Spectral properties of stationary solutions of the nonlinear heat equation

Abstract

In this paper, we prove that if Ψ is a radially symmetric, signchanging 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 λΨ blows up in finite time if |λ − 1| > 0 is sufficiently small and if α > 0 is sufficiently small. The proof depends on showing that the inner product of Ψ with the first eigenfunction of the linearized operator L = −∆ − (α + 1)|Ψ| α is nonzero.

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

“…Note that u ∈ G, so that G is not star-shaped. Let us point out that an analogous result was proven for N = 3 and p close to 1, see [8]. The results in [5] have been extended to case of general non symmetric domains in [19].…”

“…For N ≥ 3 and subcritical p < p S , it was shown in [8] that λu ∈ B if |1 − λ| and p S − p are small enough (λ = 1), independently of the number K of oscillations of the stationary solution u. We were not able to obtain here an analogous result, since p and λ depend on K in Theorem 1.1.…”

“…can be found in [6] and [7]. The results of [5] and [8] do not apply in the case N = 1. In fact, for N = 1 and p > 1 v λ (u) is global and converges uniformly to zero if |λ| < 1, while v λ (u) blows up if |λ| > 1.…”

Complexity of asymptotic behavior of the porous medium equation in $${\mathbb{R}^N}$$

“…For example, Vázquez and Zuazua [14] revealed that if u(x, t) is a solution of (1. can present more complexity of asymptotic behavior in the linear case of the Eq. (1.1), i.e., m = 1, and this has recently been discovered by Cazenave, Dickstein and Weissler in their recent works, see [3][4][5].…”

“…In our arguments, we will take all 0 < μ < 2N N (m − 1) + 2 (1.5) and β > 2 − μ(m − 1) 4 > 1 N (m − 1) + 2 (1.6) such that μ 2β is in a countable subset of 0, 2N (N (m−1)+2)(2+μ(m−1)) in (1.4) to study the complexity of asymptotic behavior. The main difficulties of this paper come from the following three aspects: Obviously, the first one is the nonlinearity of the porous medium equation comparing to the works of Cazenave et al (see [3][4][5]), where the heat equation was considered; The second one is the more general version of rescaling for the solutions. Vázquez [11] and Zuazua et al [14] only considered the rescaling for fixed μ and fixed β.…”