Analysis of Reaction-Diffusion Equations on a Time-Dependent Domain

We consider non-negative solutions to reaction-diffusion equations on time-dependent domains with zero Dirichlet boundary conditions. The reaction term is either linear or else monostable of the KPP type. For a range of different forms of the boundary motion, we use changes of variables, exact solutions, supersolutions and subsolutions to study the long-time behaviour.For a linear equation on (A(t), A(t) + L(t)), we prove that the solution can be found exactly by a separation of variables method when ¨LL3 and ¨AL3 are constants. In these cases L(t) has the form L(t) = √at2 + 2bt + l2. We also analyse the linear problem near the boundary, deriving conditions on L(t) such that the gradient at the boundary remains bounded above and below away from zero. Interesting links are observed between this ‘critical’ boundary motion and Bramson’s logarithmic term for the nonlinear KPP equation.The exact solutions and investigation of behaviour near the boundary are also extended to a ball with time-dependent radius, and a time-dependent box.We then consider time-periodic bounded domains Ω(t). The long-time be-haviour is determined by a principal periodic eigenvalue µ, for which we derive several bounds and also consider the large and small frequency limits. For the nonlinear problem, we prove that the solution converges to either zero or a unique positive periodic solution u∗.The nonlinear problem is also studied on a bounded domain in RN moving at constant velocity c, and we derive several properties of the positive stationary limit Uc.Results describing long-time behaviour for the nonlinear equation are then extended to certain other types of time-dependent domain that have non-constant velocity and non-constant length, and to time-dependent cylinders.