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A linear growth-diffusion equation is studied in a time-dependent intervalwhose location and length both vary. We prove conditions on the boundary motionfor which the solution can be found in exact form, and derive the explicitexpression in each case.  Next we prove the precise behaviour near the boundary in a `critical' case:when the endpoints of the interval move in such a way that near the boundarythere is neither exponential growth nor decay, but the solution behaves like apower law with respect to time. The proof uses a subsolution based on the Airyfunction with argument depending on both space and time. Interesting links areobserved between this result and Bramson's logarithmic term in the nonlinearFKPP equation on the real line.  Each of the main theorems is extended to higher dimensions, with acorresponding result on a ball with time-dependent radius.

Exact Solutions and Critical Behaviour for a Linear Growth-Diffusion Equation on a Time-Dependent Domain