Let $a_1,\ldots,a_n$ and $b_1,\ldots,b_m$ be fixed positive integers, let ${\mathcal S}$ denote the set of all nonnegative integer solutions equation $x_1a_1+\cdots +x_na_n=y_1b_1+\cdots...

We prove local properties of symmetry and monotonicity for nonnegative solutions of scalar eld equations with nonlinearities which are not Lipschitz. Our main tools are a local Moving Planes method and a unique continuation argument which is connected with techniques used for proving the uniqueness of radially symmetric solutions.

We consider the aggregation equation ut + ∇ · (u∇K ∗ u) = 0 in R n, n ≥ 2, where K is a rotationally symmetric, nonnegative decaying kernel with a Lipschitz point at the origin, e.g. K(x) = e−|x|. We prove finite-time blow-up of solutions from specific smooth initial data, for which the problem is known to have short time existence of smooth solutions.