Tan, Zhong The reaction-diffusion equation with Lewis function and critical Sobolev exponent. (English) Zbl 1015.35017 J. Math. Anal. Appl. 272, No. 2, 480-495 (2002). 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 \(\mathbb{R}^N\), \(N\geq 3\), \(a\in L^\infty(\Omega)\) 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 \(S=\min\{\|\nabla u\|_2\); \(u\in H^1(\mathbb{R}^N)\), \(\|u\|_{2^*}=1\}\) and \(\|\cdot\|_q\) denotes the norm in \(L^q(\mathbb{R}^N)\). 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 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. Reviewer: Pavol Quittner (Bratislava) Cited in 6 Documents MSC: 35B40 Asymptotic behavior of solutions to PDEs 35B33 Critical exponents in context of PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations Keywords:nonlinear parabolic equation; global existence; blow-up; homogeneous Dirichlet boundary condition Citations:Zbl 0278.35052 PDFBibTeX XMLCite \textit{Z. Tan}, J. Math. Anal. Appl. 272, No. 2, 480--495 (2002; Zbl 1015.35017) Full Text: DOI References: [1] Cazenave, T.; Lions, P. L., Solutions globales d’equations de la chaleur semilineaires, Comm. Partial Differential Equations, 9, 955-978 (1984) · Zbl 0555.35067 [2] Fila, M., Boundedness of global solutions of nonlinear diffusion equations, J. Differential Equations, 98, 226-240 (1992) · Zbl 0764.35010 [3] Fujita, H., On the blowing up of solutions of the Cauchy problem for \(u_t= Δu +u^{1+α} \), J. Fac. Sci. Univ. Tokyo Sect. 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