Summary
We study the asymptotic behaviour of the solutions of the equation ut=Au+λu−|u|αu. Denoting by λ0 the principal eigenvalue of the second-order differential operator A, we shall prove that if λ ⩽ λ0 the only equilibrium solution, namely zero, is asymptotically stable, whereas, if λ>λ0, the nontrivial equilibrium solutions without internal zeros are asymptotically stable. Attractivity and stability are proved both in the L2-norm and in the H 10 -norm.
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Entrata in Redazione il 15 ottobre 1976.
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de Mottoni, P., Talenti, G. & Tesei, A. Stability results for a class of non-linear parabolic equations. Annali di Matematica 115, 295–310 (1977). https://doi.org/10.1007/BF02414721
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DOI: https://doi.org/10.1007/BF02414721