Life-Span of solutions to nonlinear dissipative evolution equations: a singular perturbation approach

Abstract

We investigate the large time behavior of solutions to nonlinear dissipative wave equations of the general form \[ \varepsilon u_{tt}+u_{t}-\Delta u=F\left(x,t,u,D_{x}u,D_{x}^{2}u\right); \] in particular, we study the dependence of the solutions $u=u^{\varepsilon}$ and of their life span $T_{\varepsilon}$ on the (small'' parameter $\varepsilon$. We are interested in the behavior of $u^{\varepsilon}$ and $T_{\varepsilon}$ as $\varepsilon\rightarrow0$, and in their relations with the solution v, and its life span T$_{p}$ , of the corresponding limit equation when $\varepsilon=0$, which is of parabolic type. We look for conditions under which either $T_{\varepsilon}=+\infty,\: or\: T_{\varepsilon}\rightarrow T_{p}\leq+\infty$ as $\varepsilon=0$.