Semilinear parabolic equation in ${𝐑}^{N}$ associated with critical Sobolev exponent
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 3, p. 877-900
We consider the semilinear parabolic equation ${u}_{t}-\Delta u={|u|}^{p-1}u$ on the whole space ${𝐑}^{N}$, $N⩾3$, where the exponent $p=\left(N+2\right)/\left(N-2\right)$ is associated with the Sobolev imbedding ${H}^{1}\left({𝐑}^{N}\right)\subset {L}^{p+1}\left({𝐑}^{N}\right)$. First, we study the decay and blow-up of the solution by means of the potential-well and forward self-similar transformation. Then, we discuss blow-up in infinite time and classify the orbit.
DOI : https://doi.org/10.1016/j.anihpc.2010.01.002
Classification:  35K55
Keywords: Parabolic equation, Critical Sobolev exponent, Cauchy problem, Stable and unstable sets, Self-similarity
@article{AIHPC_2010__27_3_877_0,
author = {Ikehata, Ryo and Ishiwata, Michinori and Suzuki, Takashi},
title = {Semilinear parabolic equation in ${\mathbf{R}}^{N}$ associated with critical Sobolev exponent},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {27},
number = {3},
year = {2010},
pages = {877-900},
doi = {10.1016/j.anihpc.2010.01.002},
zbl = {1192.35099},
mrnumber = {2629884},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2010__27_3_877_0}
}

Ikehata, Ryo; Ishiwata, Michinori; Suzuki, Takashi. Semilinear parabolic equation in ${\mathbf{R}}^{N}$ associated with critical Sobolev exponent. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 3, pp. 877-900. doi : 10.1016/j.anihpc.2010.01.002. http://www.numdam.org/item/AIHPC_2010__27_3_877_0/

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