The global solution of a diffusion equation with nonlinear gradient term

Consider the viscosity solution to the initial boundary value problem of the diffusion equation u(t) = div(vertical bar del u(m)vertical bar(p-2)del u(m)) - u(q1m)vertical bar del u(m)vertical bar(p1), with p > 1, m > 0, p(1) <= 2, p > 2p(1), its initial value u(x, 0) = u(0)(x) is an element of Lq-1+1/m (Omega), 3 > q > 1 and its boundary value u(x, t) = 0, (x, t) is an element of partial derivative Omega x (0, infinity). If p > 1+ 1/m, by considering the regularized problem and using Moser's iteration technique, we get the locally uniformly bounded property of the solution and the locally bounded property of the L-p-norm of the gradient. By the compactness theorem, the existence of the viscosity solution of the equation is obtained provided that mNq(1)/Nm(p - 1) - N + mq + p1(m(p - 1) + m - 2)/m(p - 1) - 1 < 1. If 2 < p < 1+ 1/m, the existence of solution is obtained in a similar way, and the extinction of the solution is proved in this case.