Stable blowup for wave equations in odd space dimensions

Abstract

We consider semilinear wave equations with focusing power nonlinearities in odd space dimensions d ≥ 5. We prove that for every p > d+3 d−1 there exists an open set of radial initial data in Hsuch that the corresponding solution exists in a backward lightcone and approaches the ODE blowup profile. The result covers the entire range of energy supercritical nonlinearities and extends our previous work for the three-dimensional radial wave equation to higher space dimensions. 1(1.5)By setting Ψ T (τ ) := (ψ T 1 (τ, ·), ψ T 2 (τ, ·)) this can be written aswhere L 0 represents the linear part of the right hand side of Eq. (1.5). To formulate the following statement we define for k ∈ N 0 H k rad (B d ) := {u ∈ H k (B d ) : u is radial}.

Stable blowup for the supercritical Yang–Mills heat flow

“…The proof in [14] is based on a canonical method that was developed by the authors to investigate stable self-similar blowup for wave equations, cf. similar results for supercritical wave maps [21] and wave equations with focusing power-nonlinearities, [17], [20], [19]. In this paper, we generalize our method to parabolic problems.…”

“…From now on we proceed exactly as in our previous works on radial wave equations, see for example [19], to prove Theorem 1.3. For convenience of the reader, we repeat the main arguments.…”

Threshold for blowup for the supercritical cubic wave equation

“…The stability of this solution has been established in the radial setting by Donninger and the third author [20], [21] for all odd d ≥ 3, see also Donninger and Chatzikaleas [13] for a generalization to the non-radial case in d = 5.…”

Stable self-similar blowup in the supercritical heat flow of harmonic maps

“…• From now on we follow the argument introduced in our earlier works [11][12][13][14][15][16] on selfsimilar blowup for wave-type equations. We first show that the nonlinearity is locally Lipschitz on X .…”