Abstract
We consider the Cauchy problem for the one-dimensional parabolic equations
, with initial data in \(H^s (\mathbb{R}).\). We study the flow map corresponding to the integral equation. Our results complete the known results on ill-posedness in \(H^s (\mathbb{R}).\)and show the particularity of the case (k, d) = (2, 0) for which we prove that the critical space \(H^{s_c } (\mathbb{R}) = H^{ - 3/2} (\mathbb{R})\) suggesting by standard scaling arguments cannot be reached. Our results hold also in the periodic setting.
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Molinet, L., Ribaud, F., Youssfi, A. (2003). On the Flow Map for a Class of Parabolic Equations. In: Haroske, D., Runst, T., Schmeisser, HJ. (eds) Function Spaces, Differential Operators and Nonlinear Analysis. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8035-0_28
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DOI: https://doi.org/10.1007/978-3-0348-8035-0_28
Publisher Name: Birkhäuser, Basel
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