In this paper, we provide regularizing effect for continuous bounded from below viscosity solutions of a discontinuous Hamilton-Jacobi equation posed on a junction. We consider different quasi-convex and coercive time-space Hamiltonians on each branch and a flux limiter condition at the junction point. We then prove that the derivative with respect to time of the solution is bounded. As consequence, we deduce that the solution of the equation is locally Lipschitz continuous.

We classify all the blow-up solutions in self-similar form to the following reaction-diffusion equation$$\partial_tu=\Delta u^m+|x|^{\sigma}u^p,$$posed for $(x,t)\in \mathbb{R}^N\times(0,T)$, with $m > 1$, $1 \leq p < m$ and $-2(p-1)/(m-1) < \sigma < \infty$. We prove that there are several types of self-similar solutions with respect to the local behavior near the origin, and their existence depends on the magnitude of $\sigma$. In particular, these solutions have different blow-up sets and rates: some of them have $x=0$ as a blow-up point, some other only blow up at (space) infinity. We thus emphasize on the effect of the weight on the specific form of the blow-up patterns of the equation. The present study generalizes previous works by the authors limited to dimension $N=1$ and $\sigma > 0$.

For any $2{^\textrm{nd}}$ order scalar PDE $\mathcal{E}$ in one unknown function, we construct, by means of the characteristics of $\mathcal{E}$, a contact sub--bundle of the underlying contact manifold $J^1$, consisting of conic varieties, called the {contact cone structure} associated with $\mathcal{E}$. We then focus on symplectic Monge--Ampère equations in 3 independent variables, that are naturally parametrized, over ${\mathbb{C}}$, bythe projectivization of the 14--dimensional irreducible representation of the simple Lie group ${\mathsf{Sp}}(6,{\mathbb{C}})$. The associated moment map allows to define a rational map $\varpi$ from the space of symplectic 3D Monge-Ampère equations to the projectivization of the space of quadratic forms on a $6$--dimensional symplectic vector space. We study the relationship between the variety $\varpi({\mathcal{E}})=0$, herewith called the {co-characteristic variety} of ${\mathcal{E}}$, and the contact cone structure of a 3D Monge-Ampère equation ${\mathcal{E}}$, by obtaining a complete list of mutually non--equivalent quadratic forms on a $6$--dimensional symplectic space.