# Lectures on Nonlinear Hyperbolic Differential Equations

@inproceedings{Hrmander1997LecturesON, title={Lectures on Nonlinear Hyperbolic Differential Equations}, author={Lars H{\"o}rmander}, year={1997} }

In this introductory textbook, a revised and extended version of well-known lectures by L. Hormander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from that the book only studies classical solutions. Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data. Four chapters are devoted to microanalysis of the singularities…

## 703 Citations

LONG-TIME SOBOLEV STABILITY FOR SMALL SOLUTIONS OF QUASI-LINEAR KLEIN-GORDON EQUATIONS ON THE CIRCLE

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We prove that higher Sobolev norms of solutions of quasi-linear Klein-Gordon equations with small Cauchy data on S 1 remain small over intervals of time longer than the ones given by local existence…

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This text deals with the singularities of the solutions of several classes of nonlinear partial differential equations and systems. Applications of the results here obtained are given for the…

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We establish the small data solvability of suitable quasilinear wave and Klein-Gordon equations in high regularity spaces on a geometric class of spacetimes including asymptotically de Sitter spaces.…

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In this paper, we study the global nonexistence of solutions to a nonlinear wave equation with critical potential V(x)$V(x)$ on a Riemannian manifold, the form of which is more general than those in…

Remarks on decay of small solutions to systems of Klein-Gordon equations with dissipative nonlinearities

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Initial Value Problems for Wave Equations on Manifolds

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We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of…

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We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of…