LAUSR

Abstract 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 x x , 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.