Global solutions to viscous hamilton-jacobi equations with irregular initial data

摘要

We investigate the existence and uniqueness of non-negative weak solutions to u_t - #DELTA#_u + |nabla u|~p = 0 in (0, +infinity) * R~N with initial data in the space of bounded and non-negative measures when 1 < p < (N + 2)/(N + 1) and in L~1(R~N) intersect L~q (R~N) when (N + 2)/(N + 1) <= p < 2 provided that q is large enough. Non-existence of solutions with Dirac masses as initial data is also shown when p >= (N + 2)/(N + 1). We finally study the large time behaviour of the L~1-norm of the solutions to the Cauchy problem, thereby extending previous results to a wider class of initial data. One main step in our approach is the derivation of decay estimates of the L~(infinity)-norm of the gradient of u~((p - 1)/p).