Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity

Abstract

We are concerned with blow-up mechanisms in a semilinear heat equation:ut = ∆u + |x| 2a u p , x ∈ R N , t > 0, where p > 1 and a > −1 are constants. As for the Fujita equation, which corresponds to a = 0, a well-known result due to M. A. Herrero and J. J. L. Velázquez, C. R. Acad. Sci. Paris Sér. I Math. (1994), states that if N ≥ 11 andwhere T is the blow-up time. We revisit the idea of their construction and obtain refined estimates for such solutions by the techniques developed in recent works and elaborate estimates of the heat semigroup in backward similarity variables. Our method is naturally extended to the case a = 0. As a consequence, we obtain an example of solutions that blow up at x = 0, the zero point of potential |x| 2a with a > 0, and behave in non-self-similar manner for N > 10 + 8a. This last result is in contrast to backward self-similar solutions previously obtained for N < 10 + 8a, which blow up at x = 0.

Radial equivalence and applications to the qualitative theory for a class of non-homogeneous reaction-diffusion equations

“…More recently, the Hardy equation has been strongly studied in the semilinear case m = 1, see for example [4,3,7,8,66,18]. We stress here that the results in the papers [58,9,65,51] mentioned in the previous paragraph are also proved for σ > −2, but without including the limit case of equality.…”

“…As a remark, the very interesting paper [51] gives more precise asymptotic estimates on the solutions to Eq. (2.8) with m = 1 and their blow-up patterns both in the inner layer formed in a small neighborhood of the unique blow-up point s = 0 and in bounded regions, leading to a very deep description of the asymptotics of the solutions w n as τ → T 0 .…”

“…(c) The finite time blow-up of Type II for Eq. (2.8) in the semilinear case m = 1 but with σ > −2 and p > p JL (σ) has been thoroughly described in the recent work [51]. More precisely, [51,Theorem 1.1] states that, for any sufficiently large natural number n, there exists at least one radially symmetric and radially decreasing positive solution w n to Eq.…”

“…(2.8) in the semilinear case m = 1 but with σ > −2 and p > p JL (σ) has been thoroughly described in the recent work [51]. More precisely, [51,Theorem 1.1] states that, for any sufficiently large natural number n, there exists at least one radially symmetric and radially decreasing positive solution w n to Eq. (2.8) blowing up at some time T 0 ∈ (0, ∞) and at s = 0, such that their blow-up rate is given by lim Let us notice first that, taking into account the definition of p JL (σ) in (7.1b) and the expressions of σ and N in (2.6), p JL (σ) is mapped into p JL (σ 2 ) by undoing (2.6).…”

“…A series of works by Guo, Shimojo, Souplet et al [14,15,16,17] addressed the question of the blow-up set of solutions to (1.5) in the semilinear case m = 1. In a different research direction, works such as [9] and [51] studied the way finite time blow-up occurs, again in the semilinear case.…”