A doubly critical semilinear heat equation in the $$L^1$$ space

Abstract

We study the existence and nonexistence of a Cauchy problem of the semilinear heat equationHere, N ≥ 1, p = 1 + 2/N and φ ∈ L 1 (R N ) is a possibly sign-changing initial function. Since N (p − 1)/2 = 1, the L 1 space is scale critical and this problem is known as a doubly critical case. It is known that a solution does not necessarily exist for every. In this paper we construct a local-in-time mild solution in L 1 (R N ) for φ ∈ X q if q ≥ N/2. We show that, for each 0 ≤ q < N/2, there is a nonnegative initial function φ 0 ∈ X q such that the problem has no nonnegative solution, using a necessary condition given by Baras-Pierre [Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), . Since X q ⊂ X N/2 (q ≥ N/2), X N/2 becomes a sharp integrability condition. We also prove a uniqueness in a certain set of functions which guarantees the uniqueness of the solution constructed by our method.

Thresholds on growth of nonlinearities and singularity of initial functions for semilinear heat equations

“…This case is quite delicate and referred as a doubly critical case in [2, Section 7.5], since r = N/2 and q = 1 + r. See Figure 1. A sufficient condition for existence is recently studied in [18]. Combining a nonexistence result [3] and an existence result [18], we have the following:…”

“…A sufficient condition for existence is recently studied in [18]. Combining a nonexistence result [3] and an existence result [18], we have the following:…”

Solvability of a semilinear heat equation on Riemannian manifolds

“…In the case where µ is an L q -function, the study of sufficient conditions dates back to Weissler [44,45]. For the progress with functions as initial data, we refer recent papers [11,13,17,18,24,29], a book [37,Section 15] and the references therein. For sufficient conditions with measures as initial data, we refer [1,2,3,5,22,23,30,39,43].…”

on a second critical value for the local existence of solutions in lebesgue spaces

“…Is it possible to define a class of initial data where the local existence hold? As far as we know there are few results about this and only in the case f (s) = s q , q > 1 see [14], [11].…”

“…Recently, Miyamoto [11] studied the so-called doubly critical case, that is, p = p ⋆ , r = 1, considering h(t) = 1 and the set…”