Abstract
We consider the Cauchy problem for the complex valued semi-linear heat equation
where \(m\ge 2\) is an integer and the initial data belong to super-critical spaces \(E^s_\sigma \) for which the norms are defined by
If \(s<0\), then any Sobolev space \(H^{r}\) is a subspace of \(E^s_\sigma \), i.e., \(\cup _{r \in \mathbb {R}} H^r \subset E^s_\sigma \). We obtain the global existence and uniqueness of the solutions if the initial data belong to \(E^s_\sigma \) (\(s<0, \ \sigma \ge d/2-2/(m-1)\)) and their Fourier transforms are supported in the first octant, the smallness conditions on the initial data in \(E^s_\sigma \) are not required for the global solutions. Moreover, we show that the error between the solution u and the iteration solution \(u^{(j)}\) is \(C^j/(j\,!)^2\). Similar results also hold if the nonlinearity \(u^m\) is replaced by an exponential function \(e^u-1\).
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Notes
\(\chi _E\) denote the characteristic function on E.
Such a kind of initial data have a direct relation with (6) and they are complex-valued.
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Acknowledgements
The second named author is very grateful to Professor Y. Giga for his enlightening discussions and for his valuable suggestions on the paper. Also, he would like to thank the reviewers for their valuable suggestions, which enable us to give an essential improvement to Theorem 2.1. The authors are supported in part by the NSFC, Grants 11771024, 12171007.
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Chen, J., Wang, B. & Wang, Z. Complex valued semi-linear heat equations in super-critical spaces \(E^s_\sigma \). Math. Ann. 386, 1351–1389 (2023). https://doi.org/10.1007/s00208-022-02425-5
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DOI: https://doi.org/10.1007/s00208-022-02425-5