Suppose $b\in L^\infty(\mathbb{R})$ and $u\colon \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, where $(t,x)\mapsto u(t,x)$ is a solution to $$
\partial_t^2 u = \partial_x^2 u +b(x)\partial_x u,\quad (x,t)\in \mathbb{R}\times \mathbb{R}
$$ with initial conditions $u(0,\cdot)=u_0$ and $\partial_t u(0,\cdot)=u_1$.
Do we then have an $L^p$-estimate of the form
$$
\|u(t,\cdot)\|_{L^p(\mathbb{R})}\leq c_p(|t|)\bigg(\|u_0\|_{L^p(\mathbb{R})}+\|u_1\|_{L^p(\mathbb{R})}\bigg)
$$ for all $t\in \mathbb{R}$ and $p\in [1,\infty)$ for some function $c_p\colon [0,\infty)\to[0,\infty)$ which is of polynomial growth?
If $b=0$, then by D'Alembert formula, we can choose $c_p(s)=1+s$. If $b\neq 0$ is constant, then one can find an explicit representation of the solution by means of the Fourier transform. I have not worked out the details, but I suspect that it is true in this case. Can we say anything in case of general $b\in L^\infty(\mathbb{R})?$
I would appreciate any hints. Thanks in advance!