Studies of Nonlinear Hyperbolic and Elliptic-Parabolic Equations

We consider three problems on nonlinear PDE-s. First is a generalization of the Greenberg-Rascle construction of spatially and temporally periodic solutions to a nonlinear wave equation. We use the hodograph transformation to study the interactions between simple waves and obtain the desired existence for a class of systems u, _{t}-upsilon,_{x}=0 and upsilon,_{t} -c^2(u)u,_{x}=0, where c is positive, even, increasing on [ 0, infty) and has a simple jump discontinuity in {dcover du} at u=0 . The second problem concerns the continuum limits of discrete particle systems with short range repulsive forces. Using the theory of Abel equations, we establish the existence of a class of short range interparticle force laws with the property that the asymptotic trajectories of two sufficiently energetic particles of equal mass entering and leaving the region of a binary interaction are the same as the asymptotic trajectories of particles which undergo a simple point-mass elastic collision. Using such force laws, we consider the evolution of an N particle gas, each particle having mass {1over N}, for initial data which are guaranteed to generate only binary collisions. We show that such problems are exactly solvable and we characterize the continuum limit (Ntoinfty) of such solutions. These limit flows are independent of the details of the repulsive forces and are the same as these obtained if one replaces the interparticle force law by the elastic collision rule which simply interchanges particle velocities during a collision. We also study semilinear equations with a nonnegative characteristic form of the type -a^{ij }(x)u,_{ij}+b^{k}(x)u, _{k}=f(x,u) defined on a bounded, piecewise smooth domain Omegasubset R^{n} and subjected to Fichera boundary conditions on u. We construct the monotone iteration scheme and prove the existence theorem for these equations.