Global existence via singular perturbations for quasilinear evolution equations: The initial-boundary value problem (Q2733870)

Let \(\Omega\subset R^n\) be a bounded domain, and \(T>0\). The author deals with the question of global in time solvability of the initial-boundary value problem for the quasilinear dissipative wave equation NEWLINE\[NEWLINE\begin{cases} \varepsilon u_{tt}+ u_t-a_{ij} (\nabla u)\partial_i \partial_j u=f(x,t) \quad & \text{in }\Omega \times(0,t), \\ u(x,0)={\mathcal U}_0(x),\;u_t(x,0)=u_1(x) \quad & \text{in }\Omega \times \{t=0\},\\ u(\cdot,t)=0 \quad &\text{in } \partial \Omega\times (0,t)\end{cases} \tag{1}NEWLINE\]NEWLINE for small \(\varepsilon>0\), as well as with the global solvability of the formal limit parabolic problem, that is NEWLINE\[NEWLINE\begin{cases} v_t-a_{ij} (\nabla v)\partial_i \partial_jv= g(x,t)\quad & \text{in }\Omega \times(0,T),\\ v(x,0)=v_0(x) \quad & \text{in }\Omega \times\{t=0\},\\ v(\cdot t)=0, \quad & \text{in }\partial \Omega\times (0,T).\end{cases}\tag{2}NEWLINE\]NEWLINE The author shows that (1) admits global solutions iff the corresponding linear parabolic equation (2) does the same.