The Cauchy problem for <f>ut=Δu+&z.sfnc;∇u&z.sfnc;q</f>

With <f>q</f> a positive real number, the nonlinear partial differential equation in the title of the paper arises in the study of the growth of surfaces. In that context it is known as the generalized deterministic KPZ equation. The paper is concerned with the initial-value problem for the equation under the assumption that the initial-data function is bounded and continuous. Results on the existence, uniqueness, and regularity of solutions are obtained. [Copyright &y& Elsevier]