## Global existence via singular perturbations for quasilinear evolution equations.(English)Zbl 0868.35008

The paper deals with the initial value problems $\varepsilon u_{tt}+ u_t- \sum^n_{i,j=1} a_{ij}(Du)D_iD_ju= f,\;x\in\mathbb{R}^n,\;u(0,x)=u_0\quad\text{for }\varepsilon\geq 0,\;u_t(0,x)= u_1\quad\text{for }\varepsilon>0.$ The global in time smooth solutions to the above problems are established and the estimates of the difference $$u^\varepsilon- u^0$$, which give the singular convergence of the solutions if $$\varepsilon$$ tends to 0, are proved.

### MSC:

 35B25 Singular perturbations in context of PDEs 35L70 Second-order nonlinear hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs

### Keywords:

quasilinear wave equation; singular convergence