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Asymptotic analysis for a nonlinear parabolic equation on

Eva Fašangová — 1998

Commentationes Mathematicae Universitatis Carolinae

We show that nonnegative solutions of u t - u x x + f ( u ) = 0 , x , t > 0 , u = α u ¯ , x , t = 0 , supp u ¯ compact either converge to zero, blow up in L 2 -norm, or converge to the ground state when t , where the latter case is a threshold phenomenon when α > 0 varies. The proof is based on the fact that any bounded trajectory converges to a stationary solution. The function f is typically nonlinear but has a sublinear growth at infinity. We also show that for superlinear f it can happen that solutions converge to zero for any α > 0 , provided supp u ¯ is sufficiently small.

Spectral mapping inclusions for the Phillips functional calculus in Banach spaces and algebras

Eva FašangováPedro J. Miana — 2005

Studia Mathematica

We investigate the weak spectral mapping property (WSMP) μ ̂ ( σ ( A ) ) ¯ = σ ( μ ̂ ( A ) ) , where A is the generator of a ₀-semigroup in a Banach space X, μ is a measure, and μ̂(A) is defined by the Phillips functional calculus. We consider the special case when X is a Banach algebra and the operators e A t , t ≥ 0, are multipliers.

Maximal regularity, the local inverse function theorem, and local well-posedness for the curve shortening flow

Sahbi BoussandelRalph ChillEva Fašangová — 2012

Czechoslovak Mathematical Journal

Local well-posedness of the curve shortening flow, that is, local existence, uniqueness and smooth dependence of solutions on initial data, is proved by applying the Local Inverse Function Theorem and L 2 -maximal regularity results for linear parabolic equations. The application of the Local Inverse Function Theorem leads to a particularly short proof which gives in addition the space-time regularity of the solutions. The method may be applied to general nonlinear evolution equations, but is presented...

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