Abstract
We consider reaction diffusion equations of the form (*) ∂ t u = νΔu + ζ u + \( \varsigma u + \mathcal{P}\left( u \right),\mathcal{P}\left( u \right) = \sum _z^m a_k u^k \) and seek solutions on ℝn which are almost periodic in the space variables x. Such solutions are constructed in the space H 0(ℝn) of almost periodic functions f(x) subject to (**) \( f\left( x \right) = \sum f_k e^{i\nabla _k x} ,\sum \left| {fk} \right| < \infty \) , provided that the coefficients a k in (*) are also in this class. Such solutions are obtained via an instable manifold construction, which yields solutions on t ∈ (− ∞, 0] of slow exponential decay. An extension of the method to Fourier transforms of complex measures is outlined.
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References
H. Bohr: Almost periodic functions. Chelsea Publ. Co. (1947).
P. Collet: Nonlinear dynamics of extended systems. In: J.C. Robinson, P.A. Glendinning (eds.), From finite to infinite dynamical systems, Kluwer Ac. Publ. (2001).
C. Corduneanu: Almost periodic functions. Wiley, New York (1968).
D. Henri: Geometric theory of semilinear parabolic equations. Lecture Notes in Math. 840, Springer, New York (1980).
E. Hewitt, K. Ross: Abstract harmonic analysis I. Springer, Berlin, Heidelberg(1963).
K. Kirchgässner: Preference in pattern and cellular bifurcation in fluid dynamics. In: Applications of bifurcation theory, P. Rabinowitz (ed.), Academic Press, New York (1977).
G. Ladas, V. Laksmithkantam: Differential equations in abstract spaces. Academic Press, New York, London (1972).
B. Levitan, V. Zhikov: Almost periodic functions and differential equations. Cambridge Univ. Press, New York (1982).
B. Scarpellini, P. Vuillermot: Smooth manifolds for semilinear wave equations: on the existence of almost periodic breathers. J. Diff. Eq., Vol. 77, No. 1, 123–166 (1989).
B. Scarpellini: Solutions of evolution equations of slow exponential decay. “Analysis” 20, (2000), 255–283.
B. Scarpellini: Instable solutions of nonlinear parabolic equations in R 3. J. Diff. Eq., Vol. 70, (1987), 197–225.
B. Scarpellini, Stability, instability and direct integrals. Chapman & Hall/CRC, Boca Raton, (1999).
M. Shubin: Almost periodic functions and partial differential operators. Russian Math. Survey 33:2, (1978), 1–52.
A. Pazy: Semigroups of linear operators and applications to PDE’s. Appl. Math. Sci. 44, Springer, New York (1983).
V. Stepanov: Über einige Verallgemeinerungen der fast periodischen Funktionen. Math. Annalen 95, 437–498 (1926).
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© 2007 Birkhäuser Verlag Basel/Switzerland
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Scarpellini, B. (2007). Space Almost Periodic Solutions of Reaction Diffusion Equations. In: Amann, H., Arendt, W., Hieber, M., Neubrander, F.M., Nicaise, S., von Below, J. (eds) Functional Analysis and Evolution Equations. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7794-6_35
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DOI: https://doi.org/10.1007/978-3-7643-7794-6_35
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