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Global existence for quasi-linear dissipative hyperbolic equations with large data and small parameter. (English) Zbl 0728.35070

From the introduction: Consider the following dissipative quasilinear hyperbolic initial-value problem \[ (H_{\epsilon})\quad \epsilon u_{tt}+u_ t-\sum^{n}_{i,j=1}a_{ij}(\nabla u)\partial_ i\partial_ ju=0;\quad u(x,0)=u_ 0(x),\quad u_ t(x,0)=u_ 1(x) \] where \(\epsilon \in R^+\), \(u=u(x,t)\in R^ 1\), \(t\geq 0\), \(x\in R^ n\), \(\partial_ i=\partial /\partial x_ i\) and \(\nabla\) is the gradient with respect to the space variables only. The coefficients \(a_{ij}\) are smooth symmetric, and satisfy the strong ellipticity condition \[ (1)\quad \forall p\in R^ n,\quad \forall q\in R^ n,\quad \sum a_{ij}(p)q^ iq^ j\geq a(p)| q|^ 2, \] where a: \(R^ n\to R\) is a continuous function, such that \(a(0)>0\). We are interested in smooth solutions to \((H_{\epsilon})\), corresponding to smooth initial data, and to conditions that ensure that such solutions exist globally in time. Local existence results for \((H_{\epsilon})\) are well known; we refer mainly to Kato’s general theory, in the framework of the Sobolev spaces \(H^ s=H^ s(R^ n)\), of sufficiently high order s \((s>1+n/2)\). Such results are generally established under some smallness assumptions on the initial data; since \((H_{\epsilon})\) is dissipative, a further smallness assumption on the data allows us to extend these local solutions to all later times. These smallness conditions on the data refer to their \(H^ s\) norm, so that all their derivatives are usually required to have a small \(L^ 2\) norm. We have recently been interested in the question, whether such smallness requirements on the data can be replaced by a smallness condition on the parameter \(\epsilon\) ; in particular, making essential use of the assumption that the coefficients \(a_{ij}\) do not depend on \(u_ t\) (they might, however, depend on x,t and u), we have been able to prove that \((H_{\epsilon})\) is globally solvable for small \(\epsilon\) under some additional assumptions.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B25 Singular perturbations in context of PDEs
35L15 Initial value problems for second-order hyperbolic equations
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References:

[1] T. Kato: Quasi-linear Equations of Evolution, with Applications to Partial Differential Equations. Lect. Notes Math. 448, pp. 25-70; Springer-Verlag, Berlin 1975. · Zbl 0315.35077
[2] A. Malsumura: Global Existence and Asymptotics of the Solutions of Second Order Quasilinear Hyperbolic Equations with First Order Dissipation Term. Publ. R.I.M.S. Kyoto Univ., 13 (1977), 349-379. · Zbl 0371.35030
[3] A. Milani: Long Time Existence and Singular Perturbation Results for Quasi-linear Hypebolic Equations with Small Parameter and Dissipation Term. Non Linear Anal., 10/11 (1986), 1237-1248. · Zbl 0645.35064
[4] A. Milani: idem, II. Non Linear Anal., 11/12 (1987), 1371-1381.
[5] A. Milani: Global Existence for Quasi-linear Dissipative Wave Equations with Large Data and Small Parameter. Math. Zeit., 198 (1988), 291-297. · Zbl 0648.35051
[6] L. Niremberg: On Elliptic Partial Differential Equations. Ann. Sc. Norm. Pisa, 13 (1959), 115-162.
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