## 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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### References:

 [1] Friedman ( A. ). - Partial Differential Equations of Parabolic Type . Krieger , Malabar, FL 1983 . [2] Kato ( T. ). - Abstract Differential Equations and Nonlinear Mixed Problems . Fermian Lectures , Pisa , 1985 . MR 930267 | Zbl 0648.35001 · Zbl 0648.35001 [3] Lions ( J.L. ). - Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires . Dunod , Paris , 1969 . MR 259693 | Zbl 0189.40603 · Zbl 0189.40603 [4] Lions ( J.L. ), Magenes ( E. ). - Non-Homogeneous Boundary value Problems , Vol. I . Springer Verlag , New York , 1972 . Zbl 0223.35039 · Zbl 0223.35039 [5] Lions ( J.L. ), Magenes ( E. ). - Non-Homogeneous Boundary value Problems , Vol. II . Springer Verlag , New York , 1972 . Zbl 0227.35001 · Zbl 0227.35001 [6] Matsumura ( A. ). - Global Existence and Asymptotics of the Solutions of Second Order Quasilinear Hyperbolic Equations with First Order Dissipation term . , Publ. RIMS Kyoto Univ. , 13 , 349 - 379 ( 1977 ). Article | MR 470507 | Zbl 0371.35030 · Zbl 0371.35030 [7] Milani ( A. ). - Long Time Existence and Singular Perturbation Results for Quasilinear Hyperbolic Equations with Small Parameter and Dissipation Term . Non Linear An. TMA , 10 /11 ( 1986 ), 1237 - 1248 . MR 866256 | Zbl 0645.35064 · Zbl 0645.35064 [8] Milani ( A. ). - Global Existence via Singular Perturbations for Quasilinear Evolution Equations . Adv. Math. Sci. Appl. , 6 /2 ( 1996 ), 419 - 444 . MR 1411976 | Zbl 0868.35008 · Zbl 0868.35008 [9] Milani ( A. ). - A Remark on the Sobolev Regularity of Classical Solutions to Uniformly Parabolic Equations . Math. Nachr. , 199 ( 1999 ), 115 - 144 . MR 1676322 | Zbl 0962.35038 · Zbl 0962.35038 [10] Milani ( A. ). - On the Construction of Compatible Data for Hyperbolic-Parabolic Initial-Boundary Value Problems . Rend. Sem. Mat. Univ. Trieste , 29 ( 1997 ), 167 - 188 . MR 1658443 | Zbl 0921.35013 · Zbl 0921.35013 [11] Yang ( H. ), Milani ( A. ). - On the Diffusion Phenomenon of Quasilinear Hyperbolic Flows . To appear on Bull. Sc. Math. [12] Milani ( A. ). - Global Existence via Singular Perturbations for Quasilinear Evolution Equations: the Initial-Boundary Value Problem . Preprint, 1999 . MR 1807450 [12] Milani ( A. ). - Sobolev Regularity for t > 0 in Quasilinear Parabolic Equations . Preprint, 1999 . MR 1866198 [13] Racke ( R. ). - Lectures on Nonlinear Evolution Equations . Vieweg , Braunschweig , 1992 . MR 1158463 | Zbl 0811.35002 · Zbl 0811.35002
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