On the finite time blow-up for filtration problems with nonlinear reaction

TL;DR

This paper investigates conditions under which solutions to certain filtration problems with nonlinear reactions blow up in finite time, highlighting the influence of initial data and nonlinearities.

Contribution

It provides new criteria for finite time blow-up in filtration problems with nonlinear reactions, especially relating to initial data size and reaction dominance.

Findings

Solutions blow up in finite time with large initial data.

Initial data above stationary states lead to blow-up.

Reaction terms can overpower nonlinear diffusion, causing blow-up.

Abstract

We present results for finite time blow-up for filtration problems with nonlinear reaction under appropriate assumptions on the nonlinearities and the initial data. In particular, we prove first finite time blow up of solutions subject to sufficiently large initial data provided that the reaction term "overpowers" the nonlinear diffusion in a certain sense. Secondly, under related assumptions on the nonlinearities, we show that initial data above positive stationary state solutions will always lead to finite time blow up.