Blow-up and pattern formation in hyperbolic models for chemotaxis in 1-D

In this paper we study finite time blow-up of solutionsof a hyperbolic model for chemotaxis. Using appropriate scalingthis hyperbolic model leads to a parabolic model as studied byOthmer and Stevens (1997) and Levine and Sleeman (1997). In thelatter paper, explicit solutions which blow-up in finite time wereconstructed. Here, we adapt their method to construct acorresponding blow-up solution of the hyperbolic model. Thisconstruction enables us to compare the blow-up times of thecorresponding models. We find that the hyperbolic blow-up isalways later than the parabolic blow-up. Moreover, we show thatsolutions of the hyperbolic problem become negative near blow-up.We calculate the zero-turning-rate time explicitly and we showthat this time can be either larger or smaller than the parabolicblow-up time.The blow-up models as discussed here and elsewhere are limitingcases of more realistic models for chemotaxis. At the end of thepaper we discuss the relevance to biology and exhibit numericalsolutions of more realistic models.