Abstract.
The initial value problem for the conservation law \(\partial_t u + (-\Delta)^{\alpha/2}u + \nabla \cdot f(u) = 0\) is studied for \(\alpha \in (1, 2)\) and under natural polynomial growth conditions imposed on the nonlinearity. We find the asymptotic expansion as \(\mid x \mid \rightarrow \infty\) of solutions to this equation corresponding to initial conditions, decaying sufficiently fast at infinity.
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The preparation of this paper was supported in part by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389, and in part by the Polonium Project PAI EGIDE N. 09361TG. The first author gratefully thanks the Mathematical Institut of Wrocław University for the warm hospitality. The preparation of this paper by the second author was also partially supported by the grant N201 022 32 / 09 02.
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Brandolese, L., Karch, G. Far field asymptotics of solutions to convection equation with anomalous diffusion. J. evol. equ. 8, 307–326 (2008). https://doi.org/10.1007/s00028-008-0356-9
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DOI: https://doi.org/10.1007/s00028-008-0356-9