Vacuum isolating, blow up threshold, and asymptotic behavior of solutions for a nonlocal parabolic equation

Abstract

In this paper, we consider a nonlocal parabolic equation associated with initial and Dirichlet boundary conditions. Firstly, we discuss the vacuum isolating behavior of solutions with the help of a family of potential wells. Then we obtain a threshold of global existence and blow up for solutions with critical initial energy. Furthermore, for those solutions satisfy J(u 0 ) ≤ d and I(u 0 ) 0, we show that global solutions decay to zero exponentially as time tends to infinity and the norm of blow-up solutions increase exponentially.2010 Mathematics Subject Classification. 35K20, 35K55.

Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity

“…In [11], we extended above results to the critical energy level initial data and established the asymptotic behavior results. That is to say, for a regular initial value u 0 ∈ C(Ω) ∩ H 1 0 (Ω), we have • If J(u 0 ) ≤ d and I(u 0 ) > 0, then the solution of (1.1) is global and decays to 0 exponentially as t → ∞.…”

“…Due to (2.2), we get z(u)|u| p−1 u ∈ L q (Ω). If u ∈ N , then by Hölder inequality we find [11]). Let 1 < p < n+2 n−2 .…”

“…• If J(u 0 ) ≤ d and I(u 0 ) < 0, then the solution of (1.1) blows up in finite time and the norm of it increases exponentially. (See [10,11] for more details.) Here d can be interpreted as the depth of the potential well, i.e.…”

Blow-up and global existence of solutions to a parabolic equation associated with the fraction <i>p</i>-Laplacian

“…The global existence condition of problem (1) is not studied; 4. The vacuum isolating phenomenon is important for evolution equation (see [4,13,21,15,16] and references therein), whether the vacuum isolating phenomenon happens for the solutions of problem (1) is not studied.…”

“…In order to give the first result, we recall some results of Remark 4, the equation d(δ) = e admits two positive roots δ 1 and δ 2 such that 0 < δ 1 < 1 < δ 2 < q p , where d(δ) is defined in (13) and e is any constant between 0 and d.…”