NONEXISTENCE RESULTS FOR A FULLY NONLINEAR EVOLUTION INEQUALITY

In this paper, a Liouville type theorem is proved for some global fully nonlinear evolution inequality via suitable choices of test functions and the argument of integration by parts.


Introduction
In the papers [7,8], Phuc and Verbitsky deduced nonexistence results for the following Hessian inequality: for α ∈ (k, nk n−2k ], where u ∈ C 2 (R n ), u is k-convex, and σ k (−D 2 u) are the k-Hessian of (−D 2 u) as usual, i.e., the sum of all the k-th principle minors of (−D 2 u). They employed the potential theory developed by Trudinger-Wang [9,10,11] and Labutin [4], and they also showed that the power α = nk n−2k is sharp. The nonexistence results were deduced in [7,8] from sharp pointwise estimates of solutions in terms of Wolff potentials. Later in [6], the author reproved some of Phuc-Verbitsky's results by a very different method -by using the argument of integration by parts via careful choices of the test functions.
In this paper, we will use the same method as in [6] and extend the results to the evolution case, namely, for the following fully nonlinear inequalty: with u ∈ C 2 (R n ), u is k-convex, and u(x, t) > 0 ∀ (x, t) ∈ R n × (0, +∞), and u 0 = u(x, 0) ≥ 0 ∀ x ∈ R n . Denote k * := k + 2k n . We will deduce a Liouville type theorem as follows: Note that according to Caffarelli-Nirenberg-Spruck [1], we say that u is k-admissible (or k-convex) with respect to σ k (−D 2 u) if u ∈ Γ k , where Γ k is defined by Similar nonexistence results for some quasilinear evolution inequalities were proved by Mitidieri-Pohozaev [5]. In particular, in the case of equality for k = 1 , the results of our Theorem 1.1 were proved by Fujita [2] and Hayakawa [3]. Therefore, Theorem 1.1 can be viewed as a generalization of their results to the fully nonlinear case.

Proof of Theorem 1.1
Assume that u > 0 is a k-admissible solution of (1.2). In the following, we write σ k (−D 2 u) simply as σ k .
First, we will construct a suitable test function. Denote by D the gradient operator in the space directions only. Let ϕ(x), ψ 0 (s) be two C 2 cut-off function satisfying Here and in the rest of the paper, B R denotes a ball in R n centered at the origin with radius R, and we use " ", " ", etc., to drop out some positive constants independent of R and u. Take and let η(x, t) = ϕ(x)ψ(t). Denote, as in [6], for s = 1, . . . , k, where ρ, δ, θ are constants to be determined. Now we give the proof of Theorem 1.1.

16)
and (2.18) Remark 2.2. For the details of the proof of (2.18), we refer the readers to [6] (see (3.30) combining with (3.15) in [6]). But here we must point out that the term R − n α δ V k had been left out in the inequality (3.26) (and hence (3.30)) in [6]), although the final result is still valid.