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On singular perturbations for quasilinear IBV problems. (English) Zbl 0989.35021

Summary: We show that the hyperbolic singular perturbations \[ \begin{cases} \varepsilon u_{tt}+ a_t-u_{ij} (\nabla u)\partial_i \partial_ju= f(x,t) & \text{ in } \Omega\times (0,T),\\ u(x,0)= u_0(x),\;u_t(x,0)= u_1(x) & \text{ in }\Omega \times\{t=0\},\\ u(\cdot,t)=0 & \text{ in }\partial \Omega\times (0, T), \end{cases} \tag{1} \] of the quasilinear parabolic problems \[ \begin{cases} v_t-a_{ij} (\nabla u)\partial_i \partial_jv= g(x,t) & \text{ in }\Omega\times (0,T),\\ v(x, 0)= v_0(x) & \text{ in }\Omega \times\{t=0\},\\ v(\cdot,t)=0 & \text{ in } \partial \Omega\times (0,T).\end{cases} \tag{2} \] admit local Kato-Sobolev solutions defined on a common time interval, on which their singular convergence to a solution of (2) can be studied. We also find that if this limit problem admits a global Kata-Sobolev solution, the life span of the solutions of (1) grows to \(+\infty\) as \(\varepsilon\to 0\).

MSC:

35B25 Singular perturbations in context of PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
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