The parabolic logistic equation with blow-up initial and boundary values

Abstract

Abstract. In this article, we investigate the parabolic logistic equation with blow-up initial and boundary values over a smooth bounded domain Ω:where T > 0 and p > 1 are constants, a and b are continuous functions, withWe study the existence and uniqueness of positive solutions, and their asymptotic behavior near the parabolic boundary. Under the extra condition that b(for some constants c > 0, θ > 0 and β > −2, we show that such a solution stays bounded in any compact subset of Ω as t increases to T , and hence solves the equation up to t = T .

Initial and Boundary Blow-Up Problem for $$p$$ p -Laplacian Parabolic Equation with General Absorption

“…Motivated by the above works, in this paper, we study the problem (1.1)- (1.3). We are able to extend some of the results of [3,12,17]. Our method refers to Karamata's regular variation theory [4], which has been used by many authors in elliptic boundary blow-up problems.…”

“…By (1.9) and (1.10), there is a constant l > 1 such that u(x, t) ≤ ū(x, t) ≤ lu(x, t) in Ω T . The remainder of the proof is similar to that of [12,Theorem 1.4]. We omit the details.…”

“…We shall divide the proof into five steps. Some of the techniques used are based on those found in [12] and [6], though the adaptation to our setting is not straightforward.…”

Localized peaking regimes for quasilinear parabolic equations

“…where a 1 (t), a 2 (t) is positive continuous on [0, T ) functions, the existence of maximal u and minimal u positive solutions of the problem (1.16), (1.17), (1.18) was proved in [13]. Moreover the main result of [13] says that under the following additional condition on the degeneration of a 1 (t) near t = T :…”

A Diffusive Logistic Equation With Memory in Bessel Potential Spaces

“…This model has been widely utilised for many different purposes. See, for example, [5,6,8,9,11,13,14,16,17] and the references therein.…”