Abstract
We show the local well-posedness of the Cauchy problem to a nonlinear heat equation of Fujita type in lower space dimensions. It is well known that the nonnegative solution corresponding to the Fujita critical exponent \(p=1+\frac{2}{n}\) does not exist in the critical scaling invariant space \(L^1(\mathbb R^n)\). We show if the initial data is in a modified Besov spaces, then the corresponding mild solution to the equation with the Fujita critical exponent \(p=1+\frac{2}{n}\) exists and the problem is locally well-posed in the same space of the initial data. Besides we also show the problem is ill-posed in the scaling invariant Besov and inhomogeneous Besov spaces. This is known in \(L^1\) space and extension of the result known in the Lebesgue spaces.
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Acknowledgements
The authors thank Professor Kazuhiro Ishige, Professor Tsukasa Iwabuchi, and Dr. Ryuichi Sato for their stimulation discussion on the local well-posedness. The work of Takayoshi Ogawa is partially supported by JSPS grant-in-aid for Scientific Research (S) #25220702.
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Ogawa, T., Yamane, Y. (2017). Local Well-Posedness for the Cauchy Problem to Nonlinear Heat Equations of Fujita Type in Nearly Critical Besov Space. In: Maekawa, Y., Jimbo, S. (eds) Mathematics for Nonlinear Phenomena — Analysis and Computation. MNP2015 2015. Springer Proceedings in Mathematics & Statistics, vol 215. Springer, Cham. https://doi.org/10.1007/978-3-319-66764-5_10
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