I solved an initial boundary value problem, but I have been stuck trying to prove a certain limit. I will post the problem and my progress so far for completeness.
Problem
Solve the initial boundary value problem for $t>0$:
$$ \begin{array}{ll} 4u_{tt}(x,t)=u_{xx}(x,t)-7u(x,t),&0<x<\pi\\ u_x(0,t)=u_x(\pi,t)=0\\ u(x,0)=4\cos^3x,&0<x<\pi\\ u_t(x,0)=\cos(5x),&0<x<\pi. \end{array} $$
Find $\lim_{t\to\infty} u(x,t)$ as a function of $x$.
Solution to first part
For the solution to the problem, I used separation of variables and expanded $4\cos^3 x$ as $3\cos x+\cos(3x)$. The solution I found was:
$$ u(x,t)=3\cos(\sqrt{2}t)\cos x+\cos(2t)\cos(3x)+\dfrac{1}{2\sqrt{2}}\sin(2\sqrt{2}t)\cos(5x). $$
This satisfies each condition imposed above in the statement of the problem.
Question
How would one find $\lim_{t\to\infty} u(x,t)$? It seems to me that I cannot directly apply the limit to the solution $u(x,t)$ as I wrote above since $u(x,t)$ is a sum of periodic functions in $t$. I'm almost thinking there is no limit unless I'm missing something. Also, is there some physical insight into what is happening in this particular problem?