On solution of linear first order pde

Please help me in this problem. Show that the solution of the following problem in $\mathbb{R}^2$ is $U$ identically zero.

$aU_x+ bU_y = -U$ where $U$ tends to zero as $x^2 + y^2$ tends to infinity.

• Hint: what happens at local maximum or minimum points of $U$? – user360874 Oct 17 '17 at 14:09
• U is zero at local maximum/ minimum points but how can we conclude usin this and the given condition that U is entirely zero? – Jem Y Oct 17 '17 at 14:39
• If $U$ were defined on a compact domain, and $U= 0$ on the boundary, we could say $U\equiv 0$ because if it weren't, it would have a nonzero interior max or min point, leading to a contradiction. For the given problem, you can take a large ball outside which $|U|\leq \epsilon$, argue similarly to show $|U|\leq\epsilon$ inside the ball, and send $\epsilon \to 0$. – user360874 Oct 17 '17 at 16:42
• I see it now. Thank you. – Jem Y Oct 17 '17 at 19:19