LOCAL WELL-POSEDNESS AND GLOBAL EXISTENCE FOR THE BIHARMONIC HEAT EQUATION WITH EXPONENTIAL NONLINEARITY

In this paper, we prove local well-posedness in Orlicz spaces for the biharmonic heat equation partial derivative(t)u+Delta(2)u = f (u), t > 0, x is an element of R-N, with f (u) similar to e(u2) for large u. Under smallness condition on the initial data and for exponential nonlinearity f such that vertical bar f(u)vertical bar similar to vertical bar u vertical bar(m) as u -> 0, m >= 2, N(m - 1)/4 >= 2, we show that the solution is global. Moreover, we obtain decay estimates for large time for the nonlinear biharmonic heat equation as well as for the nonlinear heat equation. Our results extend to the nonlinear polyharmonic heat equation.