Why the solution of this problem can be written in this way?

Question

Consider this kind of problem

$$ v_{tt} + \Delta v + v_{t} = 0, $$

with $ x\in R^{n}, t\geq 0 $ and initial data

\begin{equation} \left\lbrace \begin{array}{ll} v(x,0) = \phi(x) \\ v_{t}(x,0) = \psi(x) \end{array} \right. \end{equation}

I read that the solution can be written in this way:

$$ v(x,t) = K_1\ast \psi + K_2\ast \phi. $$ Could anyone explain me why it is true?

Answer 1

This is a consequence of the principle of superposition.

If the initial conditions are

$$\begin{equation} \left\lbrace \begin{array}{ll} v(x,0) = \delta(x-x_0) \\ v_{t}(x,0) = 0 \end{array} \right. \end{equation}$$

or

$$\begin{equation} \left\lbrace \begin{array}{ll} v(x,0) = 0 \\ v_{t}(x,0) = \delta(x-x_0) \end{array} \right. \end{equation}$$

you get some solutions

$$v_\phi(x,t;x_0)$$ and $$v_\psi(x,t;x_0).$$

Then to get the solution corresponding to the general initial conditions, we convolve with $\phi$ and $\psi$ (the convolution is a sum of elementary solutions), and add both.

$$v(x,t)=v_\phi(x,t;x_0)\star\phi(x_0)+v_\psi(x,t;x_0)\star\psi(x_0).$$

The integration is made on $x_0$.