TY - GEN

T1 - Local well-posedness for the cauchy problem to nonlinear heat equations of Fujita type in nearly critical Besov space

AU - Ogawa, Takayoshi

AU - Yamane, Yuuki

N1 - Funding Information:
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.
Publisher Copyright:
© 2017, Springer International Publishing AG.

PY - 2017

Y1 - 2017

N2 - 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+2/n does not exist in the critical scaling invariant space L1(Rn). 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+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 L1 space and extension of the result known in the Lebesgue spaces.

AB - 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+2/n does not exist in the critical scaling invariant space L1(Rn). 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+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 L1 space and extension of the result known in the Lebesgue spaces.

KW - Fujita critical exponent

KW - Local well-posedness

KW - Modified Besov space

KW - Nonlinear heat equation

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U2 - 10.1007/978-3-319-66764-5_10

DO - 10.1007/978-3-319-66764-5_10

M3 - Conference contribution

AN - SCOPUS:85034246618

SN - 9783319667621

T3 - Springer Proceedings in Mathematics and Statistics

SP - 215

EP - 239

BT - Mathematics for Nonlinear Phenomena—Analysis and Computation - In Honor of Yoshikazu Giga’s 60th Birthday

A2 - Maekawa, Yasunori

A2 - Jimbo, Shuichi

PB - Springer New York LLC

T2 - International Conference on Mathematics for Nonlinear Phenomena: Analysis and Computation in Honor of Professor Yoshikazu Giga on his 60th Birthday, MNP 2015

Y2 - 16 August 2015 through 18 August 2015

ER -