The Global Boundedness And Traveling Wave Solution Theory Of A Class Of Nonlocal Fisher-KPP Equations

In this thesis,we will study the global boundedness of solutions and the traveling wave solutions for the non-local Fisher-KPP equation ut= ?u+ up(1-k?*uq),x? R,p?1,(*)where k?(x)=1/?k(x/?)is a ?-parametrized nonnegative kernel with k?L1(R)and?Rk(s)ds=1,K?*uq=?Ruq(x-y)k?(x-y)k?(y)dy.Due to the non-local features of the equation,the comparison principle does not hold.At the same time,the non-linear growth and the non-linear consumption of resources with distinct biological significance bring essential difficulties to mathematical research.We aim to generalize the results for non-local Fisher-KPP equations with p=q=1 to the general cases of p?1,q?1.The first chapter is an introduction.We briefly describe the research background,research status and the main results and innovations of this thesis.In the second chapter,as a preliminary of the global boundedness,we study the local existence,uniqueness and non-negativity of solutions of(*).In the third chapter,we obtain the global boundedness of solutions by means of localization method and the comparison principle of parabolic equations.In the fourth chapter,we give the necessary and sufficient conditions for the existence of the monotone traveling wave solutions.To prove the sufficiency,we first construct a pair of upper and lower solutions and monotone iteration scheme,then obtain the existence according to the fixed point theorem.To prove the necessity,we first estimate v(x):=1-u(x)and then prove the nonexistence by contradiction.