## 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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