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The reaction-diffusion equation with Lewis function and critical Sobolev exponent. (English) Zbl 1015.35017
Consider the equation $a\left(x\right){u}_{t}-{\Delta }u={|u|}^{p-1}u$, $x\in {\Omega }$, $t>0$, complemented by the homogeneous Dirichlet boundary condition and the initial condition $u\left(·,0\right)={u}_{0}$. Here ${\Omega }$ is a smoothly bounded domain in ${ℝ}^{N}$, $N\ge 3$, $a\in {L}^{\infty }\left({\Omega }\right)$ is nonnegative, $a¬\equiv 0$, and $p={2}^{*}-1$, where ${2}^{*}=2N/\left(N-2\right)$ is the critical Sobolev exponent. Let $E$ denote the corresponding energy functional and ${\Sigma }=\left\{u\in {H}_{0}^{1}\left({\Omega }\right)$; $u\ge 0$, $E\left(u\right)<{S}^{N}/N\right\}$, where $S=min\left\{\parallel \nabla u{\parallel }_{2}$; $u\in {H}^{1}\left({ℝ}^{N}\right)$, ${\parallel u\parallel }_{{2}^{*}}=1\right\}$ and ${\parallel ·\parallel }_{q}$ denotes the norm in ${L}^{q}\left({ℝ}^{N}\right)$. The author shows that the solution $u$ is global and decays to zero exponentially fast if ${u}_{0}\in {\Sigma }$, ${\int }_{{\Omega }}{|{u}_{0}|}^{{2}^{*}}dx<{S}^{N}$, while it blows up in finite time if either ${u}_{0}¬\equiv 0$, $E\left({u}_{0}\right)\le 0$, or ${u}_{0}\in {\Sigma }$, ${\int }_{{\Omega }}{|{u}_{0}|}^{{2}^{*}}dx\ge {S}^{N}$. The proof of blow-up is based on the classical concavity argument of H. A. Levine [Arch. Ration. Mech. Anal. 51, 371-386 (1973; Zbl 0278.35052)]. The condition $E\left(u\right)<{S}^{N}/N$ excludes the most interesting case of threshold solutions lying on the borderline between global existence and blow-up. It is known that these solutions may be global but unbounded. The paper also contains some general convergence results for global solutions (Theorems 1.4 and 1.5) but these results are obviously incorrect.
##### MSC:
 35B40 Asymptotic behavior of solutions of PDE 35B33 Critical exponents (PDE) 35K60 Nonlinear initial value problems for linear parabolic equations