Abstract :
In this paper we investigate the existence, multiplicity, and asymptotic behavior of solutions for a reaction diffusion equation under nonlocal boundary and nonlocal initial conditions. A distinctive feature of this type of boundary and initial conditions is that the parabolic equation may possess multiple solutions even if the reaction function is smooth including linear functions. Of special concern is the asymptotic behavior of these multiple solutions in relation to the solutions of the corresponding steady-state problem which may also possess multiple solutions. In some special cases, both the time-dependent and the steady-state problems possess an infinite number of solutions, and each of the time-dependent solutions converges to one of the steady-state solutions.
