Optimal growth rates for a viscous Hamilton-Jacobi equation

The growth of the L-m-norm, m is an element of [1, infinity], of non-negative solutions to the Cauchy problem partial derivative(t)u - Delta u = vertical bar del u vertical bar I is studied for non-negative initial data decaying at infinity. More precisely, the function t -> t(-Nm) parallel to u(t)parallel to m is shown to be bounded from above and from below by positive real numbers. This result indicates an asymptotic behaviour dominated by the hyperbolic Hamilton-Jacobi term of the equation. A one-sided estimate for Delta ln u is also established.