Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction

We perform a thorough study of the blow up profiles associated to the following second order reaction-diffusion equation with non-homogeneous reaction: $$\partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u^p,$$ in the range of exponents $10$. We classify blow up solutions in self-similar form, that are likely to represent typical blow up patterns for general solutions. We thus show that the non-homogeneous coefficient $|x|^{\sigma}$ has a strong influence on the qualitative aspects related to the finite time blow up. More precisely, for $\sigma\sim0$, blow up profiles have similar behavior to the well-established profiles for the homogeneous case $\sigma=0$, and typically \emph{global blow up} occurs, while for $\sigma>0$ sufficiently large, there exist blow up profiles for which blow up \emph{occurs only at space infinity}, in strong contrast with the homogeneous case. This work is a part of a larger program of understanding the influence of unbounded weights on the blow up behavior for reaction-diffusion equations

Mean field equations on tori: existence and uniqueness of evenly symmetric blow-up solutions

We are concerned with the blow-up analysis of mean field equations. It has been proven in [6] that solutions blowing-up at the same non-degenerate blow-up set are unique. On the other hand, the authors in [18] show that solutions with a degenerate blow-up set are in general non-unique. In this paper we first prove that evenly symmetric solutions on a flat torus with a degenerate two-point blow-up set are unique. In the second part of the paper we complete the analysis by proving the existence of such blow-up solutions by using a Lyapunov-Schmidt reduction method. Moreover, we deduce that all evenly symmetric blow-up solutions come from one-point blow-up solutions of the mean field equation on a "half" torus

Blow-up behavior of collocation solutions to Hammerstein-type volterra integral equations

We analyze the blow-up behavior of one-parameter collocation solutions for Hammerstein-type Volterra integral equations (VIEs) whose solutions may blow up in finite time. To approximate such solutions (and the corresponding blow-up time), we will introduce an adaptive stepsize strategy that guarantees the existence of collocation solutions whose blow-up behavior is the same as the one for the exact solution. Based on the local convergence of the collocation methods for VIEs, we present the convergence analysis for the numerical blow-up time. Numerical experiments illustrate the analysis