Existence of solutions for a fractional semilinear parabolic equation with singular initial data

Abstract

In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the Cauchy problemwhere N ≥ 1, 0 < θ ≤ 2, p > 1 and µ is a Radon measure or a measurable function in R N . Our conditions lead optimal estimates of the life span of the solution with µ behaving like λ|x| −A (A > 0) at the space infinity, as λ → +0.

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

“…Using their result, one can show that there is c 0 > 0 such that if φ(x) ≥ c 0 ψ(x) in a neighborhood of the origin, then (1.1) has no nonnegative integral solution. Here, [6]. For each 0 ≤ q < N/2 we will see that a modified function φ 0 , which is given by (4.1), belongs to X q .…”

“…However, the case q = q c = 1, i.e., p = 1 + 2/N, is not covered by Proposition 1.1, and it is known that there is a nonnegative initial function φ ∈ L 1 (R N ) such that (1.1) with p = 1 + 2/N has no local-in-time nonnegative solution. See Brezis-Cazenave [ [1,6,11] and references therein for existence and nonexistence results with measures as initial data. In [2, Section 7.5] the case p = 1 + 2/N is referred to as "doubly critical case".…”

“…We will see that if ρ ≥ e, then g(u) is convex for u ≥ 0 and g plays a crucial role in the construction of the solution of (1.6). In order to construct a nonnegative solution we use a method developed by Robinson-Sierżȩga [10] with the convex function g, which was also used in Hisa-Ishige [6]. We define a sequence of functions (u n ) ∞ n=0 by…”

Existence of solutions for a higher-order semilinear parabolic equation with singular initial data

“…In 1985 Baras and Pierre [3] studied necessary conditions for the existence of local-in-time solutions of (1.3) and proved the following (see also [14] and [23]). We remark that, if 1 < p < p 1 , then (1.4) is equivalent to sup x∈R N µ(B(x, T 1/2 )) ≤ cT Sufficient conditions for the existence of solutions of problem (1.3) have been studied in many papers since the pioneering work due to [25].…”

“…[1,2,6,11,14,17,20,21,22,23,26] and references therein. Among others, by [14] and [22] we have: By assertions (a) and (c) we can identify the strongest singularity of the initial data for the existence of solutions of (1.3) with p ≥ p 1 . Assertions (b) and (c) are proved by the construction of suitable supersolutions of (1.3) and the order-preserving property and the semigroup property of the heat operator are crucial in the proofs.…”