ABSTRACT

A basic question in the study of nonlinear parabolic equations is whether a time-dependent solution, as time increases, remains in a neighbourhood of an equilibrium solution and whether it converges to this equilibrium solution as t → ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003150169/0dfa0042-ed9a-4311-a778-5bb9f3b2e0db/content/math5_1.tif"/> . From a practical point of view, it is important to know that, for a given equilibrium solution u s , whether there is a set of initial data such that their corresponding time-dependent solutions converge to u s as t → ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003150169/0dfa0042-ed9a-4311-a778-5bb9f3b2e0db/content/math5_2.tif"/> . This leads to a question of stability and asymptotic stability of an equilibrium solution and its stability region.