Semilinear parabolic equation in $ R^{N} $ associated with critical Sobolev exponent

Ryo Ikehata; Michinori Ishiwata; Takashi Suzuki

doi:10.1016/j.anihpc.2010.01.002

JournalsaihpcVol. 27, No. 3pp. 877–900

Semilinear parabolic equation in $R^{N}$ associated with critical Sobolev exponent

Ryo Ikehata
Department of Mathematics, Faculty of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan
Michinori Ishiwata
Department of Mathematical Science, Common Subject Division, Muroran Institute of Technology, Muroran 050-8585, Japan
Takashi Suzuki
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka 560-0043, Japan

$Semilinear parabolic equation in $ R^{N} $ associated with critical Sobolev exponent cover$

Download PDF

Abstract

We consider the semilinear parabolic equation $u_{t} - Δ u = ∣ u ∣^{p - 1} u$ on the whole space $R^{N}$ , $N ⩾ 3$ , where the exponent $p = (N + 2) / (N - 2)$ is associated with the Sobolev imbedding $H^{1} (R^{N}) \subset L^{p + 1} (R^{N})$ . 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.

Cite this article

Ryo Ikehata, Michinori Ishiwata, Takashi Suzuki, Semilinear parabolic equation in $R^{N}$ associated with critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 3, pp. 877–900

DOI 10.1016/J.ANIHPC.2010.01.002