Derivative loss for Kirchhoff equations with non-Lipschitz nonlinear term

06/2008; DOI: 10.2422/2036-2145.2009.4.01
Source: arXiv

ABSTRACT In this paper we consider the Cauchy boundary value problem for the integro-differential equation $$u_{tt}-m(\int_{\Omega}^{}|\nabla u|^{2} dx) \Delta u=0 \hspace{3em} {in}\Omega\times[0,T)$$ with a continuous nonlinearity $m:[0,+\infty)\to[0,+\infty)$. It is well known that a local solution exists provided that the initial data are regular enough. The required regularity depends on the continuity modulus of $m$. In this paper we present some counterexamples in order to show that the regularity required in the existence results is sharp, at least if we want solutions with the same space regularity of initial data. In these examples we construct indeed local solutions which are regular at $t=0$, but exhibit an instantaneous (often infinite) derivative loss in the space variables.

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Available from: Marina Ghisi, Jul 13, 2014
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    • "Our existence results confirm the general paradigm according to which Kirchhoff equations are well-posed, at least locally in time, whenever the (DGCS)-phenomenon is excluded. What happens beyond remains an open problem, and deep new ideas are likely to be needed (see [14] for a partial result). We also hope that these results could give an indication about the regularizing effects one can expect when adding a strong dissipation to quasilinear hyperbolic equations. "
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    ABSTRACT: We consider Kirchhoff equations with strong damping, namely with a friction term which depends on a power of the "elastic" operator. We address local and global existence of solutions in two different regimes depending on the exponent in the friction term. When the exponent is greater than 1/2, the dissipation prevails, and we obtain global existence in the energy space assuming only degenerate hyperbolicity and continuity of the nonlinear term. When the exponent is less than 1/2, we assume strict hyperbolicity and we consider a phase space depending on the continuity modulus of the nonlinear term and on the exponent in the damping. In this phase space we prove local existence, and global existence if initial data are small enough. The regularity we assume both on initial data and on the nonlinear term is weaker than in the classical results for Kirchhoff equations with standard damping. Proofs exploit some recent sharp results for the linearized equation and suitably defined interpolation spaces.
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    • "These conditions do not appear explicitly in [6], where they are replaced by suitable specific choices of ω, ϕ, ψ, which of course satisfy the same relations. To our knowledge, those conditions were stated for the first time in [8], thus unifying several papers that in the last 30 years had been devoted to special cases (see for example [5] and the references quoted therein). "
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    ABSTRACT: We consider a second order linear equation with a time-dependent coefficient c(t) in front of the "elastic" operator. For these equations it is well-known that a higher space-regularity of initial data compensates a lower time-regularity of c(t). In this paper we investigate the influence of a strong dissipation, namely a friction term which depends on a power of the elastic operator. What we discover is a threshold effect. When the exponent of the elastic operator in the friction term is greater than 1/2, the damping prevails and the equation behaves as if the coefficient c(t) were constant. When the exponent is less than 1/2, the time-regularity of c(t) comes into play. If c(t) is regular enough, once again the damping prevails. On the contrary, when c(t) is not regular enough the damping might be ineffective, and there are examples in which the dissipative equation behaves as the non-dissipative one. As expected, the stronger is the damping, the lower is the time-regularity threshold. We also provide counterexamples showing the optimality of our results.
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    • "Concerning the nonlinearity m(σ), there is no hope to relax assumption (1.4) to m(σ) ≥ 0, or the Lipschitz continuity assumption to mere continuity. The reason is that some examples presented in [6] show that under these weaker assumptions the Cauchy problem (1.1), (1.2) is not even locally well posed in classes of quasi-analytic functions. "
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    ABSTRACT: In a celebrated paper (Tokyo J. Math. 1984) K. Nishihara proved global existence for Kirchhoff equations in a special class of initial data which lies in between analytic functions and Gevrey spaces. This class was defined in terms of Fourier components with weights satisfying suitable convexity and integrability conditions. In this paper we extend this result by removing the convexity constraint, and by replacing Nishihara's integrability condition with the simpler integrability condition which appears in the usual characterization of quasi-analytic functions. After the convexity assumptions have been removed, the resulting theory reveals unexpected connections with some recent global existence results for spectral-gap data. Comment: 15 pages
    Bulletin of the London Mathematical Society 03/2010; DOI:10.1112/blms/bdq109 · 0.70 Impact Factor
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