MIXED LOCAL AND NONLOCAL PARABOLIC EQUATION: GLOBAL EXISTENCE, DECAY AND BLOW-UP

In this paper, we use the modified potential well method and the Galerkin method to study the following mixed local and nonlocal parabolic equation: {ut - Delta u + (-Delta)(s )u = |u|(p-2)u in Omega x R+, u(x, 0) = u(0()x) in Omega, u(x, t) = 0 in R-N\Omega x R-0(+), where Delta is the Laplace operator, (-Delta)(s) is the fractional Laplace operator, Omega subset of R-N is a bounded domain with Lipschitz boundary partial derivative Omega, N > 2s, 2 < p <= 2* and s is an element of (0, 1). In the cases of low and critical initial energy, we not only prove the existence of global solutions and the decay rate of the L-2 norm for global solutions, but also obtain blow-up of solutions in finite time and the lower and upper bounds of the blow-up time. In the case of high initial energy, we give sufficient conditions for the global existence and blow-up of solutions, and the lower and upper bounds on the blow-up time.