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The reaction-diffusion equation with Lewis function and critical Sobolev exponent. (English) Zbl 1015.35017
Consider the equation $a(x)u_t-\Delta u=|u|^{p-1}u$, $x\in\Omega$, $t>0$, complemented by the homogeneous Dirichlet boundary condition and the initial condition $u(\cdot,0)=u_0$. Here $\Omega$ is a smoothly bounded domain in $\bbfR^N$, $N\geq 3$, $a\in L^\infty(\Omega)$ is nonnegative, $a\not\equiv 0$, and $p=2^*-1$, where $2^*=2N/(N-2)$ is the critical Sobolev exponent. Let $E$ denote the corresponding energy functional and $\Sigma=\{u\in H^1_0(\Omega)$; $u\geq 0$, $E(u)<S^N/N\}$, where $S=\min\{\|\nabla u\|_2$; $u\in H^1(\bbfR^N)$, $\|u\|_{2^*}=1\}$ and $\|\cdot\|_q$ denotes the norm in $L^q(\bbfR^N)$. 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 $u_0\not\equiv 0$, $E(u_0)\leq 0$, or $u_0\in\Sigma$, $\int_\Omega|u_0|^{2^*} dx\geq S^N$. The proof of blow-up is based on the classical concavity argument of {\it H. A. Levine} [Arch. Ration. Mech. Anal. 51, 371-386 (1973; Zbl 0278.35052)]. The condition $E(u)<S^N/N$ 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
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##### References:
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