Construction and stability of blowup solutions for a non-variational semilinear parabolic system

TL;DR

This paper constructs and analyzes finite-time blowup solutions for a non-variational semilinear parabolic system, demonstrating their stability and detailed asymptotic behavior near the blowup point.

Contribution

It introduces a method to construct stable blowup solutions for a non-variational system with detailed asymptotics, overcoming challenges like non-self-adjoint linearized operators.

Findings

Existence of initial data leading to finite-time blowup at a single point.

Explicit asymptotic profiles of solutions near blowup time.

Proof of stability of blowup behavior under initial data perturbations.

Abstract

We consider the following parabolic system whose nonlinearity has no gradient structure: $$\left\{\begin{array}{ll} \partial_t u = \Delta u + |v|^{p-1}v, \quad & \partial_t v = \mu \Delta v + |u|^{q - 1}u,\\ u(\cdot, 0) = u_0, \quad & v(\cdot, 0) = v_0, \end{array}\right. $$ in the whole space $\mathbb{R}^N$, where $p, q > 1$ and $\mu > 0$. We show the existence of initial data such that the corresponding solution to this system blows up in finite time $T(u_0, v_0)$ simultaneously in $u$ and $v$ only at one blowup point $a$, according to the following asymptotic dynamics: $$\left\{\begin{array}{c} u(x,t)\sim \Gamma\left[(T-t) \left(1 + \dfrac{b|x-a|^2}{(T-t)|\log (T-t)|}\right)\right]^{-\frac{(p + 1)}{pq - 1}},\\ v(x,t)\sim \gamma\left[(T-t) \left(1 + \dfrac{b|x-a|^2}{(T-t)|\log (T-t)|}\right)\right]^{-\frac{(q + 1)}{pq - 1}}, \end{array}\right.$$ with $b = b(p,q,\mu) > 0$ and $(\Gamma, \gamma) = (\Gamma(p,q), \gamma(p,q))$. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case $\mu = 1$; and the fact that the case $\mu \ne 1$ breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data.