The Existence Of Solutions Of General Initial Data For A Class Of Parabolic Equations

This paper mainly studies the existence of solutions of Cauchy problem for a class of non-uniformly parabolic equations for initial data with less regularity.It is well known that when the initial value is smooth,the smooth solution of the Cauchy problem(1.1)in this paper is locally existent.Therefore,whether there is a solution for the initial value which is not smooth is the starting point of this paper.For the equation When the initial value u0?Llocp(R),p>1,as long as m E(1,2],there exists a solution u in C?(R ×(0,?))which converges to uo in Lp-norm as t?0+.More precisely,our first work is to prove the local uniform bound of u and ux.For the former,we can get a more general conclusion.For a class of equation ut=(a(ux))x-u,where a(0)=0,|a(P)|?C0,a'(P)>0,we can get the local uniform bound of u.By using the method of Gagliardo-Nirenberg interpolation inequality and iteration,we will prove that for any q>p,u is uniformly bounded on Llocq.Then,we will prove that ux is uniformly bounded on Llox1,and it is easy to obtain the local uniform bound of u follows from combing them.For ux,we construct a special function g,which turns the study of the upper bound of ux into the study of the upper bound of g,and then we get the bound of g.Then,according to the fact that the rest of the constructed function is bounded,we can get the upper bound of ux.Finally,we can prove the existence of solution by selecting convergence diagonal subsequence combining two conclusions.When m>2,the same result holds when the initial data belongs to Wloc l,p,p?m-1.Secondly,for the more general ut=(a(ux))x-u,we adopt the similar method.Finally,we prove that when the original value is u0?Wloc 1,p(R),the equation can be proved to have a weak solution in the given range of a'(p).