Weak solutions for a nonlinear dispersive equation

Abstract In this paper we study the existence, uniqueness, and regularity of the solutions for the Cauchy problem for the evolution equation u t + (f (u)) x − u xxt = g(x, t), ( ∗ ) where u = u(x, t), x is in (0, 1), 0 ⩽ t ⩽ T, T is an arbitrary positive real number,f(s)ϵC1 R , and g(x, t)ϵ L∞(0, T; L2(0, 1)). We prove the existence and uniqueness of the weak solutions for (∗) using the Galerkin method and a compactness argument such as that of J. L. Lions. We obtain regular solutions using eigenfunctions of the one-dimensional Laplace operator as a basis in the Galerkin method.