Nonradial type II blow up for the energy-supercritical semilinear heat equation

Abstract

Abstract. 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 rangewe prove the existence of a countable family (u ℓ ) ℓ∈N of solutions blowing-up at time T > 0 with type II blow up:. They concentrate the ground state Q being the only radially and decaying solution of ∆Q + Q p = 0: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.

Optimal condition for blow-up of the critical L norm for the semilinear heat equation

“…• p = p S (see [46], [12]); • p = (N + 1)/(N − 3) with N ≥ 7 (see [11]); • p ≥ p JL ; see [30], [29], [39], [47] in radial case and [6], [9] in nonradial case. We note that one cannot expect general conclusions regarding the blow-up of the critical L q norm for type II blow-up solutions, since there are examples of boundedness of the critical L q norm as stated above (cf.…”

“…In fact, for p > p S , radial type II blow-up solutions converge to the singular stationary solution c * |x| −2/(p−1) as t tends to T (see [35]), and this implies unboundedness of L q * norm. As for the nonradial solutions in [6], [9], [11], the unboundedness of L q * norm can be directly seen by simple computations, using the asymptotic form of the solution obtained in those works.…”

Liouville theorems for superlinear parabolic problems with gradient structure

“…Let us also mention that if p = p S , 3 ≤ n ≤ 6, and we allow sign-changing solution, then the blow-up may be of type II : This is indicated by formal 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 ≥ p JL , where p JL := 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 implies the following theorem (cf.…”

On the stability of type II blowup for the 1-corotational energy-supercritical harmonic heat flow

“…The formal result was clarified in [37], [35] and [9]. The collection of these works yields a complete classification of the type II blowup scenario for the radially symmetric energy supercritical case.…”