Renormalized Solutions of a Nonlinear Parabolic Equation with Double Degeneracy

with p ≥ 2 and b(s) ≥ 0 appropriately smooth. The equation (1.1) presents two kinds of degeneracy, since it is degenerate not only at points where b(u) = 0 but also at points where ∂ ∂x B(u) = 0 if p > 2. Using the method depending on the properties of convex functions, Kalashnikov [10] established the existence of continuous solutions of the Cauchy problem of the equation (1.1) with f ≡ 0 under some convexity assumption on A(s) and B(s). Under such assumption, the equation degenerates only at the zero value of the solutions or their spacial derivatives. The more interesting case is that the equation may present strong degeneracy, namely, the set E = {s ∈ R : b(s) = 0} may have interior points. Generally, the equation may have no classical solutions and even continuous solutions for this case and it is necessary to formulate some suitable weak solutions. Supported by the NNSF of China and Graduate Innovation Lab of Jilin University. Corresponding author. Tel.:+86 431 5167051.