Sign-changing stationary solutions and blowup for the nonlinear heat equation in a ball

Abstract

In this paper, we prove that if φ is a radially symmetric, sign-changing stationary solution of the nonlinear heat equationin the unit ball of R N , N ≥ 3, with Dirichlet boundary conditions, then the solution of (NLH) with initial value λφ blows up in finite time if |λ − 1| > 0 is sufficiently small and if α is subcritical and sufficiently close to 4/(N − 2).

Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations

“…. }, there exists a unique radial equilibrium u satisfying z (0,R) (u) = Z and u(0) > 0 (see [7]). On the other hand the proof of Theorem 1.8 shows that there exists a radial equilibrium u satisfying z (0,R) (u) = Z on the boundary of the domain of attraction of the zero solution.…”

Spectral properties of stationary solutions of the nonlinear heat equation

“…Recall that if Ψ is radially symmetric, then so is Φ. In [1], we proved the following result which concerns values of α close to 4/(N − 2) for N ≥ 3.…”

“…The results in [1], [2], [3] show that there is a more subtle relationship between stationary solutions and blowup. More precisely, let Ψ ∈ C 0 (Ω) be a stationary solution of (1.1), i.e.…”

“…In this paper, we continue our study [1] [2], [3] of the instability of sign-changing stationary solutions of the nonlinear heat equation.…”

On the structure of global solutions of the nonlinear heat equation in a ball

“…For general solutions, not just positive solutions, the situation is different and more complicated. Indeed, in a recent paper [2] the authors have shown that for N 3 and α sufficiently close to α , G is not star-shaped around 0 (and in particular not convex). We do not know if this is true in general.…”