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

Abstract

In this paper, we consider the nonlinear heat equationin the unit ball Ω of R N with Dirichlet boundary conditions, in the subcritical case. More precisely, we study the set G of initial values in C 0 (Ω) for which the resulting solution of (NLH) is global. We obtain very precise information about a specific two-dimensional slice of G, which (necessarily) contains sign-changing initial values. As a consequence of our study, we show that G is not convex. This contrasts with the case of nonnegative initial values, where the analogous set G + is known to be convex.

Sign-changing stationary solutions and blowup for a nonlinear heat equation in dimension two

“…In fact, the proofs are based on similar strategies. They are both consequences of the following proposition, which is a particular case of Theorem 2.3 of [7]. Proposition 1.2.…”

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

“…This result is well known for positive solutions, but seems to be new for nodal solutions (cf. [8]). …”

Spectral properties of stationary solutions of the nonlinear heat equation

“…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.…”