This paper investigates the existence of a uniform in time $L^{\infty}$ bounded weak entropy solution for the quasilinear parabolic-parabolic Keller-Segel model with the supercritical diffusion exponent $0 < m < 2-\frac{2}{d}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(2-m)}{2}}$ norm of initial data is smaller than a universal constant. Moreover, the weak entropy solution $u(x,t)$ satisfies mass conservation when $m > 1-\frac{2}{d}$. We also prove the local existence of weak entropy solutions and a blow-up criterion for general $L^1\cap L^{\infty}$ initial data.

In this paper, we introduce a random particle blob method for the Keller-Segel equation (with dimension ) and establish a rigorous convergence analysis.

We study the stability of traveling waves of the nonlinear Schrödinger equation with nonzero condition at infinity obtained via a constrained variational approach. Two important physical models for this are the Gross–Pitaevskii (GP) equation and the cubic-quintic equation. First, under a non-degeneracy condition we prove a sharp instability criterion for 3D traveling waves of (GP), which had been conjectured in the physical literature. This result is also extended for general nonlinearity and higher dimensions, including 4D (GP) and 3D cubic-quintic equations. Second, for cubic-quintic type nonlinearity, we construct slow traveling waves and prove their nonlinear instability in any dimension. For dimension two, the non-degeneracy condition is also proved for these slow traveling waves. For general traveling waves without vortices (that is nonvanishing) and with general nonlinearity in any dimension, we find a sharp condition for linear instability. Third, we prove that any 2D traveling wave of (GP) is transversally unstable, and we find the sharp interval of unstable transversal wave numbers. Near unstable traveling waves of all of the above cases, we construct unstable and stable invariant manifolds.