Existence And Uniqueness Of Weak Solutions To A Class Of Doublely Degenerate Parabolic Equations

This paper is divided two chapters. The first chapter discusses the existence and uniqueness of weak solution for the first boundary value problem of the following strongly degenerate parabolic equation in one dimension.u(0,t) = u(1,t) = 0, t∈(0,T), (2) C(u(x,0)) =u₀(x) x∈ (0,1).(3)whereand T > 0 is a constant,c(s)≥ 0,a(s) ≥ 0,b(s), f(u,x,t),u₀(x) are appropriately smooth functions.In the case that f(u,x,t) = 0, equation (1) was considered by Liu Qiang [16], their interests center on the uniqueness of the weak solution, and prove the uniqueness under the assumption |b(s)|²â‰¤ α(s)c(s). In this paper, we consider the cast that f(u, x, t) ≠ 0. Due to the strongly degeneracy, that is the degenerate set E_c = {s;c(s) = 0} contains interior points, the solutions may not be classical, so we formulate an approximate definition of this problem. By means of Holmgren's aaproach, we convert the uniqueness of the weak solutions to some estimates of solutions of the adjoint problem. There are not any relation between A(s) andB(s). While due to the strongly degeneracy of C(s), we require a condition that exist 1 < px < 2,1 < p2 < 2 such that \b(s)\pi < c(s)h1(s),\f's(s,x,t)\P2 < c(s)h,2(s,x,t), where hi(s) and h,2(s,x,t) are non-negative continuous functions. By this condition and A(s) increasing strictly function, we obtain the uniqueness of bounded and measurable weak solutions,also.About the existence of weak solutions of problem (l)-(3), the basic idea is the parabolic regularization. The key step is inspired by Prof. Zhao Junning's article [9] in which he used the theorem of Young measure and the method of compensated compactness. Last, we prove the existence of the bounded and measurable weak solutions by the limit. And requires that there are some relations among c(s) and f(s,x,t): \f's(s,x,t)\ 0, a(s) > 0 ,bl(s)(i = 1,2, ? ? ? , n), uo(x)are appropriately smooth functions. This is a class of equations with double degeneracy. While c(s) ^ 0, the equation (4) is the typical parabolic-hyperbolic mixed equation, which degenerates when a(s) = 0. Similar, while a(s) ^ 0, the equation. (4) is the typical elliptic-parabolic-hyperbolic mixed types, which degenerates when c(s) = 0. We need to introduce a kind of weak solution for the study of the problem when the double degeneracy appear.The first boundary value problem of the equation (4) was considered by [17] in one dimension, their interests center on existence and stability of the renormalized solution under the assumption \b(s)\ < all2(s) . In this chapter, we consider the existence and uniqueness of the weak solution of the Cauchy problem (4)-(5). Due to the strongly degeneracy, that is the degenerate sets Ec = {s;c(s) = 0} and E& = {s;a(s) = 0} contain interior points, the solutions may not be classical. So we formulate a definition of entropy solution of this problem, and prove the existence of entropy solution under the assumptions a(s) < c(s)hi(s), \b(s)\ < c(s)/i2(s), since the double degeneracy of equation (4), we not obtain the uniqueness of the entropy solution. But we obtain the result of stability about the entropy solution.