Comparison principle for some nonlocal problems

Abstract

Abstract. In this paper, for the parabolic equation ut = Au + g(x, u), (x, t) e Qx(OJ),with nonlocal boundary conditions u\da = faf(x,y)u(y, t)dy, we establish the comparison theorem and local existence of the solution. We also discuss its long time behavior.

Blow-up for a degenerate and singular parabolic equation with nonlocal boundary condition

“…They obtained some results on the existence and nonexistence of the global solutions, and derived the uniform blow-up profile estimate under some assumptions. For other works on this topic, we refer the readers to [8,9,10,19,21] and the references therein.…”

A Parabolic System With Nonlocal Boundary Conditions and Nonlocal Sources

“…To obtain the blow-up rate estimates, we need the following positivity lemma, whose proof is much the same as that of [9].…”

“…Local (in time) existence of positive classical solutions to Problem (1.1) can be obtained by using fixed point theorem (see [9,24]). By the above comparison principle and Remark 2.1, we can get the uniqueness of classical solution to Problem (1.1) in the case of p, q ≥ 1.…”

“…He established the global existence of solution, and showed that the unique solution tends to 0 monotonically and exponentially as t → +∞ in the case of g(x, u) = c(x)u with c(x) ≤ 0 and Ω |k(x, y)|dy < 1 for all x ∈ ∂Ω. In 1992, Deng [9] gave the comparison principle and local existence of classical solution to (1.2) with general g(x, u). For the case g(x, u) = c(x)u, he showed that the solution exists globally and may increase at most exponentially with t under some weaker assumptions than those in [13].…”

On a Non-Local Parabolic Problem

“…In [4], Deng studied the following nonlocal boundary-value problem He first established the comparison principle for (-Pi). Then he showed the local existence of the solution and he discussed its long time behavior, assuming (…”