Simonett, Gieri

EPrint Series of Department of Mathematics, Hokkaido University

ON NORMAL STABILITY FOR NONLINEAR PARABOLIC
EQUATIONS
Gieri Simonett
Department of Mathematics, Vanderbilt University
Nashville, TN 37240, USA
In this lecture we study convergence of solutions for quasilinear and fully nonlin-
ear parabolic equations towards equilibria in situations where the set of equilibria
is non-discrete but forms a C1-manifold.
Our main result can be summarized as follows: suppose that for a nonlinear
evolution equation we have a C1-manifold of equilibria E such that at a point u∗ ∈ E ,
the kernel N(A) of the linearization A is isomorphic to the tangent space of E at
u∗, the eigenvalue 0 of A is semi-simple, and the remaining spectral part of the
linearization A is stable. Then solutions starting nearby u∗ exist globally and
converge to some point on E . We call this situation the generalized principle of
linearized stability, and the equilibrium u∗ is then termed normally stable.
A typical example for this situation to occur is the case where the equations under
consideration involve symmetries, i.e. are invariant under the action of a Lie-group.
The situation where the set of equlibria forms a C1-manifold occurs for instance
in phase transitions, geometric evolution equations, free boundary problems in fluid
dynamics, stability of traveling waves, and models of tumor growth, to mention just
a few. (Joint work with J. Pru¨ss and R. Zacher).
References
[1] J. Pru¨ss and G. Simonett, Stability of equilibria for the Stefan problem with surface tension,
SIAM J. Math. Anal., 40 (2008), 675–698.
[2] J. Pru¨ss, G. Simonett, and R. Zacher, On convergence of solutions to equilibria for quasilinear
parabolic problems, arXiv:0807.1539v2. To appear in J. Differential Equations.
[3] J. Pru¨ss, G. Simonett, and R. Zacher, On normal stability for nonlinear parabolic equations,
arXiv:0811.1784v1. Submitted.
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On normal stability for nonlinear parabolic equations

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EPrint Series of Department of Mathematics, Hokkaido University