MATHEMATICAL TRIPOS Part II PAPER 3 Before you begin read ...

1730CPartial Differential EquationsDefine the parabolic boundary ∂ par Ω T of the domain Ω T = [0, 1] × (0, T ] for T > 0.Let u = u(x, t) be a smooth real-valued function on Ω T which satisfies the inequalityu t − au xx + bu x + cu 0 .Assume that the coefficients a, b and c are smooth functions and that there exist positiveconstants m, M such that m a M everywhere, and c 0. Prove thatmax u(x, t) max u + (x, t) .(x,t)∈Ω T (x,t)∈∂ parΩ T(∗)[Here u + = max{u, 0} is the positive part of the function u.]Consider a smooth real-valued function φ on Ω T such thatφ t − φ xx − (1 − φ 2 )φ = 0 ,φ(x, 0) = f(x)everywhere, and φ(0, t) = 1 = φ(1, t) for all t 0. Deduce from (∗) that if f(x) 1 forall x ∈ [0, 1] then φ(x, t) 1 for all (x, t) ∈ Ω T . [Hint: Consider u = φ 2 − 1 and computeu t − u xx .]31BAsymptotic MethodsLetI(x) =∫ π0f(t)e ixψ(t) dt ,where f(t) and ψ(t) are smooth, and ψ ′ (t) ≠ 0 for t > 0; also f(0) ≠ 0, ψ(0) = a,ψ ′ (0) = ψ ′′ (0) = 0 and ψ ′′′ (0) = 6b > 0. Show that, as x → +∞,Consider the Bessel function( ) 1 1/3I(x) ∼ f(0)e i(xa+π/6) Γ (1/3) .27bxJ n (x) = 1 π∫ π0cos(nt − x sin t) dt .Show that, as n → +∞,J n (n) ∼ Γ (1/3)π1(48) 1/6 1n 1/3 .Part II, Paper 3[TURN OVER