On the existence and nonexistence of global solutions of semilinear parabolic equations with slowly decaying initial data
The author discusses the existence and nonexistence of global solutions of \[ u_t= \Delta u + |x |^{\sigma}u^p \text{ in }D \times (0, T), \qquad u(x, t) = 0 \text{ on }\partial D \times (0, T), \qquad u(x, 0) = a(x) \geq 0 \text{ in }D, \] where \(\sigma \geq 0\), \(p >1\), \(D = \{ x \in R^N \setminus\{0\}\); \(x/ |x |\in \Omega \}\), \(\Omega \subseteq S^{N - 1}\) is an open connected subset with smooth boundary, \(a\) is continuous, \( \langle x \rangle^{ \sigma/(p - 1)} a(x)\) is bounded in \(\overline D\), and \(a = 0\) on \(\partial D\) \((\langle x \rangle = \sqrt{1 + |x |^2})\). Let \(\{ \psi_n \}_{n \geq 1}\) be a normalized orthogonal system for the Laplace-Beltrami operator in \(\Omega\) corresponding to \(\{ \omega_n \}_{n \geq 1}\), the Dirichlet eigenvalues for this problem \((\psi_1 > 0\) in \( \Omega)\). Let \(\gamma_+\) be the positive root of \( \gamma(\gamma + N -2) = \omega_1\), and let \(M(\alpha)\) be such that i) if \( \alpha < (2 + \sigma) / (p -1)\), then \(M(\alpha)\) is any positive real, ii) if \( \alpha = (2 + \sigma) / (p -1)\), then \[ M(\alpha)= \frac{2^{\gamma_+ + (3\alpha)/2} \exp(1/4)\Gamma(N/2 + \gamma_+)\int_{\Omega} \psi_{1} (\theta)dS_{\theta}} {(p-1)^{1/(p-1) \Gamma((N + \gamma_+ - \alpha)/2) \int_{\Omega_1} \psi_{1}^{2}(\theta)dS_{\theta}}} \] where \(\Omega_1 \subseteq \Omega\) and \(|\Omega_1 |\not= 0\). The results of this paper are: 1) if \(0 \leq \alpha \leq \frac{p-1}{N-2}\), \(N \geq 2\), \(|a |_{\sigma/(p-1)} < \infty\) and \(a(x) \geq M \langle x \rangle^{- \alpha} \psi_{1}(x/|x |)\) for \( x \in \{ |x |> 1\) and \(x/|x |\in \Omega_1 \}\), where \(0 \leq \sigma \leq \frac{2 + \alpha} {p-1}\) and \(M > M(\alpha)\), then the solution is not global. 2) if \(\alpha > \frac{2 + \sigma}{p-1}\) and if there is \(m>0\) such that \(0 \leq a(x) \leq m \langle x \rangle^{- \alpha} \psi_{1} (x/|x |)\) for \(x \in D\), then there exists a unique nontrivial global solution.
scientific article; zbMATH DE number 1102903