On the behavior of the solution of a boundary value problem when $t\to\infty$

This paper investigates the solution of the boundary value problem $\partial\Delta u/\partial t+\partial u/\partial x=f(x,y)$, $u(x,y,0)=u_0(x,y)$, $u\mid_\Gamma=0$ for the rectangle $0<x<a$, $0<y<b$. It is shown that everywhere outside of neighborhoods of the boundaries $y=0$, $y=b$ and $x=a$ the solution converges uniformly to $-\int_x^a f(\xi,y)\,d\xi$ as $t\to\infty$. Near the indicated boundaries there are boundary layers of width $t^{-1/2}$ and $t^{-1}$ respectively. Explicit formulas are given for the first term of an asymptotic expansion of the solution in each of these boundary layers.
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