Burgers equation in two dimensions. When α ∈ (0, 1), the same problem has been studied in one dimensions, we can refer to [1, 17, 24]. In this paper, we study well-posedness of solutions to the Burgers equation with supercritical dissipation. We prove the local existence with large initial data and global existence with small initial data in critical Besov space by energy method. Furthermore, we show that solutions can blow up in finite time if initial data is not small by contradiction method.

In this paper, we consider the Cauchy problem for a generalized parabolic-elliptic Keller-Segel equation with fractional dissipation and the additional mixing effect of advection by an incompressible flow. Under suitable mixing condition on the advection, we study wellposedness of solution with large initial data. We establish the global L ∞ estimate of the solution through nonlinear maximum principle, and obtain the global classical solution.

In this paper, we are concerned with the monotonicity and the estimates of x v andx