Uniqueness of Bounded Solutions to a Viscous Diffusion Equation

In this paper we prove the uniqueness of bounded solutions to a viscous diusion equation based on approximate Holmgren’s approach.


Introduction
We consider the uniqueness of bounded solutions to the following viscous diffusion equation in one dimension of the form with the initial and boundary condition u(x, 0) = u 0 (x), x ∈ [0, 1], (1.3) where λ > 0 is the viscosity coefficient, Q T = (0, 1) × (0, T ), A(s), B(s is a constant, and f is a function only of x and t. If λ = 0, then the equation (1.1) becomes In the case that A (s) ≥ 0, the equation (1.4) is the one dimensional form of the wellknown nonlinear diffusion equation, which is degenerate at the points where A (u) = 0 and has been studied extensively.In particular, the discussion of the uniqueness of solutions can be found in many papers, see for example have been discussed by Chen, Gurtin [11] and Ting, Showalter [12].
In this paper, we establish the uniqueness of the solutions to the initial-boundary problem of the equation (1.1) by using an approximate Holmgren's approach.It is worth recalling the work of [1] concerning related parabolic problems (1.4).Due to the degeneracy, the problem (1.1)-(1.3)admits only weak solutions in general.So our result is concerned with the generalized solutions to the problem (1.1)- (1.3).
Our main result is the following theorem.
3) has at most one generalized solution in the sense of Definition 1.1.

Preliminaries
Let u 1 , u 2 ∈ L ∞ (Q T ) be solutions of the boundary value problem (1.1)- (1.3).By the definition of generalized solutions, we have where For small η > 0, let Let Ãε and λ η,ε be a C ∞ approximation of Ã and λ η respectively, such that For given g ∈ C ∞ 0 (Q T ), consider the approximate adjoint problem
EJQTDE, 2003 No. 17, p. 5 3 Proof of Theorem 1.1 Given g ∈ C ∞ 0 (Q T ).Let ϕ be a solution of (2.1)-(2.3).Then As indicated above, from the definition of generalized solutions, we have Now we are ready to estimate all terms on the right side of (3.1).First, from Lemma 2.1 We also have [1],[3]-[7].While if A (s)EJQTDE, 2003  No. 17, p. 1 is permitted to change sign, (1.4) is called the forward-backward nonlinear diffusion equation.For the case of λ > 0, Cohen and Pego[10] considered the equation (1.1) with B(s) = 0 and f = 0s) has no monotonicity.Their interests center on the steady state solution for the equation (1.5), and the uniqueness of the solution of the Neumann initial-boundary value problem and the Dirichlet initial-boundary value problem of the linear case of the equation (1.5),
Here and in the sequel, we use C to denote a universal constant, indenpent of η and ε, which may take different value on different occasions.
.3) It is easily to see that the solution to the problem (2.1)-(2.3) is in C ∞ from the smooth of g in (2.1).Lemma 2.1 The solution ϕ of the problem (2.1)-(2.3)satisfies