Nonradial type II blow up for the energy-supercritical semilinear heat equation
By Charles Collot. 2016
46 citations recorded by the source corpus.
We consider the semilinear heat equation in large dimension d ≥ 11 ∂tu = ∆u + |u| p−1 u, p = 2q + 1, q ∈ N on a smooth bounded domain Ω ⊂ R d with Dirichlet boundary condition. In the supercritical range p ≥ p(d) > 1 + 4 d−2 we prove the existence of a countable family (u l) l∈N of solutions blowing-up at time T > 0 with type II blow up: u l (t) L∞ ∼ C(T − t) −c l with blow-up speed c l > 1 p−1. They concentrate the ground state Q being the only radially and decaying solution of ∆Q + Q p = 0: u(x, t) ∼ 1 λ(t) 2 p−1 Q x − x0 λ(t) , λ ∼ C(un)(T − t) c l (p−1) 2 at some point x0 ∈ Ω. The result generalizes previous works on the existence of type II blow-up solutions, either constructive [14, 15, 35] or nonconstructive [25, 36], which only existed in the radial setting and relied on parabolic arguments. The present proof uses robust nonlinear tools instead, based on energy methods and modulation techniques in the continuity of [3, 32]. This is the first non-radial construction of a solution blowing up by concentration of a stationary state in the supercritical regime, and provides a general strategy to prove similar results for dispersive equations or parabolic systems and to extend it to multiple blow ups.
Subfields: Applied Mathematics, Control and Systems Engineering
View Paper: https://www.semanticscholar.org/paper/81a1d987b94f0992f9234b9b51bbe666229ca983
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In Applied Mathematics
In Control and Systems Engineering
Citations in the corpus
- Optimal Liouville theorems for superlinear parabolic problems (2020, background): “arguments in [12] and proved in [30, 7, 8, 9, 17, 18]. Type II blow-up can also occur for positive radial and non-radial solutions if n ≥ 11 and p ≥ pJL, where pJL := 1+4 n−4+2 √ n−1 (n−2)(n−10) (see [19, 20, 3, 4, 31]). In the case of smooth domains and Dirichlet boundary conditions, Theorem 1 impl”