Consider the 1-dimensional Burger's equation on a finite interval $I=(0,1)$,
$$u\_t+uu\_x=u\_{xx}$$
with initial condition
$$u(x,0)=f(x)$$
and boundary conditions
$$u(0,t)=A(t) \qquad u(1,t)=B(t).$$



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> **QUESTION.** Suppose that $A(t)$ and $B(t)$ are periodic functions, of integer period $m$ and $n$, respectively. Is it true that the solution to the above IBVP should be periodic in time? If so, what is the period in terms of $m$ and $n$?
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Periodicity and Burger's equation
Of course **not**. For instance if $A=B\equiv0$ (these are periodic), then the solution decays to $0$ as $t\to+\infty$.


Instead, and this is classical in dynamical system theory, if $A$ and $B$ are periodic as in your assumption, then you might be able to prove that there exists an initial data $f$ for which the solution is periodic with the same period $p:={\rm lcm}(m,n$). And it should be unique, because the solution mapping $S(t):u(0)\mapsto u(t)$ is $L^1$-contracting. The existence relies upon two ingredients: 1) denoting $M=\max\{\|A\|\_\infty,\|B\|\_\infty\}$, the set of measurable functions $a$ such that $|a(x)|\le M$ a.e. is sent into itself by $S(p)$, 2) $S(p)$ is compact in the $L^1$-topology.


