Blowup solutions for the shadow limit model of singular Gierer-Meinhardt system with critical parameters

Abstract

We consider a nonlocal parabolic PDE, which may be regarded as the standard semilinear heat equation with power nonlinearity, where the nonlinear term is divided by some Sobolev norm of the solution. Unlike the earlier work in [13] where we consider a subcritical regime of parameters, we focus here on the critical regime, which is much more complicated. Our main result concerns the construction of a blow-up solution with the description of its asymptotic behavior. Our method relies on a formal approach, where we find an approximate solution. Then, adopting a rigorous approach, we linearize the equation around that approximate solution, and reduce the question to a finite dimensional problem. Using an argument based on index theory, we solve that finite-dimensional problem, and derive an exact solution to the full problem. We would like to point out that our constructed solution has a new blowup speed with a log correction term, which makes it different from the speed in the subcritical range of parameters and the standard heat equation.