简介：ThepurposeofthispaperistouseaveryrecentthreecriticalpointstheoremduetoBonannoandMaranotoestablishtheexistenceofatleastthreesolutionsforthequasilinearsecondorderdifferentialequationonacompactinterval[a,b]?R{-u’’=(λf(x,u)+g(u))h(u’),in(a,b),u(a)=u(b)=0underppropriatehypotheses.Weexhibittheexistenceofatleastthree(weak)solutionsand,andtheresultsareillustratedbyexamples.

简介：Thequasilinearhyperbolicequationwithnonlineardampingisconsideredinthispaper,anewasymptoticprofileforthesolutiontotheequationisobtainedbysuitablychoosingtheinitialdataofthecorrespondingparabolicequation,theconvergenceratesofthenewprofilearebetterthanthatobtainedbyNishihara(1997,J.DifferentialEquations137,384-395)andH.-J.Zhao(2000,J.DifferentialEquations167,467-494).

简介：Forfirst-orderquasilinearhyperbolicsystemswithzeroeigenvalues,theauthorestablishesthelocalexactcontrollabilityinashortertime-periodbymeansofinternalcontrolsactingonsuitabledomains.Inparticular,undercertainspecialbutreasonablehypotheses,thelocalexactcontrollabilitycanberealizedonlybyinternalcontrols,andthecontroltimecanbearbitrarilysmall.