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Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term. (English) Zbl 1423.35186

This article studies the equation \[ u_t = \Delta u + |u|^{p-1} u + \mu |\nabla u|^q \] with \(\mu>0\) in \(\mathbb{R}^N\times(0,T)\), which in some sense is an intermediate between semilinear heat and Hamilton-Jacobi equations. More precisely, the article is concerned with the critical case \(q=\frac{2p}{p+1}\) for \(p>3\).
Here, a new blow-up profile (different from those occurring in the case of \(\mu=0\)) appears. In this article it is first motivated by a formal approach, afterwards the existence of a solution approaching this profile is proven: If initial data are suitably chosen, the difference between solution and profile is ‘trapped’ in a set shrinking to zero. (The choice of this trap, along with parabolic estimates adapted to its geometry, is one of the novelties of the paper, but also causes the restriction on \(p\), whose necessity remains unknown.)
The method also gives stability of this solution.
Proofs are presented such that first an outline is given and technical results are provided in more detail later.

MSC:

35K55 Nonlinear parabolic equations
35B44 Blow-up in context of PDEs
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