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
UR - http://www.scopus.com/inward/record.url?scp=85034246618&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85034246618&partnerID=8YFLogxK
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 -