Solvability to some strongly degenerate parabolic problems

Nonlinear parabolic equations of “divergence form,” u _t =(φ(u)ψ(u _x )) _x , are considered under the assumption that the “material flux,” φ(u)ψ(v), is bounded for all values of arguments, u and v. In literature such equations have been referred to as “strongly degenerate” equations. This is due to the fact that the coefficient, φ(u)ψ ^′ (u _x ), of the second derivative, u _xx , can be arbitrarily small for large value of the gradient, u _x . The “hyperbolic phenomena” (unbounded growth of space derivatives within a finite time) have been established in literature for solutions to Cauchy problem for the above-mentioned equations. Accordingly one can expect a correct statement of the initial-boundary value problem for such equations only under additional assumptions on the problem data. In this paper we describe several restrictions, under which the initial-boundary value problems for strongly degenerate parabolic equations are well-posed.