Partial Differential Equations

This Research Note features recent results in the theory and practice of partial differential equations. The articles should be of interest to many mathematicians, scientists, and engineers. The papers cover topics such as mathematical problems in elasticity, asymptotic stability of partial differential equations, reaction-diffusion equations, existence theorems and the number of solutions for elliptic equations, localization of dependence for hyperbolic equations and decay of solutions for hyperbolic systems, existence of solutions for periodic parabolic problems, and finite extinction time for nonlinear parabolic problems. Some of the equations represent mathematical models in the life and natural sciences: reaction-diffusion equations in ecology, problems in climate modelling, diffusively perturbed predator-prey models. There are also articles concerned with optimal control, nonlinear evolution equations, variational methods, and numerical techniques. Specialists in mathematical and theoretical physics will certainly enjoy the reports on limit sets of scalar conservation laws, resonant nonlinear systems of conservation laws, incompressible flow in porous media, shock transition zones, travelling fronts in platelet aggregation, and models of superconductivity. The papers employ a wide array of deep mathematical theories and techniques. These include methods from linear and nonlinear functional analysis, a number of topological and topological degree techniques, as well as asymptotic and other classical analysis tools. We hope the book will become a valuable source of information and new ideas for any researcher concerned with partial differential equations and mathematical physics.