Wave Maps and Ill-posedness of their Cauchy Problem.- On the Global Behavior of Classical Solutions to Coupled Systems of Semilinear Wave Equations.- Decay and Global Existence for Nonlinear Wave Equations with Localized Dissipations in General Exterior Domains.- Global Existence in the Cauchy Problem for Nonlinear Wave Equations with Variable Speed of Propagation.- On the Nonlinear Cauchy Problem.- Sharp Energy Estimates for a Class of Weakly Hyperbolic Operators.
Progress in Partial Differential Equations is devoted to modern topics in the theory of partial differential equations. It consists of both original articles and survey papers covering a wide scope of research topics in partial differential equations and their applications. The contributors were participants of the 8th ISAAC congress in Moscow in 2011 or are members of the PDE interest group of the ISAAC society.This volume is addressed to graduate students at various levels as well as researchers in partial differential equations and related fields. The readers will find this an excellent resource of both introductory and advanced material. The key topics are:o Linear hyperbolic equations and systems (scattering, symmetrisers)
o Non-linear wave models (global existence, decay estimates, blow-up)
o Evolution equations (control theory, well-posedness, smoothing)
o Elliptic equations (uniqueness, non-uniqueness, positive solutions)
o Special models from applications (Kirchhoff equation, Zakharov-Kuznetsov equation, thermoelasticity)
The contributions contained in the volume, written by leading experts in their respective fields, are expanded versions of talks given at the INDAM Workshop "Anomalies in Partial Differential Equations" held in September 2019 at the Istituto Nazionale di Alta Matematica, Dipartimento di Matematica "Guido Castelnuovo", Università di Roma "La Sapienza". The volume contains results for well-posedness and local solvability for linear models with low regular coefficients. Moreover, nonlinear dispersive models (damped waves, p-evolution models) are discussed from the point of view of critical exponents, blow-up phenomena or decay estimates for Sobolev solutions. Some contributions are devoted to models from applications as traffic flows, Einstein-Euler systems or stochastic PDEs as well. Finally, several contributions from Harmonic and Time-Frequency Analysis, in which the authors are interested in the action of localizing operators or the description of wave front sets, complete the volume.