Solution. \(c'(\lambda ) = 1 - \lambda ^{-2} = 0\) at \(\lambda = 1\), where \(c = 2\); \(c \to \infty \) at both ends. In physical units \(c^* = 2\sqrt {rD} = 2\sqrt {5} = 4.47\) km per year.

Solution. \(c = \sqrt 2(0.5 - 0.25) = 0.354\), with the state \(1\) invading. \(U = 0.9\) at \(e^{z/\sqrt 2} = \frac 19\), \(z = -\sqrt 2\ln 9\), and \(U = 0.1\) at \(z = \sqrt 2\ln 9\), so the width is \(2\sqrt 2\ln 9 = 6.21\) in scaled units.

Solution. With \(f_u = 0.8\), \(g_v = -1\), \(\det J = 1\), the condition is \(0.8d - 1 > 2\sqrt d\). Put \(s = \sqrt d\): \(0.8s^2 - 2s - 1 > 0\) for \(s > (2 + \sqrt {4 + 3.2})/1.6 = 2.927\), so \(d_c = 8.57\). At threshold the minimum of \(h(q)\) is at \(q = (0.8d_c - 1)/2d_c = 0.342\), so \(k_c = 0.585\), consistent with the narrow band near \(k = 0.56\) found at \(d = 10\).

Solution. If \(d = 1\) then \(J_k = J - |k|^2I\), whose eigenvalues are those of \(J\) shifted by \(-|k|^2\). Since those of \(J\) have negative real parts, so do all those of \(J_k\). Equivalently, the condition of Theorem 14.3.1 would need \(f_u + g_v > 0\), contradicting \(\operatorname {tr}J < 0\).

Solution. The source is a single eigenfunction, so try \(u = T(t)\sin x\): \(T' = -T + e^{-t}\), \(T(0) = 0\), giving \(T = te^{-t}\). The forcing decays at exactly the rate of the mode it drives, a resonance, which produces the factor \(t\). So \(u = te^{-t}\sin x\).

Solution. (a) Not well posed: Hadamard’s example (Section 3) has data tending to zero and solutions blowing up. (b) Well posed: existence by Fourier series, uniqueness and stability by the maximum principle. (c) Not well posed: \(u = \sin \pi x\sin \pi t\) vanishes on all four sides but is not zero, so solutions are not unique; boundary data on a closed curve are the wrong data for a hyperbolic equation.