Abstract
We consider parabolic equations of the form. ut=δu+f(u)+h(x,t),(x,t)∈R{double-struck}N×(0,∞), where f is a C1 function with f(0)=0, f<(0)<0, and h is a suitable function on RN×[0,∈) which decays to zero as t→∈ (hence the equation is asymptotically autonomous). We show that, as t→∈, each bounded localized solution u≥0 approaches a set of steady states of the limit autonomous equation ut=δu+f(u). Moreover, if the decay of h is exponential, then u converges to a single steady state. We also prove a convergence result for abstract asymptotically autonomous parabolic equations.
Original language | English (US) |
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Pages (from-to) | 1903-1922 |
Number of pages | 20 |
Journal | Journal of Differential Equations |
Volume | 251 |
Issue number | 7 |
DOIs | |
State | Published - Oct 1 2011 |
Bibliographical note
Funding Information:E-mail address: polacik@math.umn.edu (P. Polácˇik). 1 Supported in part by NSF Grant DMS-0900947.
Keywords
- Asymptotically autonomous
- Convergence
- Parabolic equation
- Quasiconvergence