Schedule-at-a-Glance

56 PRIMA 2013 AbstractsBaili ChenGustavus Adolphus College, USAbchen@gustavus.eduWe prove the existence of periodic solutions of systemsof nonlinear impulsive differential equations arising fromchemotherapeutic treatment for cancer. Our analysis isbased on Mawhin’s continuation theorem of coincidencedegree theorem.Analysis of some nonlinear PDEs from multiscalegeophysical applicationsBin ChengUniversity of Surrey, UKb.cheng@surrey.ac.ukThis talk is regarding PDE systems from geophysical applicationswith multiple time scales, in which linear skewself-adjointoperators of size 1/epsilon gives rise to highlyoscillatory solutions. Analysis is performed in justifyingthe limiting dynamics as epsilon goes to zero; furthermore,the analysis yields estimates on the difference betweenthe multiscale solution and the limiting solution.We will introduce a simple yet effective time-averagingtechnique which is especially useful in general domainswhere Fourier analysis is not applicable.Some piston problems in fluid dynamicsMin DingShanghai Jiao Tong University, Chinaminding@sjtu.edu.cnThe piston problem is analyzed from the mathematicalpoint of view. Some features and phenomena caused bythe motion of the piston are revealed. We discuss somepiston problems for both classical Euler equations andrelativistic Euler equations of compressible fluids. In particular,we focus on strong shock front solutions.Over-compressive shock profile associatedwith a simplified system from MHDLinglong DuNational University of Singapore, Singaporeg0901229@nus.edu.sgIn this talk, we present the wave propagation around anover-compressive shock profile with large amplitude for asimplified system from MHD, which is a rotationally invariantsystem of viscous conservation laws. We showthat the solution converges pointwise to another overcompressiveprofile exponentially, when the perturbationsof the initial data to a given profile are sufficiently small.We also give the explicit structure of the solution forthe linearized system. The approach begins with an extractionof non-decaying component stacked at the shockfront, followed by an iterated approximation process forthe remainder. This sharp estimate is strong enough tostudy the nonlinear problem and conclude the stability ofprofile.This is a joint work with Professor Shih-Hsien Yu.Global stability of E-H type regular refractionof shocks on the interface between two mediaBeixiang FangShanghai Jiao Tong University, Chinabxfang@sjtu.edu.cnIn this talk I will discuss the refraction of shocks on theinterface for 2-d steady compressible flow. Particularly,the class of E-H type regular refraction is defined and itsglobal stability of the wave structure is verified. The 2-dsteady potential flow equations is employed to describethe motion of the fluid. The stability problem of the E-Htype regular refraction can be reduced to a free boundaryproblem of nonlinear mixed type equations in an unboundeddomain. The corresponding linearized problemhas similarities to a generalized Tricomi problem of thelinear Lavrentiev-Bitsadze mixed type equation, and itcan be reduced to a nonlocal boundary value problemof an elliptic system. The latter is finally solved by establishingthe bijection of the corresponding nonlocal operatorin a weighted Hölder space via careful harmonicanalysis.This is a joint work with CHEN Shuxing and HU Dian.Multiplicity of positive solutions for a p-q-Laplacian equation with critical nonlinearitiesTsing-San HsuChang Gung University, Taiwan R.O.C.tshsu@mail.cgu.edu.twIn this talk, we study the effect of the coefficient f(x)of the critical nonlinearity on the number of positive solutionsfor a p-q-Laplacian equation. Under suitable assumptionsfor f(x) and g(x), we should prove that forsufficiently small λ > 0, there exist at least k positivesolutions of the following p-q-Laplacian equation−∆ pu−∆ qu=f(x)|u| p∗ −2 u+λg(x)|u| r−2 u in Ω,u = 0 on ∂Ω,where Ω ⊂ R N is a bounded smooth domain, N > p,1 < q < N(p−1) < p ≤ max{p, p N−1 ∗ − qp−1 } < r < p∗ ,p ∗ =Np is the critical Sobolev exponent and ∆su =N−pdiv(|∇u| s−2 ∇u) is the s-Laplacian of u.The linear hyperbolic initial boundary valueproblems in a domain with cornersAimin HuangIndiana University, USAaimhuang@indiana.eduIn this article, we consider linear hyperbolic InitialBoundary Value Problems (IBVP) in a rectangle in boththe constant and variable coefficients cases. We use semigroupmethod instead of Fourier analysis to achieve thewell-posedness of the linear hyperbolic system, and wefind by diagonalization that there are only two simplemodes in the system which we call hyperbolic and ellipticmodes. The hyperbolic system in consideration is eithersymmetric or Friedrichs-symmetrizable.On quasilinear elliptic equations with variableexponentsYun-Ho KimSangmyung University, Koreakyh1213@smu.ac.krWe study the following elliptic equations with variableexponents−div(φ(x, |∇u|)∇u) = λf(x, u)in Ωwhich is subject to Dirichlet boundary condition. Undersuitable conditions on φ and f, employing the variationalmethods, in particular, mountain pass theorem