On the Stability of Type I Blow Up For the Energy Super Critical Heat Equation

Abstract

We consider the energy super critical semilinear heat equationWe first revisit the construction of radially symmetric self similar solutions performed through an ode approach in [51], [2], and propose a bifurcation type argument suggested in [3] which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. We then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional non radial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self similar blow up in non radial energy super critical settings. (T − t)1 p−1 2010 Mathematics Subject Classification. primary, 35K58 35B44 35B35, secondary 35J61 35B32.

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

“…In the supercritical and critical range in the Sobolev sense, type I blow-up is also known to occur for some solutions which are neither radial nor increasing in time: see [10] for N = 3, p > p S , [38] for N = 4, p > 5, [7] for N ≥ 7 and p = p S and [8] for p = p S . (ii) Type II blow-up.…”

On Strongly Anisotropic Type I Blowup

“…Our analysis revisits the stability analysis of the self similar ODE blow up [1,29,30] and combines it with the study of the Type I self similar blow up [7]. This provides a robust canonical framework for the construction of strongly anisotropic blow up bubbles.…”

“…Let Φ(r) be a three dimensional radial self similar solution for the three supercritical probmem as exhibited and studied in [7]. We show the finite codimensional transversal stability of the corresponding blow up solution by exhibiting a manifold of finite energy blow up solutions of the four dimensional problem with cylindrical symmetry which blows up asOur analysis revisits the stability analysis of the self similar ODE blow up [1,29,30] and combines it with the study of the Type I self similar blow up [7]. This provides a robust canonical framework for the construction of strongly anisotropic blow up bubbles.…”

“…A countable class of such solutions has been constructed using either a direct Lyapunov functional approach [19,37,2,3] or a bifurcation argument [4,7], and these solutions satisfy…”

“…Since p > 5, p JL = +∞ and the only known blow up bubbles correspond to type I blow up bubbles with either the ODE or a non trivial 4 dimensional self similar profile. A basic observation however is now that any three dimensional radially symmetric type I self similar solution u(t, r) as constructed in [7] is formally a solution to the four dimensional equation (1.9), but this solution is constant in the z direction. Hence it has infinite energy and its dynamical role among solutions to (1.9) is unclear.…”