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27r-periodic solution AMS Subject Classifications apply argument assume assumption asymptotic asymptotic stability axisymmetric Bellman equation blow-up boundary condition boundary value problem bounded Cauchy problem compact completes the proof consider continuous function converges decay defined denote Department of Mathematics Differential Equations Dirichlet problem dissipative domain eigenvalue elliptic equations energy estimate Evans function exists a constant Hence holds implies inequality initial data integro-differential large solutions Lemma linear Lipschitz continuous main result mapping Math maximum principle metasolution monotone Moreover mountain-pass Navier-Stokes equations nonlinear nonlinear Schrodinger equation nonnegative norm obtain paper parabolic positive constant positive solutions proof of Theorem Proposition radial Remark respectively satisfies Section semilinear sequence Sobolev space solutions of 1.1 space stability subset subsolution super solution symmetric Theorem 1.1 traveling wave solutions uniformly vector viscosity solutions vortex rings wave equation well-posedness