Decay of waves IIb: Minkowski space, with right-hand side

In the first half of this second part of the series, we considered solutions to the linear, homogeneous wave equation on flat Minkowski space, and showed that for compactly supported initial data, we have strong Huygens’ principle. We further made references to the fact that this behaviour is expected to be unstable. In this post, we will further illustrate this instability by looking at Equation 1 first with a fixed source F = F(t,x), and then with a nonlinearity F = F(t,x, \phi, \partial\phi).

Duhamel’s Principle

To study how one can incorporate inhomogeneous terms into a linear equation, and to get a qualitative grasp of how the source term contributes to the solution, we need to discuss the abstract method known as Duhamel’s Principle. We start by illustrating this for a very simple ordinary differential equation.

Consider the ODE satisfied by a scalar function \alpha:

Equation 13
\displaystyle \frac{d}{ds}\alpha(s) = k(s)\alpha(s) + \beta(s)

when \beta\equiv 0, we can easily solve the equation with integration factors

\displaystyle \alpha(s) = \alpha(0) e^{\int_0^s k(t) dt}

Using this as a sort of an ansatz, we can solve the inhomogeneous equation as follows. For convenience we denote by K(s) = \int_0^s k(t) dt the anti-derivative of k. Then multiplying Equation 13 through by \exp -K(s), we have that

Equation 14
\displaystyle \frac{d}{ds} \left( e^{-K(s)}\alpha(s)\right) = e^{-K(s)}\beta(s)

which we solve by integrating

Equation 15
\displaystyle \alpha(s) = e^{K(s)}\alpha(0) + e^{K(s)} \int_0^s e^{-K(t)}\beta(t) dt

If we write K(s;t) = \int_t^s k(u) du, then we can rewrite Equation 15 as given by an integral operator

Equation 15′
\displaystyle \alpha(s) = e^{K(s)}\alpha(0) + \int_0^s e^{K(s;t)}\beta(t) dt

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