Test 2
Math 490
A. Consider a reaction diffusion equation
∂u
∂t =∂2u
∂x2+λu(1 −u)−au
1 + u, t > 0, x ∈(0,1),
u(t, 0) = 0, u(t, 1) = 0,
u(0, x) = u0(x), x ∈(0,1),
(1)
where λ > 0 and a > 0.
1. u(t, x) = 0 is an equilibrium solution. Consider
(φ00(x) + λf 0(0)φ(x) = µφ(x), x ∈(0,1),
φ(0) = φ(1) = 0,(2)
where f(u) = u(1 −u)−au
1 + u. Determine for which (λ, a), u= 0 is asymptotically stable,
and for which (λ, a), u= 0 is unstable.
2. For fixed a > 0, we define a bifurcation point as the value of λwhere u= 0 changes from
stable to unstable. From part 1, determine the bifurcation point when a=a0(a positive
number). (Note that for some a, there is no bifurcation point.)
3. When a= 0.5, the bifurcation point is λ= 2π2. Near the bifurcation point, if ε=λ−2π2,
then u(ε, x) = εu1(x) + ε2u2(x) + · · · . Find u1(x). (Hint: in expansion of au/(1 + u), you
should use the Taylor expansion: 1
1 + u= 1 −u+u2−u3+· · · .)
B. Suppose that u(x) is a positive solution of
(u00 +u2= 0, x ∈(0,1),
u(0) = u(1) = 0.(3)
Prove that u(x) is unstable. (Hint: use the idea in proof of Proposition 3.9.)
C. Consider a reaction diffusion equation
∂u
∂t =∂2u
∂x2+λu(3 −u)−au
1 + u, t > 0, x ∈(0,1),
u(t, 0) = 0, u(t, 1) = 0,
u(0, x) = u0(x), x ∈(0,1),
(4)
where λ > 0 and a > 0.
1. When a= 3.5, use Maple to draw the global bifurcation diagram of the positive equilibrium
solutions. What is the smallest value of umax =u(1/2)? And what is the smallest λsuch
that there is a positive equilibrium solution?
