Attempting PDE with inequality in boundary condition

I am attempting to solve the equation below, which requires v[y,0]==0 for all y greater than 0. I have followed the /; approach to enforce this definition, as outlined in the DSolve/Delay Differential Equations page in the language documentation.

DSolve[{Derivative[2,0][v][y,t]==Derivative[0,1][v][y,t],v[0,t]==1,v[y/;y>0]==0},v[y,t],{y,t}]

Wolfram simply outputs the cell back without an error message. I believe that my inequality restriction on y in the second boundary condition is the issue, but I am not sure how to proceed.

• I want to solve on {y, 0, Infinity}. The problem is startup of Couette flow. – eoncarlyle Feb 21 '19 at 16:29
• Do you want an analytical solution or a numerical one? – Alex Trounev Feb 21 '19 at 16:31
• I want an analytical solution, which I am confident exists – eoncarlyle Feb 21 '19 at 16:34
• The solution exists on the half-line. See answer. – Alex Trounev Feb 21 '19 at 17:13

sol= DSolve[{Derivative[2, 0][v][y, t] == Derivative[0, 1][v][y, t],