Question

Find a formula for the solution of the initial value problem for for t>0, -oc < x < oo ut = uzz-u a(1:0) = g(z) -x < 1 < x where g is continuous and bounded.( Hint: use v(x, t) = et u(z. t).)

Answer 1

Every solution of the wave equation utt = c2uxx has the form u(x,t) = F(x−ct) + G(x + ct) for some functions F,G. In particular, the initial value problem on the real line
utt = c2uxx, u(x,0) = ϕ(x), ut(x,0) = ψ(x) (5.1)
has a unique solution which is given by d’Alembert’s formula
u(x,t) =
ϕ(x + ct) + ϕ(x−ct) 2
+
1 2cZx+ct x−ct
ψ(s)ds.

. Let u1 be the unique solution of the Cauchy problem with initial data ϕ1,ψ1 and let u2 be the unique solution with initial data ϕ2,ψ2. Then the difference of these two solutions satisfies the estimate |u1(x,t)−u2(x,t)|≤||ϕ1 −ϕ2||∞ + t·||ψ1 −ψ2||∞ at all points (x,t). Thus, the Cauchy problem (5.1) is well-posed on [0,T] for any T > 0.