English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

A typical example is that f(u) = u−μ1u −μ2 with constants p > max(q, 0). This kind of semilinear equation appears in several applications in mechanics and physics. In particular, it has been used to model the dynamics of thin films for viscous liquids. Some detailed physics background can be found in [1]-[3], [9] and [12]-[14]. Some recent mathematical analysis can be found in [4, 6, 7, 8, 10, 11, 15, 17] and the references therein. It was proved in [8] (see also [11]) that in dimension N ≥ 3, for each α ∈ (0, t0), (1.1) has a unique positive solution uα. Moreover, uα oscillates around the constant t0, that is, there is an increasing positive sequence {rn α} such that {r ∈ (0,∞) : uα(r) = 0} = {rn α}, and limn→∞ r α = ∞. Here r α are local maxima of uα with uα(r 2i+1 α ) > t0 for any i ∈ N; while r α are local minima with uα(r α) 0 for r ∈ (0,∞) and f(u0) ∈ Lloc(R ). Moreover, any singular radial solution of (1.1) is oscillatory around t0 and converges to t0 as r → ∞.

Asymptotic Behavior of Oscillating Radial Solutions to Certain Nonlinear Equations