Life-span of solutions to nonlinear dissipative evolution equations: A singular perturbation approach

We investigate the large time behavior of solutions to nonlinear dissipative wave equations of the general form εu_u + u_t - Δu = F(x, t, u, D_xu, D_x²u); in particular, we study the dependence of the solutions u = u^ε and of their life span T_ε on the (small) parameter ε. We are interested in the behavior of u^ε and T_ε as ε → 0, and in their relations with the solution v, and its life span T_p, of the corresponding limit equation when ε = 0, which is of parabolic type. We look for conditions under which either T_ε = +∞, or T_ε → T_p < +∞ as ε → 0.