Some effects of the noise intensity upon non-linear stochastic heat equations on [0,1][0,1]

Various effects of the noise intensity upon the solution u(t,x)u(t,x) of the stochastic heat equation with Dirichlet boundary conditions on [0,1][0,1] are investigated. We show that for small noise intensity, the ppth moment of supx∈[0,1]|u(t,x)|supx∈[0,1]|u(t,x)| is exponentially stable, however, for large one, it grows at least exponentially. We also prove that the noise excitation of the ppth energy of u(t,x)u(t,x) is 44, as the noise intensity goes to infinity. We formulate a common method to investigate the lower bounds of the above two different behaviors for large noise intensity, which are hard parts in Foondun and Joseph (2014), Foondun and Nualart (2015) and Khoshnevisan and Kim (2015).