Finite time blow-up for an inhomogeneous parabolic equation

Abstract

We consider the inhomogeneous nonlinear heat equation \[ \partial_t u-\Delta u=|u|^{p-1}u+f(x),\qquad x\in\mathbb{R}^3,\quad p>5, \] where \(f\in L^\infty\cap C^{0,1}(\mathbb{R}^3)\). For every sufficiently large integer \(n\), we construct a codimension-\(n\) Lipschitz manifold of non-radial initial data whose corresponding solutions blow up in finite time and whose rescaled profiles converge to the prescribed self-similar profile \(\Phi_n\) of the homogeneous equation. The main novelty is to show that the finite-codimensional stability mechanism for self-similar blow-up, developed in the work of Collot, Rapha\"el and Szeftel [Mem. Amer. Math. Soc. (2019)] for the homogeneous equation, is robust under the addition of a bounded, Lipschitz spatially inhomogeneous source term. In contrast with the homogeneous problem, the equation considered here has no exact scaling invariance, which is a key ingredient in many previous constructions. We expect that the framework developed in this work may also be useful for related problems in which exact scaling invariance is broken.