A mathematical study of diffusive logistic equations with mixed type boundary conditions

Figure 1.  The bounded domain $ D $ and the unit outward normal $ \mathbf{n} $ to $ \partial D $

Figure 2.  The boundary portion $ M $ is deadly and its complement $ {\partial D} \setminus M $ is a barrier

Figure 3.  The structural condition (Z.1) on the function $ h(x) $

Figure 4.  The bifurcation diagram of Theorem 1.5

Figure 5.  The bifurcation diagram of Theorem 1.5: Malthus versus Verhulst

Figure 6.  The positive solution curve (16) for $ \lambda > \lambda_{1}(m) $ under condition (Z.3) via the Semenov approximation

Figure 7.  The bifurcation diagram of Remark 1.4 under condition (Z.3) (Verhulst theory)

Figure 8.  Conditions (b) and (d) in Theorem 3.3

Figure 9.  The bifurcation curves $ \varGamma_{1} $ and $ \varGamma_{2} $ of the nonlinear equation (20) in Theorem 3.3

Figure 10.  The point $ \left(1/ \mathop{\mathrm{spr}}(B), 0\right) $ is a bifurcation point of the nonlinear equation (21) to the trivial solution in Theorem 3.4

Figure 11.  The mapping properties of the resolvent $ R_{c} $ in the spaces $ C(\overline{D}) $, $ W^{2,p}(D) $ and $ C^{1}_{B}(\overline{D}) $

Figure 12.  The mapping properties of the resolvent $ R_{c} = \left(- \varDelta + c(x)\right)^{-1} $ in the spaces $ C(\overline{D}) $, $ C_{e}(\overline{D}) $ and $ C^{1}_{B}(\overline{D}) $

Figure 13.  The first eigenvalues $ \mu_{1}(\lambda) = \gamma_{1}(\lambda) - \lambda $, $ \mu_{1}(0) = \gamma_{1} $ and $ \mu_{1}\left(\lambda_{1}(m)\right) = 0 $

Figure 14.  The first eigenvalues $ \mu_{D}(\lambda) $, $ \mu_{N}(\lambda) $ and $ \mu_{1}(\lambda) $ in the case $ \int_{D}m(x)\,dx < 0 $

Figure 15.  The first eigenvalues $ \mu_{D}(\lambda) $, $ \mu_{N}(\lambda) $ and $ \mu_{1}(\lambda) $ in the case $ \int_{D}m(x)\,dx = 0 $

Figure 16.  The first eigenvalues $ \mu_{D}(\lambda) $, $ \mu_{N}(\lambda) $ and $ \mu_{1}(\lambda) $ in the case $ \int_{D}m(x)\,dx > 0 $

Figure 17.  A flowchart of proof of Theorem 1.5, part (i)

Figure 18.  A flowchart of proof of Lemma 7.2

Figure 19.  The set of solutions of the semilinear problem (1) consists of a pitchfork near $ \lambda = \lambda_{1}(m) $

Figure 20.  The critical value $ \overline{\lambda}(h) $ of the positive bifurcation solution curve $ \mathcal{C} = \{(\lambda, u(\lambda))\} $

Figure 21.  The mapping properties of the resolvent $ R_{\lambda} = (\lambda I - \varDelta)^{-1} $ in the spaces $ C(\overline{D}) $, $ C_{e}(\overline{D}) $ and $ C^{1}_{B}(\overline{D}) $

Figure 22.  A positive bifurcation solution curve $ (\lambda, u(\lambda)) $ of the nonlinear operator equation $ u = H(\lambda,u) $ can be continued beyond the point $ (\lambda^{\ast}, u^{\ast}) $ via the implicit function theorem (Theorem 3.1)

Figure 23.  The mapping properties of the negative Laplacian $ - \varDelta $ and the resolvent $ R_{0} = \left(- \varDelta\right)^{-1} $

Figure 24.  A flowchart of proof of Theorem 1.5, part (ii)

Figure 25.  The bifurcation diagram of Theorem 8.1 (the Dirichlet case)

Figure 26.  The bifurcation diagram of Theorem 8.2 in the case $ \int_{D}m(x)\,dx < 0 $ and $ \nu_{1}(m) > 0 $ (the Neumann case)

Figure 27.  The bifurcation diagram of Theorem 8.2 in the case $ \int_{D}m(x)\,dx = 0 $ and $ \nu_{1}(m) = 0 $ (the Neumann case)

Figure 28.  The bifurcation diagram of Theorem 8.2 in the case $ \int_{D}m(x)\,dx > 0 $ and $ \nu_{1}(m) < 0 $ (the Neumann case)

Figure 30.  The bifurcation diagrams of Theorem 8.3 in the case $ \int_{D}m(x)\,dx < 0 $ and $ \nu_{1}(m) > 0 $

Figure 31.  The bifurcation diagrams of Theorem 8.3 in the case $ \int_{D}m(x)\,dx = 0 $ and $ \nu_{1}(m) = 0 $

Figure 32.  The bifurcation diagrams of Theorem 8.3 in the case $ \int_{D}m(x)\,dx > 0 $ and $ \nu_{1}(m) < 0 $

Figure 29.  The open subset $ D^{+} $ with boundary $ \partial D^{+} $