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.


Available from: Marina Ghisi, Jul 13, 2014
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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.