QUALITATIVE PROPERTIES OF WEAK SOLUTIONS FOR p-LAPLACIAN EQUATIONS WITH NONLOCAL SOURCE AND GRADIENT ABSORPTION

We consider the following Dirichlet initial boundary value problem with a gradient absorption and a nonlocal source partial derivative u/partial derivative t - div(vertical bar del u vertical bar(p-2)del u) = lambda u(k) integral(Omega) u(s)dx - mu u(l)vertical bar del u vertical bar(q) in a bounded domain Omega subset of R-N, where p > 1, the parameters k, s, l, q, lambda > 0 and mu >= 0. Firstly, we establish local existence for weak solutions; the aim of this part is to prove a crucial priori estimate on vertical bar del u vertical bar. Then, we give appropriate conditions in order to have existence and uniqueness or nonexistence of a global solution in time. Finally, depending on the choices of the initial data, ranges of the coefficients and exponents and measure of the domain, we show that the non-negative global weak solution, when it exists, must extinct after a finite time.