Existence and asymptotic behavior for $$L^2$$-norm preserving nonlinear heat equations

Abstract

We consider a nonlinear parabolic equation with a nonlocal term which preserves the $$L^2$$ L 2 -norm of the solution. We study the local and global well-posedness on a bounded domain, as well as the whole Euclidean space, in $$H^1$$ H 1 . Then we study the asymptotic behavior of solutions. In general, we obtain weak convergence in $$H^1$$ H 1 to a stationary state. For a ball, we prove strong convergence to the ground state when the initial condition is positive.