Blowup versus global in time existence of solutions for nonlinear heat equations

Abstract

This note is devoted to a simple proof of blowup of solutions for a nonlinear heat equation. The criterion for a blowup is expressed in terms of a Morrey space norm and is in a sense complementary to conditions guaranteeing the global in time existence of solutions. The method goes back to H. Fujita and extends to other nonlinear parabolic equations.

Around a singular solution of a nonlocal nonlinear heat equation

“…2] for α ∈ (0, 2).These are counterparts of results in [5, Remark 7, Theorem 2] for the classical nonlinear heat equation. Together with results of Section 2, this leads to the following partial dichotomy result, similarly as was in[5, Corollary 11]…”

“…[29, Th. A]) but the proof in [5] seems be somewhat novel. Similar pointwise arguments are powerful tools and, as such, they have been used in different contexts as e.g.…”

Blowup of solutions for nonlinear nonlocal heat equations

“…This is a natural extension of properties of the Cauchy problem (4), (6) studied in, e.g., [15,16] and [3].…”

“…a sufficient condition for the blowup of (4) which was derived in [11], and has been analyzed in a recent paper [3].…”

“…The proof of (ii) for p = p F and e tA = e −t(−∆) α/2 , α ∈ (0, 2), is in [18]. A rather simple new proof of the result (ii) for α = 2 and p = p F is in [3].…”