Let us consider the following linear system with damping: $$ \begin{cases} u_t - u_x = -\frac{1}{2} v\\ v_t + v_x = -\frac{1}{2} v \end{cases} $$ Let's write the solution as $w=(u,v)$ corresponding to initial data $w_0 := w(0,\cdot) \in L^1(\mathbb R) \cap L^2(\mathbb R)$. Can we prove that $$w=w_1+w_2$$ where $$\|w_1\|_{L^2} \lesssim e^{-\alpha t}\|w_0\|_{L^2}, \qquad \|w_2\|_{L^\infty} \lesssim t^{-1/2}\|w_0\|_{L^1}$$ or something similar? Does more hold?

Let us consider the following linear system with damping: $$ \begin{cases} u_t - u_x = -\frac{1}{2} (u+v)\\ v_t + v_x = -\frac{1}{2} (u+v) \end{cases} $$ Let's write the solution as $w=(u,v)$ corresponding to initial data $w_0 := w(0,\cdot) \in L^1(\mathbb R) \cap L^2(\mathbb R)$. Can we prove that $$w=w_1+w_2$$ where $$\|w_1\|_{L^2} \lesssim e^{-\alpha t}\|w_0\|_{L^2}, \qquad \|w_2\|_{L^\infty} \lesssim t^{-1/2}\|w_0\|_{L^1}$$ or something similar? Does more hold?