Existence Of Solutions To Cauchy Problem Of A Class Of Parabolic Equations With General Initial Data

This paper mainly discusses the existence of solutions of Cauchy problem for one dimensional non-uniformly parabolic equations for initial data with less regularity.We study a class of equations ut=(a(ux))x+b(x,ux),(x,t)?R×(0,T),(0.1)where a,b satisfy(1.2)?(1.5).When the initial value satisfies u0 ? Wlocp,p>1,we prove that there exists a weak solution of(0.1).We also study a special case of the equation(?),(x,t)? R ×(0,T),m>1.(0.2)where C(x)and its derivative are bounded.When m ?(1,2],there exists a solution u in C?(R ×(0,?))of(0.2)which converges to u0 in Lp-norm as t?0+ as long as the initial value satisfies u0?Llocp(R),p>1.When m>2,the similar result holds as long as the initial value satisfies u0?Wlocl,p(R),p>m-1.More precisely,for(0.2),our first work is to prove the local uniform bound of u and ux.By using the method of Gagliardo-Nirenberg interpolation inequality and iteration,we prove that u is uniformly bounded on Llocq for any q?p,and then we will prove that ux is uniformly bounded on Llocl.Thus it is easy to obtain the local uniform bound of u.For the local uniform bound of 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.According to the bound of g and the rest of the constructed function,the result is obtained.Finally,we can prove the existence of solution by selecting convergence diagonal subsequence combining two estimates.For(0.1),we discuss the Lp-norm of u and ux and the L2-norm of ut.Similarly,we can prove the existence of weak solution by combining these integral estimates.