The reaction-diffusion equation with Lewis function and critical Sobolev exponent
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 \textit{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.
homogeneous Dirichlet boundary condition
scientific article; zbMATH DE number 1846001
scientific article