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.